Theorem bdayfinbndlem1

Description: Lemma for bdayfinbnd . Show the first half of the inductive step. (Contributed by Scott Fenton, 26-Feb-2026)

	Ref	Expression
Hypotheses	bdayfinbndlem.1	⊢ ( 𝜑 → 𝑁 ∈ ℕ0s )
	bdayfinbndlem.2	⊢ ( 𝜑 → ∀ 𝑧 ∈ No ( ( ( bday ‘ 𝑧 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑧 ) → ( 𝑧 = 𝑁 ∨ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) )
Assertion	bdayfinbndlem1	⊢ ( 𝜑 → ∀ 𝑤 ∈ No ( ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 0s ≤s 𝑤 ) → ( 𝑤 = ( 𝑁 +s 1s ) ∨ ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		bdayfinbndlem.1	⊢ ( 𝜑 → 𝑁 ∈ ℕ0s )
2		bdayfinbndlem.2	⊢ ( 𝜑 → ∀ 𝑧 ∈ No ( ( ( bday ‘ 𝑧 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑧 ) → ( 𝑧 = 𝑁 ∨ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) )
3		bdayn0p1	⊢ ( 𝑁 ∈ ℕ0s → ( bday ‘ ( 𝑁 +s 1s ) ) = suc ( bday ‘ 𝑁 ) )
4	1 3	syl	⊢ ( 𝜑 → ( bday ‘ ( 𝑁 +s 1s ) ) = suc ( bday ‘ 𝑁 ) )
5	4	adantr	⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( bday ‘ ( 𝑁 +s 1s ) ) = suc ( bday ‘ 𝑁 ) )
6	5	sseq2d	⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ ( 𝑁 +s 1s ) ) ↔ ( bday ‘ 𝑤 ) ⊆ suc ( bday ‘ 𝑁 ) ) )
7		bdayon	⊢ ( bday ‘ 𝑤 ) ∈ On
8		bdayon	⊢ ( bday ‘ 𝑁 ) ∈ On
9	8	onsuci	⊢ suc ( bday ‘ 𝑁 ) ∈ On
10	7 9	onsseli	⊢ ( ( bday ‘ 𝑤 ) ⊆ suc ( bday ‘ 𝑁 ) ↔ ( ( bday ‘ 𝑤 ) ∈ suc ( bday ‘ 𝑁 ) ∨ ( bday ‘ 𝑤 ) = suc ( bday ‘ 𝑁 ) ) )
11		onsssuc	⊢ ( ( ( bday ‘ 𝑤 ) ∈ On ∧ ( bday ‘ 𝑁 ) ∈ On ) → ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ 𝑁 ) ↔ ( bday ‘ 𝑤 ) ∈ suc ( bday ‘ 𝑁 ) ) )
12	7 8 11	mp2an	⊢ ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ 𝑁 ) ↔ ( bday ‘ 𝑤 ) ∈ suc ( bday ‘ 𝑁 ) )
13	12	orbi1i	⊢ ( ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ 𝑁 ) ∨ ( bday ‘ 𝑤 ) = suc ( bday ‘ 𝑁 ) ) ↔ ( ( bday ‘ 𝑤 ) ∈ suc ( bday ‘ 𝑁 ) ∨ ( bday ‘ 𝑤 ) = suc ( bday ‘ 𝑁 ) ) )
14	10 13	bitr4i	⊢ ( ( bday ‘ 𝑤 ) ⊆ suc ( bday ‘ 𝑁 ) ↔ ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ 𝑁 ) ∨ ( bday ‘ 𝑤 ) = suc ( bday ‘ 𝑁 ) ) )
15		fveq2	⊢ ( 𝑧 = 𝑤 → ( bday ‘ 𝑧 ) = ( bday ‘ 𝑤 ) )
16	15	sseq1d	⊢ ( 𝑧 = 𝑤 → ( ( bday ‘ 𝑧 ) ⊆ ( bday ‘ 𝑁 ) ↔ ( bday ‘ 𝑤 ) ⊆ ( bday ‘ 𝑁 ) ) )
17		breq2	⊢ ( 𝑧 = 𝑤 → ( 0s ≤s 𝑧 ↔ 0s ≤s 𝑤 ) )
18	16 17	anbi12d	⊢ ( 𝑧 = 𝑤 → ( ( ( bday ‘ 𝑧 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑧 ) ↔ ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑤 ) ) )
19		eqeq1	⊢ ( 𝑧 = 𝑤 → ( 𝑧 = 𝑁 ↔ 𝑤 = 𝑁 ) )
20		eqeq1	⊢ ( 𝑧 = 𝑤 → ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ↔ 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ) )
21	20	3anbi1d	⊢ ( 𝑧 = 𝑤 → ( ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ↔ ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) )
22	21	rexbidv	⊢ ( 𝑧 = 𝑤 → ( ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ↔ ∃ 𝑝 ∈ ℕ0s ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) )
23	22	2rexbidv	⊢ ( 𝑧 = 𝑤 → ( ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ↔ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) )
24	19 23	orbi12d	⊢ ( 𝑧 = 𝑤 → ( ( 𝑧 = 𝑁 ∨ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ↔ ( 𝑤 = 𝑁 ∨ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) )
25	18 24	imbi12d	⊢ ( 𝑧 = 𝑤 → ( ( ( ( bday ‘ 𝑧 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑧 ) → ( 𝑧 = 𝑁 ∨ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) ↔ ( ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑤 ) → ( 𝑤 = 𝑁 ∨ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) ) )
26	25	rspccva	⊢ ( ( ∀ 𝑧 ∈ No ( ( ( bday ‘ 𝑧 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑧 ) → ( 𝑧 = 𝑁 ∨ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) ∧ 𝑤 ∈ No ) → ( ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑤 ) → ( 𝑤 = 𝑁 ∨ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) )
27	2 26	sylan	⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑤 ) → ( 𝑤 = 𝑁 ∨ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) )
28	1	adantr	⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ 𝑤 = 𝑁 ) ) → 𝑁 ∈ ℕ0s )
29		0n0s	⊢ 0s ∈ ℕ0s
30	29	a1i	⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ 𝑤 = 𝑁 ) ) → 0s ∈ ℕ0s )
31		simprr	⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ 𝑤 = 𝑁 ) ) → 𝑤 = 𝑁 )
32	1	n0nod	⊢ ( 𝜑 → 𝑁 ∈ No )
33	32	addsridd	⊢ ( 𝜑 → ( 𝑁 +s 0s ) = 𝑁 )
34	33	adantr	⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ 𝑤 = 𝑁 ) ) → ( 𝑁 +s 0s ) = 𝑁 )
35	31 34	eqtr4d	⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ 𝑤 = 𝑁 ) ) → 𝑤 = ( 𝑁 +s 0s ) )
36		0lt1s	⊢ 0s <s 1s
37	36	a1i	⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ 𝑤 = 𝑁 ) ) → 0s <s 1s )
38	32	ltsp1d	⊢ ( 𝜑 → 𝑁 <s ( 𝑁 +s 1s ) )
39	33 38	eqbrtrd	⊢ ( 𝜑 → ( 𝑁 +s 0s ) <s ( 𝑁 +s 1s ) )
40	39	adantr	⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ 𝑤 = 𝑁 ) ) → ( 𝑁 +s 0s ) <s ( 𝑁 +s 1s ) )
41		oveq1	⊢ ( 𝑎 = 𝑁 → ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) = ( 𝑁 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) )
42	41	eqeq2d	⊢ ( 𝑎 = 𝑁 → ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ↔ 𝑤 = ( 𝑁 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ) )
43		oveq1	⊢ ( 𝑎 = 𝑁 → ( 𝑎 +s 𝑞 ) = ( 𝑁 +s 𝑞 ) )
44	43	breq1d	⊢ ( 𝑎 = 𝑁 → ( ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ↔ ( 𝑁 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) )
45	42 44	3anbi13d	⊢ ( 𝑎 = 𝑁 → ( ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ( 𝑤 = ( 𝑁 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑁 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
46		oveq1	⊢ ( 𝑏 = 0s → ( 𝑏 /su ( 2s ↑s 𝑞 ) ) = ( 0s /su ( 2s ↑s 𝑞 ) ) )
47	46	oveq2d	⊢ ( 𝑏 = 0s → ( 𝑁 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) = ( 𝑁 +s ( 0s /su ( 2s ↑s 𝑞 ) ) ) )
48	47	eqeq2d	⊢ ( 𝑏 = 0s → ( 𝑤 = ( 𝑁 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ↔ 𝑤 = ( 𝑁 +s ( 0s /su ( 2s ↑s 𝑞 ) ) ) ) )
49		breq1	⊢ ( 𝑏 = 0s → ( 𝑏 <s ( 2s ↑s 𝑞 ) ↔ 0s <s ( 2s ↑s 𝑞 ) ) )
50	48 49	3anbi12d	⊢ ( 𝑏 = 0s → ( ( 𝑤 = ( 𝑁 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑁 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ( 𝑤 = ( 𝑁 +s ( 0s /su ( 2s ↑s 𝑞 ) ) ) ∧ 0s <s ( 2s ↑s 𝑞 ) ∧ ( 𝑁 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
51		oveq2	⊢ ( 𝑞 = 0s → ( 2s ↑s 𝑞 ) = ( 2s ↑s 0s ) )
52		2no	⊢ 2s ∈ No
53		exps0	⊢ ( 2s ∈ No → ( 2s ↑s 0s ) = 1s )
54	52 53	ax-mp	⊢ ( 2s ↑s 0s ) = 1s
55	51 54	eqtrdi	⊢ ( 𝑞 = 0s → ( 2s ↑s 𝑞 ) = 1s )
56	55	oveq2d	⊢ ( 𝑞 = 0s → ( 0s /su ( 2s ↑s 𝑞 ) ) = ( 0s /su 1s ) )
57		0no	⊢ 0s ∈ No
58		divs1	⊢ ( 0s ∈ No → ( 0s /su 1s ) = 0s )
59	57 58	ax-mp	⊢ ( 0s /su 1s ) = 0s
60	56 59	eqtrdi	⊢ ( 𝑞 = 0s → ( 0s /su ( 2s ↑s 𝑞 ) ) = 0s )
61	60	oveq2d	⊢ ( 𝑞 = 0s → ( 𝑁 +s ( 0s /su ( 2s ↑s 𝑞 ) ) ) = ( 𝑁 +s 0s ) )
62	61	eqeq2d	⊢ ( 𝑞 = 0s → ( 𝑤 = ( 𝑁 +s ( 0s /su ( 2s ↑s 𝑞 ) ) ) ↔ 𝑤 = ( 𝑁 +s 0s ) ) )
63	55	breq2d	⊢ ( 𝑞 = 0s → ( 0s <s ( 2s ↑s 𝑞 ) ↔ 0s <s 1s ) )
64		oveq2	⊢ ( 𝑞 = 0s → ( 𝑁 +s 𝑞 ) = ( 𝑁 +s 0s ) )
65	64	breq1d	⊢ ( 𝑞 = 0s → ( ( 𝑁 +s 𝑞 ) <s ( 𝑁 +s 1s ) ↔ ( 𝑁 +s 0s ) <s ( 𝑁 +s 1s ) ) )
66	62 63 65	3anbi123d	⊢ ( 𝑞 = 0s → ( ( 𝑤 = ( 𝑁 +s ( 0s /su ( 2s ↑s 𝑞 ) ) ) ∧ 0s <s ( 2s ↑s 𝑞 ) ∧ ( 𝑁 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ( 𝑤 = ( 𝑁 +s 0s ) ∧ 0s <s 1s ∧ ( 𝑁 +s 0s ) <s ( 𝑁 +s 1s ) ) ) )
67	45 50 66	rspc3ev	⊢ ( ( ( 𝑁 ∈ ℕ0s ∧ 0s ∈ ℕ0s ∧ 0s ∈ ℕ0s ) ∧ ( 𝑤 = ( 𝑁 +s 0s ) ∧ 0s <s 1s ∧ ( 𝑁 +s 0s ) <s ( 𝑁 +s 1s ) ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) )
68	28 30 30 35 37 40 67	syl33anc	⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ 𝑤 = 𝑁 ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) )
69	68	expr	⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( 𝑤 = 𝑁 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
70		idd	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) ) → ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) → 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ) )
71		idd	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) ) → ( 𝑦 <s ( 2s ↑s 𝑝 ) → 𝑦 <s ( 2s ↑s 𝑝 ) ) )
72		n0addscl	⊢ ( ( 𝑥 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) → ( 𝑥 +s 𝑝 ) ∈ ℕ0s )
73	72	n0nod	⊢ ( ( 𝑥 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) → ( 𝑥 +s 𝑝 ) ∈ No )
74	73	3adant2	⊢ ( ( 𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) → ( 𝑥 +s 𝑝 ) ∈ No )
75	74	adantl	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) ) → ( 𝑥 +s 𝑝 ) ∈ No )
76	75	adantr	⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) → ( 𝑥 +s 𝑝 ) ∈ No )
77	32	adantr	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) ) → 𝑁 ∈ No )
78	77	adantr	⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) → 𝑁 ∈ No )
79		peano2no	⊢ ( 𝑁 ∈ No → ( 𝑁 +s 1s ) ∈ No )
80	78 79	syl	⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) → ( 𝑁 +s 1s ) ∈ No )
81		simpr	⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) → ( 𝑥 +s 𝑝 ) <s 𝑁 )
82	77	ltsp1d	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) ) → 𝑁 <s ( 𝑁 +s 1s ) )
83	82	adantr	⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) → 𝑁 <s ( 𝑁 +s 1s ) )
84	76 78 80 81 83	ltstrd	⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) → ( 𝑥 +s 𝑝 ) <s ( 𝑁 +s 1s ) )
85	84	ex	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) ) → ( ( 𝑥 +s 𝑝 ) <s 𝑁 → ( 𝑥 +s 𝑝 ) <s ( 𝑁 +s 1s ) ) )
86	70 71 85	3anim123d	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) ) → ( ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) → ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s ( 𝑁 +s 1s ) ) ) )
87		oveq1	⊢ ( 𝑎 = 𝑥 → ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) = ( 𝑥 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) )
88	87	eqeq2d	⊢ ( 𝑎 = 𝑥 → ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ↔ 𝑤 = ( 𝑥 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ) )
89		oveq1	⊢ ( 𝑎 = 𝑥 → ( 𝑎 +s 𝑞 ) = ( 𝑥 +s 𝑞 ) )
90	89	breq1d	⊢ ( 𝑎 = 𝑥 → ( ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ↔ ( 𝑥 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) )
91	88 90	3anbi13d	⊢ ( 𝑎 = 𝑥 → ( ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ( 𝑤 = ( 𝑥 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑥 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
92		oveq1	⊢ ( 𝑏 = 𝑦 → ( 𝑏 /su ( 2s ↑s 𝑞 ) ) = ( 𝑦 /su ( 2s ↑s 𝑞 ) ) )
93	92	oveq2d	⊢ ( 𝑏 = 𝑦 → ( 𝑥 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑞 ) ) ) )
94	93	eqeq2d	⊢ ( 𝑏 = 𝑦 → ( 𝑤 = ( 𝑥 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ↔ 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑞 ) ) ) ) )
95		breq1	⊢ ( 𝑏 = 𝑦 → ( 𝑏 <s ( 2s ↑s 𝑞 ) ↔ 𝑦 <s ( 2s ↑s 𝑞 ) ) )
96	94 95	3anbi12d	⊢ ( 𝑏 = 𝑦 → ( ( 𝑤 = ( 𝑥 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑥 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑥 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
97		oveq2	⊢ ( 𝑞 = 𝑝 → ( 2s ↑s 𝑞 ) = ( 2s ↑s 𝑝 ) )
98	97	oveq2d	⊢ ( 𝑞 = 𝑝 → ( 𝑦 /su ( 2s ↑s 𝑞 ) ) = ( 𝑦 /su ( 2s ↑s 𝑝 ) ) )
99	98	oveq2d	⊢ ( 𝑞 = 𝑝 → ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑞 ) ) ) = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) )
100	99	eqeq2d	⊢ ( 𝑞 = 𝑝 → ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑞 ) ) ) ↔ 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ) )
101	97	breq2d	⊢ ( 𝑞 = 𝑝 → ( 𝑦 <s ( 2s ↑s 𝑞 ) ↔ 𝑦 <s ( 2s ↑s 𝑝 ) ) )
102		oveq2	⊢ ( 𝑞 = 𝑝 → ( 𝑥 +s 𝑞 ) = ( 𝑥 +s 𝑝 ) )
103	102	breq1d	⊢ ( 𝑞 = 𝑝 → ( ( 𝑥 +s 𝑞 ) <s ( 𝑁 +s 1s ) ↔ ( 𝑥 +s 𝑝 ) <s ( 𝑁 +s 1s ) ) )
104	100 101 103	3anbi123d	⊢ ( 𝑞 = 𝑝 → ( ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑥 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s ( 𝑁 +s 1s ) ) ) )
105	91 96 104	rspc3ev	⊢ ( ( ( 𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) ∧ ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s ( 𝑁 +s 1s ) ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) )
106	105	ex	⊢ ( ( 𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) → ( ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s ( 𝑁 +s 1s ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
107	106	adantl	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) ) → ( ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s ( 𝑁 +s 1s ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
108	86 107	syld	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s ) ) → ( ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
109	108	rexlimdvvva	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
110	109	adantr	⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
111	69 110	jaod	⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ( 𝑤 = 𝑁 ∨ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑤 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
112	27 111	syld	⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑤 ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
113	112	impr	⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑤 ) ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) )
114	113	olcd	⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑤 ) ) ) → ( 𝑤 = ( 𝑁 +s 1s ) ∨ ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
115	114	expr	⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑤 ) → ( 𝑤 = ( 𝑁 +s 1s ) ∨ ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) )
116	115	expd	⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ 𝑁 ) → ( 0s ≤s 𝑤 → ( 𝑤 = ( 𝑁 +s 1s ) ∨ ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) ) )
117	5	eqeq2d	⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ↔ ( bday ‘ 𝑤 ) = suc ( bday ‘ 𝑁 ) ) )
118		df-ne	⊢ ( 𝑤 ≠ ( 𝑁 +s 1s ) ↔ ¬ 𝑤 = ( 𝑁 +s 1s ) )
119		simprl	⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) → 𝑤 ∈ No )
120		lesloe	⊢ ( ( 0s ∈ No ∧ 𝑤 ∈ No ) → ( 0s ≤s 𝑤 ↔ ( 0s <s 𝑤 ∨ 0s = 𝑤 ) ) )
121	57 119 120	sylancr	⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) → ( 0s ≤s 𝑤 ↔ ( 0s <s 𝑤 ∨ 0s = 𝑤 ) ) )
122		simprrl	⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) → ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) )
123	122	adantr	⊢ ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) → ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) )
124	1	peano2n0sd	⊢ ( 𝜑 → ( 𝑁 +s 1s ) ∈ ℕ0s )
125		n0bday	⊢ ( ( 𝑁 +s 1s ) ∈ ℕ0s → ( bday ‘ ( 𝑁 +s 1s ) ) ∈ ω )
126	124 125	syl	⊢ ( 𝜑 → ( bday ‘ ( 𝑁 +s 1s ) ) ∈ ω )
127	126	adantr	⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) → ( bday ‘ ( 𝑁 +s 1s ) ) ∈ ω )
128	127	adantr	⊢ ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) → ( bday ‘ ( 𝑁 +s 1s ) ) ∈ ω )
129	123 128	eqeltrd	⊢ ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) → ( bday ‘ 𝑤 ) ∈ ω )
130		oldfi	⊢ ( ( bday ‘ 𝑤 ) ∈ ω → ( O ‘ ( bday ‘ 𝑤 ) ) ∈ Fin )
131	129 130	syl	⊢ ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) → ( O ‘ ( bday ‘ 𝑤 ) ) ∈ Fin )
132		leftssold	⊢ ( L ‘ 𝑤 ) ⊆ ( O ‘ ( bday ‘ 𝑤 ) )
133		ssfi	⊢ ( ( ( O ‘ ( bday ‘ 𝑤 ) ) ∈ Fin ∧ ( L ‘ 𝑤 ) ⊆ ( O ‘ ( bday ‘ 𝑤 ) ) ) → ( L ‘ 𝑤 ) ∈ Fin )
134	131 132 133	sylancl	⊢ ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) → ( L ‘ 𝑤 ) ∈ Fin )
135		simplrl	⊢ ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) → 𝑤 ∈ No )
136		simpr	⊢ ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) → 0s <s 𝑤 )
137	135 136	0elleft	⊢ ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) → 0s ∈ ( L ‘ 𝑤 ) )
138	137	ne0d	⊢ ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) → ( L ‘ 𝑤 ) ≠ ∅ )
139		leftssno	⊢ ( L ‘ 𝑤 ) ⊆ No
140		ltsso	⊢ <s Or No
141		soss	⊢ ( ( L ‘ 𝑤 ) ⊆ No → ( <s Or No → <s Or ( L ‘ 𝑤 ) ) )
142	139 140 141	mp2	⊢ <s Or ( L ‘ 𝑤 )
143		fimax2g	⊢ ( ( <s Or ( L ‘ 𝑤 ) ∧ ( L ‘ 𝑤 ) ∈ Fin ∧ ( L ‘ 𝑤 ) ≠ ∅ ) → ∃ 𝑐 ∈ ( L ‘ 𝑤 ) ∀ 𝑒 ∈ ( L ‘ 𝑤 ) ¬ 𝑐 <s 𝑒 )
144	142 143	mp3an1	⊢ ( ( ( L ‘ 𝑤 ) ∈ Fin ∧ ( L ‘ 𝑤 ) ≠ ∅ ) → ∃ 𝑐 ∈ ( L ‘ 𝑤 ) ∀ 𝑒 ∈ ( L ‘ 𝑤 ) ¬ 𝑐 <s 𝑒 )
145		leftno	⊢ ( 𝑒 ∈ ( L ‘ 𝑤 ) → 𝑒 ∈ No )
146		leftno	⊢ ( 𝑐 ∈ ( L ‘ 𝑤 ) → 𝑐 ∈ No )
147		lenlts	⊢ ( ( 𝑒 ∈ No ∧ 𝑐 ∈ No ) → ( 𝑒 ≤s 𝑐 ↔ ¬ 𝑐 <s 𝑒 ) )
148	145 146 147	syl2anr	⊢ ( ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ 𝑒 ∈ ( L ‘ 𝑤 ) ) → ( 𝑒 ≤s 𝑐 ↔ ¬ 𝑐 <s 𝑒 ) )
149	148	ralbidva	⊢ ( 𝑐 ∈ ( L ‘ 𝑤 ) → ( ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ↔ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) ¬ 𝑐 <s 𝑒 ) )
150	149	adantl	⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ 𝑐 ∈ ( L ‘ 𝑤 ) ) → ( ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ↔ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) ¬ 𝑐 <s 𝑒 ) )
151		rightssold	⊢ ( R ‘ 𝑤 ) ⊆ ( O ‘ ( bday ‘ 𝑤 ) )
152		ssfi	⊢ ( ( ( O ‘ ( bday ‘ 𝑤 ) ) ∈ Fin ∧ ( R ‘ 𝑤 ) ⊆ ( O ‘ ( bday ‘ 𝑤 ) ) ) → ( R ‘ 𝑤 ) ∈ Fin )
153	131 151 152	sylancl	⊢ ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) → ( R ‘ 𝑤 ) ∈ Fin )
154	153	adantr	⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) → ( R ‘ 𝑤 ) ∈ Fin )
155	135	adantr	⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) → 𝑤 ∈ No )
156		simprrr	⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) → 𝑤 ≠ ( 𝑁 +s 1s ) )
157	156	adantr	⊢ ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) → 𝑤 ≠ ( 𝑁 +s 1s ) )
158	157	adantr	⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) → 𝑤 ≠ ( 𝑁 +s 1s ) )
159	158	neneqd	⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) → ¬ 𝑤 = ( 𝑁 +s 1s ) )
160		simpr	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) ∧ 𝑤 ∈ Ons ) → 𝑤 ∈ Ons )
161	124	ad4antr	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) ∧ 𝑤 ∈ Ons ) → ( 𝑁 +s 1s ) ∈ ℕ0s )
162		n0on	⊢ ( ( 𝑁 +s 1s ) ∈ ℕ0s → ( 𝑁 +s 1s ) ∈ Ons )
163	161 162	syl	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) ∧ 𝑤 ∈ Ons ) → ( 𝑁 +s 1s ) ∈ Ons )
164	123	adantr	⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) → ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) )
165	164	adantr	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) ∧ 𝑤 ∈ Ons ) → ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) )
166		bday11on	⊢ ( ( 𝑤 ∈ Ons ∧ ( 𝑁 +s 1s ) ∈ Ons ∧ ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ) → 𝑤 = ( 𝑁 +s 1s ) )
167	160 163 165 166	syl3anc	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) ∧ 𝑤 ∈ Ons ) → 𝑤 = ( 𝑁 +s 1s ) )
168	159 167	mtand	⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) → ¬ 𝑤 ∈ Ons )
169		elons	⊢ ( 𝑤 ∈ Ons ↔ ( 𝑤 ∈ No ∧ ( R ‘ 𝑤 ) = ∅ ) )
170	169	notbii	⊢ ( ¬ 𝑤 ∈ Ons ↔ ¬ ( 𝑤 ∈ No ∧ ( R ‘ 𝑤 ) = ∅ ) )
171		imnan	⊢ ( ( 𝑤 ∈ No → ¬ ( R ‘ 𝑤 ) = ∅ ) ↔ ¬ ( 𝑤 ∈ No ∧ ( R ‘ 𝑤 ) = ∅ ) )
172	170 171	bitr4i	⊢ ( ¬ 𝑤 ∈ Ons ↔ ( 𝑤 ∈ No → ¬ ( R ‘ 𝑤 ) = ∅ ) )
173	168 172	sylib	⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) → ( 𝑤 ∈ No → ¬ ( R ‘ 𝑤 ) = ∅ ) )
174	155 173	mpd	⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) → ¬ ( R ‘ 𝑤 ) = ∅ )
175	174	neqned	⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) → ( R ‘ 𝑤 ) ≠ ∅ )
176		rightssno	⊢ ( R ‘ 𝑤 ) ⊆ No
177		soss	⊢ ( ( R ‘ 𝑤 ) ⊆ No → ( <s Or No → <s Or ( R ‘ 𝑤 ) ) )
178	176 140 177	mp2	⊢ <s Or ( R ‘ 𝑤 )
179		fimin2g	⊢ ( ( <s Or ( R ‘ 𝑤 ) ∧ ( R ‘ 𝑤 ) ∈ Fin ∧ ( R ‘ 𝑤 ) ≠ ∅ ) → ∃ 𝑑 ∈ ( R ‘ 𝑤 ) ∀ 𝑓 ∈ ( R ‘ 𝑤 ) ¬ 𝑓 <s 𝑑 )
180	178 179	mp3an1	⊢ ( ( ( R ‘ 𝑤 ) ∈ Fin ∧ ( R ‘ 𝑤 ) ≠ ∅ ) → ∃ 𝑑 ∈ ( R ‘ 𝑤 ) ∀ 𝑓 ∈ ( R ‘ 𝑤 ) ¬ 𝑓 <s 𝑑 )
181		rightno	⊢ ( 𝑑 ∈ ( R ‘ 𝑤 ) → 𝑑 ∈ No )
182		rightno	⊢ ( 𝑓 ∈ ( R ‘ 𝑤 ) → 𝑓 ∈ No )
183		lenlts	⊢ ( ( 𝑑 ∈ No ∧ 𝑓 ∈ No ) → ( 𝑑 ≤s 𝑓 ↔ ¬ 𝑓 <s 𝑑 ) )
184	181 182 183	syl2an	⊢ ( ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ 𝑓 ∈ ( R ‘ 𝑤 ) ) → ( 𝑑 ≤s 𝑓 ↔ ¬ 𝑓 <s 𝑑 ) )
185	184	ralbidva	⊢ ( 𝑑 ∈ ( R ‘ 𝑤 ) → ( ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ↔ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) ¬ 𝑓 <s 𝑑 ) )
186	185	adantl	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) ∧ 𝑑 ∈ ( R ‘ 𝑤 ) ) → ( ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ↔ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) ¬ 𝑓 <s 𝑑 ) )
187		simp2l	⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → 𝑐 ∈ ( L ‘ 𝑤 ) )
188		simp2r	⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 )
189		simp3l	⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → 𝑑 ∈ ( R ‘ 𝑤 ) )
190		simp3r	⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 )
191	187 188 189 190	cutminmax	⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → 𝑤 = ( { 𝑐 } |s { 𝑑 } ) )
192		simpl2l	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → 𝑐 ∈ ( L ‘ 𝑤 ) )
193	132 192	sselid	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → 𝑐 ∈ ( O ‘ ( bday ‘ 𝑤 ) ) )
194	192	leftnod	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → 𝑐 ∈ No )
195		oldbday	⊢ ( ( ( bday ‘ 𝑤 ) ∈ On ∧ 𝑐 ∈ No ) → ( 𝑐 ∈ ( O ‘ ( bday ‘ 𝑤 ) ) ↔ ( bday ‘ 𝑐 ) ∈ ( bday ‘ 𝑤 ) ) )
196	7 194 195	sylancr	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ( 𝑐 ∈ ( O ‘ ( bday ‘ 𝑤 ) ) ↔ ( bday ‘ 𝑐 ) ∈ ( bday ‘ 𝑤 ) ) )
197	193 196	mpbid	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ( bday ‘ 𝑐 ) ∈ ( bday ‘ 𝑤 ) )
198	123	3ad2ant1	⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) )
199	198	adantr	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) )
200	1	adantr	⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) → 𝑁 ∈ ℕ0s )
201	200	adantr	⊢ ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) → 𝑁 ∈ ℕ0s )
202	201	3ad2ant1	⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → 𝑁 ∈ ℕ0s )
203	202	adantr	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → 𝑁 ∈ ℕ0s )
204	203 3	syl	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ( bday ‘ ( 𝑁 +s 1s ) ) = suc ( bday ‘ 𝑁 ) )
205	199 204	eqtrd	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ( bday ‘ 𝑤 ) = suc ( bday ‘ 𝑁 ) )
206	197 205	eleqtrd	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ( bday ‘ 𝑐 ) ∈ suc ( bday ‘ 𝑁 ) )
207		bdayon	⊢ ( bday ‘ 𝑐 ) ∈ On
208		onsssuc	⊢ ( ( ( bday ‘ 𝑐 ) ∈ On ∧ ( bday ‘ 𝑁 ) ∈ On ) → ( ( bday ‘ 𝑐 ) ⊆ ( bday ‘ 𝑁 ) ↔ ( bday ‘ 𝑐 ) ∈ suc ( bday ‘ 𝑁 ) ) )
209	207 8 208	mp2an	⊢ ( ( bday ‘ 𝑐 ) ⊆ ( bday ‘ 𝑁 ) ↔ ( bday ‘ 𝑐 ) ∈ suc ( bday ‘ 𝑁 ) )
210	206 209	sylibr	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ( bday ‘ 𝑐 ) ⊆ ( bday ‘ 𝑁 ) )
211		breq1	⊢ ( 𝑒 = 0s → ( 𝑒 ≤s 𝑐 ↔ 0s ≤s 𝑐 ) )
212		simpl2r	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 )
213		simpl1	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) )
214	213 137	syl	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → 0s ∈ ( L ‘ 𝑤 ) )
215	211 212 214	rspcdva	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → 0s ≤s 𝑐 )
216		simp1ll	⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → 𝜑 )
217	216	adantr	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → 𝜑 )
218		n0on	⊢ ( 𝑁 ∈ ℕ0s → 𝑁 ∈ Ons )
219	1 218	syl	⊢ ( 𝜑 → 𝑁 ∈ Ons )
220	217 219	syl	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → 𝑁 ∈ Ons )
221		simpl3l	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → 𝑑 ∈ ( R ‘ 𝑤 ) )
222	151 221	sselid	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → 𝑑 ∈ ( O ‘ ( bday ‘ 𝑤 ) ) )
223		oldbdayim	⊢ ( 𝑑 ∈ ( O ‘ ( bday ‘ 𝑤 ) ) → ( bday ‘ 𝑑 ) ∈ ( bday ‘ 𝑤 ) )
224	222 223	syl	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ( bday ‘ 𝑑 ) ∈ ( bday ‘ 𝑤 ) )
225	224 205	eleqtrd	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ( bday ‘ 𝑑 ) ∈ suc ( bday ‘ 𝑁 ) )
226		bdayon	⊢ ( bday ‘ 𝑑 ) ∈ On
227		onsssuc	⊢ ( ( ( bday ‘ 𝑑 ) ∈ On ∧ ( bday ‘ 𝑁 ) ∈ On ) → ( ( bday ‘ 𝑑 ) ⊆ ( bday ‘ 𝑁 ) ↔ ( bday ‘ 𝑑 ) ∈ suc ( bday ‘ 𝑁 ) ) )
228	226 8 227	mp2an	⊢ ( ( bday ‘ 𝑑 ) ⊆ ( bday ‘ 𝑁 ) ↔ ( bday ‘ 𝑑 ) ∈ suc ( bday ‘ 𝑁 ) )
229	225 228	sylibr	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ( bday ‘ 𝑑 ) ⊆ ( bday ‘ 𝑁 ) )
230	221	rightnod	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → 𝑑 ∈ No )
231		madebday	⊢ ( ( ( bday ‘ 𝑁 ) ∈ On ∧ 𝑑 ∈ No ) → ( 𝑑 ∈ ( M ‘ ( bday ‘ 𝑁 ) ) ↔ ( bday ‘ 𝑑 ) ⊆ ( bday ‘ 𝑁 ) ) )
232	8 230 231	sylancr	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ( 𝑑 ∈ ( M ‘ ( bday ‘ 𝑁 ) ) ↔ ( bday ‘ 𝑑 ) ⊆ ( bday ‘ 𝑁 ) ) )
233	229 232	mpbird	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → 𝑑 ∈ ( M ‘ ( bday ‘ 𝑁 ) ) )
234		onsbnd	⊢ ( ( 𝑁 ∈ Ons ∧ 𝑑 ∈ ( M ‘ ( bday ‘ 𝑁 ) ) ) → 𝑑 ≤s 𝑁 )
235	220 233 234	syl2anc	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → 𝑑 ≤s 𝑁 )
236	203	n0nod	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → 𝑁 ∈ No )
237	230 236	lesnltd	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ( 𝑑 ≤s 𝑁 ↔ ¬ 𝑁 <s 𝑑 ) )
238	235 237	mpbid	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ¬ 𝑁 <s 𝑑 )
239		lltr	⊢ ( L ‘ 𝑤 ) <<s ( R ‘ 𝑤 )
240	239	a1i	⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → ( L ‘ 𝑤 ) <<s ( R ‘ 𝑤 ) )
241	240 187 189	sltssepcd	⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → 𝑐 <s 𝑑 )
242	241	adantr	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → 𝑐 <s 𝑑 )
243		breq1	⊢ ( 𝑐 = 𝑁 → ( 𝑐 <s 𝑑 ↔ 𝑁 <s 𝑑 ) )
244	242 243	syl5ibcom	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ( 𝑐 = 𝑁 → 𝑁 <s 𝑑 ) )
245	238 244	mtod	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ¬ 𝑐 = 𝑁 )
246		fveq2	⊢ ( 𝑧 = 𝑐 → ( bday ‘ 𝑧 ) = ( bday ‘ 𝑐 ) )
247	246	sseq1d	⊢ ( 𝑧 = 𝑐 → ( ( bday ‘ 𝑧 ) ⊆ ( bday ‘ 𝑁 ) ↔ ( bday ‘ 𝑐 ) ⊆ ( bday ‘ 𝑁 ) ) )
248		breq2	⊢ ( 𝑧 = 𝑐 → ( 0s ≤s 𝑧 ↔ 0s ≤s 𝑐 ) )
249	247 248	anbi12d	⊢ ( 𝑧 = 𝑐 → ( ( ( bday ‘ 𝑧 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑧 ) ↔ ( ( bday ‘ 𝑐 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑐 ) ) )
250		eqeq1	⊢ ( 𝑧 = 𝑐 → ( 𝑧 = 𝑁 ↔ 𝑐 = 𝑁 ) )
251		eqeq1	⊢ ( 𝑧 = 𝑐 → ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ↔ 𝑐 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ) )
252	251	3anbi1d	⊢ ( 𝑧 = 𝑐 → ( ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ↔ ( 𝑐 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) )
253	252	rexbidv	⊢ ( 𝑧 = 𝑐 → ( ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ↔ ∃ 𝑝 ∈ ℕ0s ( 𝑐 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) )
254	253	2rexbidv	⊢ ( 𝑧 = 𝑐 → ( ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ↔ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑐 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) )
255		oveq1	⊢ ( 𝑥 = 𝑔 → ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) = ( 𝑔 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) )
256	255	eqeq2d	⊢ ( 𝑥 = 𝑔 → ( 𝑐 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ↔ 𝑐 = ( 𝑔 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ) )
257		oveq1	⊢ ( 𝑥 = 𝑔 → ( 𝑥 +s 𝑝 ) = ( 𝑔 +s 𝑝 ) )
258	257	breq1d	⊢ ( 𝑥 = 𝑔 → ( ( 𝑥 +s 𝑝 ) <s 𝑁 ↔ ( 𝑔 +s 𝑝 ) <s 𝑁 ) )
259	256 258	3anbi13d	⊢ ( 𝑥 = 𝑔 → ( ( 𝑐 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ↔ ( 𝑐 = ( 𝑔 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑔 +s 𝑝 ) <s 𝑁 ) ) )
260	259	rexbidv	⊢ ( 𝑥 = 𝑔 → ( ∃ 𝑝 ∈ ℕ0s ( 𝑐 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ↔ ∃ 𝑝 ∈ ℕ0s ( 𝑐 = ( 𝑔 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑔 +s 𝑝 ) <s 𝑁 ) ) )
261		oveq1	⊢ ( 𝑦 = ℎ → ( 𝑦 /su ( 2s ↑s 𝑝 ) ) = ( ℎ /su ( 2s ↑s 𝑝 ) ) )
262	261	oveq2d	⊢ ( 𝑦 = ℎ → ( 𝑔 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑝 ) ) ) )
263	262	eqeq2d	⊢ ( 𝑦 = ℎ → ( 𝑐 = ( 𝑔 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ↔ 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑝 ) ) ) ) )
264		breq1	⊢ ( 𝑦 = ℎ → ( 𝑦 <s ( 2s ↑s 𝑝 ) ↔ ℎ <s ( 2s ↑s 𝑝 ) ) )
265	263 264	3anbi12d	⊢ ( 𝑦 = ℎ → ( ( 𝑐 = ( 𝑔 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑔 +s 𝑝 ) <s 𝑁 ) ↔ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑝 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑝 ) ∧ ( 𝑔 +s 𝑝 ) <s 𝑁 ) ) )
266	265	rexbidv	⊢ ( 𝑦 = ℎ → ( ∃ 𝑝 ∈ ℕ0s ( 𝑐 = ( 𝑔 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑔 +s 𝑝 ) <s 𝑁 ) ↔ ∃ 𝑝 ∈ ℕ0s ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑝 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑝 ) ∧ ( 𝑔 +s 𝑝 ) <s 𝑁 ) ) )
267		oveq2	⊢ ( 𝑝 = 𝑖 → ( 2s ↑s 𝑝 ) = ( 2s ↑s 𝑖 ) )
268	267	oveq2d	⊢ ( 𝑝 = 𝑖 → ( ℎ /su ( 2s ↑s 𝑝 ) ) = ( ℎ /su ( 2s ↑s 𝑖 ) ) )
269	268	oveq2d	⊢ ( 𝑝 = 𝑖 → ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑝 ) ) ) = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) )
270	269	eqeq2d	⊢ ( 𝑝 = 𝑖 → ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑝 ) ) ) ↔ 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ) )
271	267	breq2d	⊢ ( 𝑝 = 𝑖 → ( ℎ <s ( 2s ↑s 𝑝 ) ↔ ℎ <s ( 2s ↑s 𝑖 ) ) )
272		oveq2	⊢ ( 𝑝 = 𝑖 → ( 𝑔 +s 𝑝 ) = ( 𝑔 +s 𝑖 ) )
273	272	breq1d	⊢ ( 𝑝 = 𝑖 → ( ( 𝑔 +s 𝑝 ) <s 𝑁 ↔ ( 𝑔 +s 𝑖 ) <s 𝑁 ) )
274	270 271 273	3anbi123d	⊢ ( 𝑝 = 𝑖 → ( ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑝 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑝 ) ∧ ( 𝑔 +s 𝑝 ) <s 𝑁 ) ↔ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) )
275	274	cbvrexvw	⊢ ( ∃ 𝑝 ∈ ℕ0s ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑝 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑝 ) ∧ ( 𝑔 +s 𝑝 ) <s 𝑁 ) ↔ ∃ 𝑖 ∈ ℕ0s ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) )
276	266 275	bitrdi	⊢ ( 𝑦 = ℎ → ( ∃ 𝑝 ∈ ℕ0s ( 𝑐 = ( 𝑔 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑔 +s 𝑝 ) <s 𝑁 ) ↔ ∃ 𝑖 ∈ ℕ0s ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) )
277	260 276	cbvrex2vw	⊢ ( ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑐 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ↔ ∃ 𝑔 ∈ ℕ0s ∃ ℎ ∈ ℕ0s ∃ 𝑖 ∈ ℕ0s ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) )
278	254 277	bitrdi	⊢ ( 𝑧 = 𝑐 → ( ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ↔ ∃ 𝑔 ∈ ℕ0s ∃ ℎ ∈ ℕ0s ∃ 𝑖 ∈ ℕ0s ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) )
279	250 278	orbi12d	⊢ ( 𝑧 = 𝑐 → ( ( 𝑧 = 𝑁 ∨ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ↔ ( 𝑐 = 𝑁 ∨ ∃ 𝑔 ∈ ℕ0s ∃ ℎ ∈ ℕ0s ∃ 𝑖 ∈ ℕ0s ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) )
280	249 279	imbi12d	⊢ ( 𝑧 = 𝑐 → ( ( ( ( bday ‘ 𝑧 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑧 ) → ( 𝑧 = 𝑁 ∨ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) ↔ ( ( ( bday ‘ 𝑐 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑐 ) → ( 𝑐 = 𝑁 ∨ ∃ 𝑔 ∈ ℕ0s ∃ ℎ ∈ ℕ0s ∃ 𝑖 ∈ ℕ0s ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ) )
281	2	adantr	⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) → ∀ 𝑧 ∈ No ( ( ( bday ‘ 𝑧 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑧 ) → ( 𝑧 = 𝑁 ∨ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) )
282	281	adantr	⊢ ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) → ∀ 𝑧 ∈ No ( ( ( bday ‘ 𝑧 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑧 ) → ( 𝑧 = 𝑁 ∨ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) )
283	282	3ad2ant1	⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → ∀ 𝑧 ∈ No ( ( ( bday ‘ 𝑧 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑧 ) → ( 𝑧 = 𝑁 ∨ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) )
284	283	adantr	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ∀ 𝑧 ∈ No ( ( ( bday ‘ 𝑧 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑧 ) → ( 𝑧 = 𝑁 ∨ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) )
285	280 284 194	rspcdva	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ( ( ( bday ‘ 𝑐 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑐 ) → ( 𝑐 = 𝑁 ∨ ∃ 𝑔 ∈ ℕ0s ∃ ℎ ∈ ℕ0s ∃ 𝑖 ∈ ℕ0s ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) )
286		orel1	⊢ ( ¬ 𝑐 = 𝑁 → ( ( 𝑐 = 𝑁 ∨ ∃ 𝑔 ∈ ℕ0s ∃ ℎ ∈ ℕ0s ∃ 𝑖 ∈ ℕ0s ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) → ∃ 𝑔 ∈ ℕ0s ∃ ℎ ∈ ℕ0s ∃ 𝑖 ∈ ℕ0s ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) )
287	245 285 286	sylsyld	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ( ( ( bday ‘ 𝑐 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑐 ) → ∃ 𝑔 ∈ ℕ0s ∃ ℎ ∈ ℕ0s ∃ 𝑖 ∈ ℕ0s ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) )
288		simp3l1	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → 𝑔 ∈ ℕ0s )
289	288	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → 𝑔 ∈ ℕ0s )
290	289	n0nod	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → 𝑔 ∈ No )
291		simp3l3	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → 𝑖 ∈ ℕ0s )
292	291	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → 𝑖 ∈ ℕ0s )
293		n0addscl	⊢ ( ( 𝑔 ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) → ( 𝑔 +s 𝑖 ) ∈ ℕ0s )
294	289 292 293	syl2anc	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ( 𝑔 +s 𝑖 ) ∈ ℕ0s )
295	294	n0nod	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ( 𝑔 +s 𝑖 ) ∈ No )
296	216	3ad2ant1	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → 𝜑 )
297	296	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → 𝜑 )
298	297 32	syl	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → 𝑁 ∈ No )
299		n0sge0	⊢ ( 𝑖 ∈ ℕ0s → 0s ≤s 𝑖 )
300	292 299	syl	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → 0s ≤s 𝑖 )
301	292	n0nod	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → 𝑖 ∈ No )
302	290 301	addsge01d	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ( 0s ≤s 𝑖 ↔ 𝑔 ≤s ( 𝑔 +s 𝑖 ) ) )
303	300 302	mpbid	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → 𝑔 ≤s ( 𝑔 +s 𝑖 ) )
304		simp3r3	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → ( 𝑔 +s 𝑖 ) <s 𝑁 )
305	304	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ( 𝑔 +s 𝑖 ) <s 𝑁 )
306	290 295 298 303 305	leltstrd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → 𝑔 <s 𝑁 )
307	297 1	syl	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → 𝑁 ∈ ℕ0s )
308		n0ltsp1le	⊢ ( ( 𝑔 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s ) → ( 𝑔 <s 𝑁 ↔ ( 𝑔 +s 1s ) ≤s 𝑁 ) )
309	289 307 308	syl2anc	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ( 𝑔 <s 𝑁 ↔ ( 𝑔 +s 1s ) ≤s 𝑁 ) )
310	306 309	mpbid	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ( 𝑔 +s 1s ) ≤s 𝑁 )
311		ltsirr	⊢ ( 𝑁 ∈ No → ¬ 𝑁 <s 𝑁 )
312	298 311	syl	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ¬ 𝑁 <s 𝑁 )
313	289	peano2n0sd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ( 𝑔 +s 1s ) ∈ ℕ0s )
314	313	n0nod	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ( 𝑔 +s 1s ) ∈ No )
315	314 298	ltsnled	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ( ( 𝑔 +s 1s ) <s 𝑁 ↔ ¬ 𝑁 ≤s ( 𝑔 +s 1s ) ) )
316	296	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 𝜑 )
317	124	n0nod	⊢ ( 𝜑 → ( 𝑁 +s 1s ) ∈ No )
318	316 317	syl	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑁 +s 1s ) ∈ No )
319	316 32	syl	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 𝑁 ∈ No )
320	52	a1i	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 2s ∈ No )
321	319 320	subscld	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑁 -s 2s ) ∈ No )
322		1no	⊢ 1s ∈ No
323	322	a1i	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 1s ∈ No )
324	321 323 323	addsassd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( ( 𝑁 -s 2s ) +s 1s ) +s 1s ) = ( ( 𝑁 -s 2s ) +s ( 1s +s 1s ) ) )
325		1p1e2s	⊢ ( 1s +s 1s ) = 2s
326	325	oveq2i	⊢ ( ( 𝑁 -s 2s ) +s ( 1s +s 1s ) ) = ( ( 𝑁 -s 2s ) +s 2s )
327		npcans	⊢ ( ( 𝑁 ∈ No ∧ 2s ∈ No ) → ( ( 𝑁 -s 2s ) +s 2s ) = 𝑁 )
328	319 52 327	sylancl	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( 𝑁 -s 2s ) +s 2s ) = 𝑁 )
329	326 328	eqtrid	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( 𝑁 -s 2s ) +s ( 1s +s 1s ) ) = 𝑁 )
330	324 329	eqtrd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( ( 𝑁 -s 2s ) +s 1s ) +s 1s ) = 𝑁 )
331	330 319	eqeltrd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( ( 𝑁 -s 2s ) +s 1s ) +s 1s ) ∈ No )
332	321 323	addscld	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( 𝑁 -s 2s ) +s 1s ) ∈ No )
333	198	3ad2ant1	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) )
334	333	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) )
335		simpl2	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 𝑤 = ( { 𝑐 } |s { 𝑑 } ) )
336	187	leftnod	⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → 𝑐 ∈ No )
337	336	3ad2ant1	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → 𝑐 ∈ No )
338	337	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 𝑐 ∈ No )
339	288	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 𝑔 ∈ ℕ0s )
340	339	peano2n0sd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑔 +s 1s ) ∈ ℕ0s )
341	340	n0nod	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑔 +s 1s ) ∈ No )
342		subscl	⊢ ( ( 𝑁 ∈ No ∧ 1s ∈ No ) → ( 𝑁 -s 1s ) ∈ No )
343	32 322 342	sylancl	⊢ ( 𝜑 → ( 𝑁 -s 1s ) ∈ No )
344	316 343	syl	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑁 -s 1s ) ∈ No )
345		simp3r1	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) )
346	345	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) )
347		simp3r2	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → ℎ <s ( 2s ↑s 𝑖 ) )
348	347	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ℎ <s ( 2s ↑s 𝑖 ) )
349	291	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 𝑖 ∈ ℕ0s )
350		expscl	⊢ ( ( 2s ∈ No ∧ 𝑖 ∈ ℕ0s ) → ( 2s ↑s 𝑖 ) ∈ No )
351	52 349 350	sylancr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 2s ↑s 𝑖 ) ∈ No )
352	351	mulslidd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 1s ·s ( 2s ↑s 𝑖 ) ) = ( 2s ↑s 𝑖 ) )
353	348 352	breqtrrd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ℎ <s ( 1s ·s ( 2s ↑s 𝑖 ) ) )
354		simp3l2	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → ℎ ∈ ℕ0s )
355	354	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ℎ ∈ ℕ0s )
356	355	n0nod	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ℎ ∈ No )
357	356 323 349	pw2ltdivmuls2d	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( ℎ /su ( 2s ↑s 𝑖 ) ) <s 1s ↔ ℎ <s ( 1s ·s ( 2s ↑s 𝑖 ) ) ) )
358	353 357	mpbird	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ℎ /su ( 2s ↑s 𝑖 ) ) <s 1s )
359	356 349	pw2divscld	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ℎ /su ( 2s ↑s 𝑖 ) ) ∈ No )
360	339	n0nod	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 𝑔 ∈ No )
361	359 323 360	ltadds2d	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( ℎ /su ( 2s ↑s 𝑖 ) ) <s 1s ↔ ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) <s ( 𝑔 +s 1s ) ) )
362	358 361	mpbid	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) <s ( 𝑔 +s 1s ) )
363	346 362	eqbrtrd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 𝑐 <s ( 𝑔 +s 1s ) )
364		n0ltsp1le	⊢ ( ( ( 𝑔 +s 1s ) ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s ) → ( ( 𝑔 +s 1s ) <s 𝑁 ↔ ( ( 𝑔 +s 1s ) +s 1s ) ≤s 𝑁 ) )
365	313 307 364	syl2anc	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ( ( 𝑔 +s 1s ) <s 𝑁 ↔ ( ( 𝑔 +s 1s ) +s 1s ) ≤s 𝑁 ) )
366	365	biimpd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ( ( 𝑔 +s 1s ) <s 𝑁 → ( ( 𝑔 +s 1s ) +s 1s ) ≤s 𝑁 ) )
367	366	impr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( 𝑔 +s 1s ) +s 1s ) ≤s 𝑁 )
368		npcans	⊢ ( ( 𝑁 ∈ No ∧ 1s ∈ No ) → ( ( 𝑁 -s 1s ) +s 1s ) = 𝑁 )
369	319 322 368	sylancl	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( 𝑁 -s 1s ) +s 1s ) = 𝑁 )
370	367 369	breqtrrd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( 𝑔 +s 1s ) +s 1s ) ≤s ( ( 𝑁 -s 1s ) +s 1s ) )
371	341 344 323	leadds1d	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( 𝑔 +s 1s ) ≤s ( 𝑁 -s 1s ) ↔ ( ( 𝑔 +s 1s ) +s 1s ) ≤s ( ( 𝑁 -s 1s ) +s 1s ) ) )
372	370 371	mpbird	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑔 +s 1s ) ≤s ( 𝑁 -s 1s ) )
373	338 341 344 363 372	ltlestrd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 𝑐 <s ( 𝑁 -s 1s ) )
374	325	oveq2i	⊢ ( 𝑁 -s ( 1s +s 1s ) ) = ( 𝑁 -s 2s )
375	374	oveq1i	⊢ ( ( 𝑁 -s ( 1s +s 1s ) ) +s 1s ) = ( ( 𝑁 -s 2s ) +s 1s )
376	319 323 323	subsubs4d	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( 𝑁 -s 1s ) -s 1s ) = ( 𝑁 -s ( 1s +s 1s ) ) )
377	376	oveq1d	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( ( 𝑁 -s 1s ) -s 1s ) +s 1s ) = ( ( 𝑁 -s ( 1s +s 1s ) ) +s 1s ) )
378		npcans	⊢ ( ( ( 𝑁 -s 1s ) ∈ No ∧ 1s ∈ No ) → ( ( ( 𝑁 -s 1s ) -s 1s ) +s 1s ) = ( 𝑁 -s 1s ) )
379	344 322 378	sylancl	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( ( 𝑁 -s 1s ) -s 1s ) +s 1s ) = ( 𝑁 -s 1s ) )
380	377 379	eqtr3d	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( 𝑁 -s ( 1s +s 1s ) ) +s 1s ) = ( 𝑁 -s 1s ) )
381	375 380	eqtr3id	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( 𝑁 -s 2s ) +s 1s ) = ( 𝑁 -s 1s ) )
382	373 381	breqtrrd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 𝑐 <s ( ( 𝑁 -s 2s ) +s 1s ) )
383	338 332 382	sltssn	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → { 𝑐 } <<s { ( ( 𝑁 -s 2s ) +s 1s ) } )
384	189	rightnod	⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → 𝑑 ∈ No )
385	384	3ad2ant1	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → 𝑑 ∈ No )
386	385	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 𝑑 ∈ No )
387		simprl	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 𝑑 = 𝑁 )
388	387	oveq1d	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑑 -s 1s ) = ( 𝑁 -s 1s ) )
389	386	ltsm1d	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑑 -s 1s ) <s 𝑑 )
390	388 389	eqbrtrrd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑁 -s 1s ) <s 𝑑 )
391	381 390	eqbrtrd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( 𝑁 -s 2s ) +s 1s ) <s 𝑑 )
392	332 386 391	sltssn	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → { ( ( 𝑁 -s 2s ) +s 1s ) } <<s { 𝑑 } )
393	335 332 383 392	sltsbday	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( bday ‘ 𝑤 ) ⊆ ( bday ‘ ( ( 𝑁 -s 2s ) +s 1s ) ) )
394	334 393	eqsstrrd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( bday ‘ ( 𝑁 +s 1s ) ) ⊆ ( bday ‘ ( ( 𝑁 -s 2s ) +s 1s ) ) )
395	124 162	syl	⊢ ( 𝜑 → ( 𝑁 +s 1s ) ∈ Ons )
396	316 395	syl	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑁 +s 1s ) ∈ Ons )
397	319 323 320	addsubsd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( 𝑁 +s 1s ) -s 2s ) = ( ( 𝑁 -s 2s ) +s 1s ) )
398		n0sge0	⊢ ( 𝑔 ∈ ℕ0s → 0s ≤s 𝑔 )
399	339 398	syl	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 0s ≤s 𝑔 )
400	323 360	addsge01d	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 0s ≤s 𝑔 ↔ 1s ≤s ( 1s +s 𝑔 ) ) )
401	399 400	mpbid	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 1s ≤s ( 1s +s 𝑔 ) )
402	360 323	addscomd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑔 +s 1s ) = ( 1s +s 𝑔 ) )
403	401 402	breqtrrd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 1s ≤s ( 𝑔 +s 1s ) )
404		simprr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑔 +s 1s ) <s 𝑁 )
405	323 341 319 403 404	leltstrd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 1s <s 𝑁 )
406	323 319 405	ltlesd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 1s ≤s 𝑁 )
407	323 319 323	leadds1d	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 1s ≤s 𝑁 ↔ ( 1s +s 1s ) ≤s ( 𝑁 +s 1s ) ) )
408	406 407	mpbid	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 1s +s 1s ) ≤s ( 𝑁 +s 1s ) )
409	325 408	eqbrtrrid	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 2s ≤s ( 𝑁 +s 1s ) )
410		2nns	⊢ 2s ∈ ℕs
411		nnn0s	⊢ ( 2s ∈ ℕs → 2s ∈ ℕ0s )
412	410 411	ax-mp	⊢ 2s ∈ ℕ0s
413	296 124	syl	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → ( 𝑁 +s 1s ) ∈ ℕ0s )
414	413	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑁 +s 1s ) ∈ ℕ0s )
415		n0subs	⊢ ( ( 2s ∈ ℕ0s ∧ ( 𝑁 +s 1s ) ∈ ℕ0s ) → ( 2s ≤s ( 𝑁 +s 1s ) ↔ ( ( 𝑁 +s 1s ) -s 2s ) ∈ ℕ0s ) )
416	412 414 415	sylancr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 2s ≤s ( 𝑁 +s 1s ) ↔ ( ( 𝑁 +s 1s ) -s 2s ) ∈ ℕ0s ) )
417	409 416	mpbid	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( 𝑁 +s 1s ) -s 2s ) ∈ ℕ0s )
418	397 417	eqeltrrd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( 𝑁 -s 2s ) +s 1s ) ∈ ℕ0s )
419		n0on	⊢ ( ( ( 𝑁 -s 2s ) +s 1s ) ∈ ℕ0s → ( ( 𝑁 -s 2s ) +s 1s ) ∈ Ons )
420	418 419	syl	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( 𝑁 -s 2s ) +s 1s ) ∈ Ons )
421	396 420	onlesd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( 𝑁 +s 1s ) ≤s ( ( 𝑁 -s 2s ) +s 1s ) ↔ ( bday ‘ ( 𝑁 +s 1s ) ) ⊆ ( bday ‘ ( ( 𝑁 -s 2s ) +s 1s ) ) ) )
422	394 421	mpbird	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑁 +s 1s ) ≤s ( ( 𝑁 -s 2s ) +s 1s ) )
423	332	ltsp1d	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( ( 𝑁 -s 2s ) +s 1s ) <s ( ( ( 𝑁 -s 2s ) +s 1s ) +s 1s ) )
424	318 332 331 422 423	leltstrd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑁 +s 1s ) <s ( ( ( 𝑁 -s 2s ) +s 1s ) +s 1s ) )
425	318 331 424	ltlesd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑁 +s 1s ) ≤s ( ( ( 𝑁 -s 2s ) +s 1s ) +s 1s ) )
426	425 330	breqtrd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑁 +s 1s ) ≤s 𝑁 )
427	316 1	syl	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 𝑁 ∈ ℕ0s )
428		n0ltsp1le	⊢ ( ( 𝑁 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s ) → ( 𝑁 <s 𝑁 ↔ ( 𝑁 +s 1s ) ≤s 𝑁 ) )
429	427 427 428	syl2anc	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → ( 𝑁 <s 𝑁 ↔ ( 𝑁 +s 1s ) ≤s 𝑁 ) )
430	426 429	mpbird	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) <s 𝑁 ) ) → 𝑁 <s 𝑁 )
431	430	expr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ( ( 𝑔 +s 1s ) <s 𝑁 → 𝑁 <s 𝑁 ) )
432	315 431	sylbird	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ( ¬ 𝑁 ≤s ( 𝑔 +s 1s ) → 𝑁 <s 𝑁 ) )
433	312 432	mt3d	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → 𝑁 ≤s ( 𝑔 +s 1s ) )
434	314 298	lestri3d	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ( ( 𝑔 +s 1s ) = 𝑁 ↔ ( ( 𝑔 +s 1s ) ≤s 𝑁 ∧ 𝑁 ≤s ( 𝑔 +s 1s ) ) ) )
435	310 433 434	mpbir2and	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ( 𝑔 +s 1s ) = 𝑁 )
436	304	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ) → ( 𝑔 +s 𝑖 ) <s 𝑁 )
437		simprr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ) → ( 𝑔 +s 1s ) = 𝑁 )
438	436 437	breqtrrd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ) → ( 𝑔 +s 𝑖 ) <s ( 𝑔 +s 1s ) )
439	291	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ) → 𝑖 ∈ ℕ0s )
440	439	n0nod	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ) → 𝑖 ∈ No )
441	322	a1i	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ) → 1s ∈ No )
442	288	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ) → 𝑔 ∈ ℕ0s )
443	442	n0nod	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ) → 𝑔 ∈ No )
444	440 441 443	ltadds2d	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ) → ( 𝑖 <s 1s ↔ ( 𝑔 +s 𝑖 ) <s ( 𝑔 +s 1s ) ) )
445	438 444	mpbird	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ) → 𝑖 <s 1s )
446		n0lts1e0	⊢ ( 𝑖 ∈ ℕ0s → ( 𝑖 <s 1s ↔ 𝑖 = 0s ) )
447	439 446	syl	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ) → ( 𝑖 <s 1s ↔ 𝑖 = 0s ) )
448	445 447	mpbid	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ) → 𝑖 = 0s )
449	345	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ) → 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) )
450	347	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ) → ℎ <s ( 2s ↑s 𝑖 ) )
451		oveq2	⊢ ( 𝑖 = 0s → ( 2s ↑s 𝑖 ) = ( 2s ↑s 0s ) )
452	451	adantl	⊢ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) → ( 2s ↑s 𝑖 ) = ( 2s ↑s 0s ) )
453	452	adantl	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ) → ( 2s ↑s 𝑖 ) = ( 2s ↑s 0s ) )
454	453 54	eqtrdi	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ) → ( 2s ↑s 𝑖 ) = 1s )
455	450 454	breqtrd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ) → ℎ <s 1s )
456	354	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ) → ℎ ∈ ℕ0s )
457		n0lts1e0	⊢ ( ℎ ∈ ℕ0s → ( ℎ <s 1s ↔ ℎ = 0s ) )
458	456 457	syl	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ) → ( ℎ <s 1s ↔ ℎ = 0s ) )
459	455 458	mpbid	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ) → ℎ = 0s )
460	459 454	oveq12d	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ) → ( ℎ /su ( 2s ↑s 𝑖 ) ) = ( 0s /su 1s ) )
461	460 59	eqtrdi	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ) → ( ℎ /su ( 2s ↑s 𝑖 ) ) = 0s )
462	461	oveq2d	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ) → ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) = ( 𝑔 +s 0s ) )
463	288	n0nod	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → 𝑔 ∈ No )
464	463	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ) → 𝑔 ∈ No )
465	464	addsridd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ) → ( 𝑔 +s 0s ) = 𝑔 )
466	449 462 465	3eqtrd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ) → 𝑐 = 𝑔 )
467		simpl2	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → 𝑤 = ( { 𝑐 } |s { 𝑑 } ) )
468	54	oveq2i	⊢ ( 𝑔 /su ( 2s ↑s 0s ) ) = ( 𝑔 /su 1s )
469	463	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → 𝑔 ∈ No )
470	469	divs1d	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( 𝑔 /su 1s ) = 𝑔 )
471		simprr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → 𝑐 = 𝑔 )
472	470 471	eqtr4d	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( 𝑔 /su 1s ) = 𝑐 )
473	468 472	eqtrid	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( 𝑔 /su ( 2s ↑s 0s ) ) = 𝑐 )
474	473	sneqd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → { ( 𝑔 /su ( 2s ↑s 0s ) ) } = { 𝑐 } )
475	54	oveq2i	⊢ ( ( 𝑔 +s 1s ) /su ( 2s ↑s 0s ) ) = ( ( 𝑔 +s 1s ) /su 1s )
476		simpllr	⊢ ( ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) → ( 𝑔 +s 1s ) = 𝑁 )
477	476	adantl	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( 𝑔 +s 1s ) = 𝑁 )
478	288	peano2n0sd	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → ( 𝑔 +s 1s ) ∈ ℕ0s )
479	478	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( 𝑔 +s 1s ) ∈ ℕ0s )
480	479	n0nod	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( 𝑔 +s 1s ) ∈ No )
481	480	divs1d	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( ( 𝑔 +s 1s ) /su 1s ) = ( 𝑔 +s 1s ) )
482		simplll	⊢ ( ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) → 𝑑 = 𝑁 )
483	482	adantl	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → 𝑑 = 𝑁 )
484	477 481 483	3eqtr4d	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( ( 𝑔 +s 1s ) /su 1s ) = 𝑑 )
485	475 484	eqtrid	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( ( 𝑔 +s 1s ) /su ( 2s ↑s 0s ) ) = 𝑑 )
486	485	sneqd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → { ( ( 𝑔 +s 1s ) /su ( 2s ↑s 0s ) ) } = { 𝑑 } )
487	474 486	oveq12d	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( { ( 𝑔 /su ( 2s ↑s 0s ) ) } |s { ( ( 𝑔 +s 1s ) /su ( 2s ↑s 0s ) ) } ) = ( { 𝑐 } |s { 𝑑 } ) )
488	288	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → 𝑔 ∈ ℕ0s )
489	488	n0zsd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → 𝑔 ∈ ℤs )
490	29	a1i	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → 0s ∈ ℕ0s )
491	489 490	pw2cutp1	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( { ( 𝑔 /su ( 2s ↑s 0s ) ) } |s { ( ( 𝑔 +s 1s ) /su ( 2s ↑s 0s ) ) } ) = ( ( ( 2s ·s 𝑔 ) +s 1s ) /su ( 2s ↑s ( 0s +s 1s ) ) ) )
492	467 487 491	3eqtr2d	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → 𝑤 = ( ( ( 2s ·s 𝑔 ) +s 1s ) /su ( 2s ↑s ( 0s +s 1s ) ) ) )
493		mulscl	⊢ ( ( 2s ∈ No ∧ 𝑔 ∈ No ) → ( 2s ·s 𝑔 ) ∈ No )
494	52 469 493	sylancr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( 2s ·s 𝑔 ) ∈ No )
495	322	a1i	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → 1s ∈ No )
496		addslid	⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
497	322 496	ax-mp	⊢ ( 0s +s 1s ) = 1s
498		1n0s	⊢ 1s ∈ ℕ0s
499	497 498	eqeltri	⊢ ( 0s +s 1s ) ∈ ℕ0s
500	499	a1i	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( 0s +s 1s ) ∈ ℕ0s )
501	494 495 500	pw2divsdird	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( ( ( 2s ·s 𝑔 ) +s 1s ) /su ( 2s ↑s ( 0s +s 1s ) ) ) = ( ( ( 2s ·s 𝑔 ) /su ( 2s ↑s ( 0s +s 1s ) ) ) +s ( 1s /su ( 2s ↑s ( 0s +s 1s ) ) ) ) )
502		exps1	⊢ ( 2s ∈ No → ( 2s ↑s 1s ) = 2s )
503	52 502	ax-mp	⊢ ( 2s ↑s 1s ) = 2s
504	503	oveq1i	⊢ ( ( 2s ↑s 1s ) ·s 𝑔 ) = ( 2s ·s 𝑔 )
505	504	oveq1i	⊢ ( ( ( 2s ↑s 1s ) ·s 𝑔 ) /su ( 2s ↑s ( 0s +s 1s ) ) ) = ( ( 2s ·s 𝑔 ) /su ( 2s ↑s ( 0s +s 1s ) ) )
506	498	a1i	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → 1s ∈ ℕ0s )
507	469 490 506	pw2divscan4d	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( 𝑔 /su ( 2s ↑s 0s ) ) = ( ( ( 2s ↑s 1s ) ·s 𝑔 ) /su ( 2s ↑s ( 0s +s 1s ) ) ) )
508	468 507 470	3eqtr3a	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( ( ( 2s ↑s 1s ) ·s 𝑔 ) /su ( 2s ↑s ( 0s +s 1s ) ) ) = 𝑔 )
509	505 508	eqtr3id	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( ( 2s ·s 𝑔 ) /su ( 2s ↑s ( 0s +s 1s ) ) ) = 𝑔 )
510	497	oveq2i	⊢ ( 2s ↑s ( 0s +s 1s ) ) = ( 2s ↑s 1s )
511	510 503	eqtri	⊢ ( 2s ↑s ( 0s +s 1s ) ) = 2s
512	511	oveq2i	⊢ ( 1s /su ( 2s ↑s ( 0s +s 1s ) ) ) = ( 1s /su 2s )
513	512	a1i	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( 1s /su ( 2s ↑s ( 0s +s 1s ) ) ) = ( 1s /su 2s ) )
514	509 513	oveq12d	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( ( ( 2s ·s 𝑔 ) /su ( 2s ↑s ( 0s +s 1s ) ) ) +s ( 1s /su ( 2s ↑s ( 0s +s 1s ) ) ) ) = ( 𝑔 +s ( 1s /su 2s ) ) )
515	501 514	eqtrd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( ( ( 2s ·s 𝑔 ) +s 1s ) /su ( 2s ↑s ( 0s +s 1s ) ) ) = ( 𝑔 +s ( 1s /su 2s ) ) )
516	515	eqeq2d	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( 𝑤 = ( ( ( 2s ·s 𝑔 ) +s 1s ) /su ( 2s ↑s ( 0s +s 1s ) ) ) ↔ 𝑤 = ( 𝑔 +s ( 1s /su 2s ) ) ) )
517	288	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ∧ 𝑤 = ( 𝑔 +s ( 1s /su 2s ) ) ) ) → 𝑔 ∈ ℕ0s )
518	498	a1i	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ∧ 𝑤 = ( 𝑔 +s ( 1s /su 2s ) ) ) ) → 1s ∈ ℕ0s )
519		simprr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ∧ 𝑤 = ( 𝑔 +s ( 1s /su 2s ) ) ) ) → 𝑤 = ( 𝑔 +s ( 1s /su 2s ) ) )
520		ltadds1	⊢ ( ( 0s ∈ No ∧ 1s ∈ No ∧ 1s ∈ No ) → ( 0s <s 1s ↔ ( 0s +s 1s ) <s ( 1s +s 1s ) ) )
521	57 322 322 520	mp3an	⊢ ( 0s <s 1s ↔ ( 0s +s 1s ) <s ( 1s +s 1s ) )
522	36 521	mpbi	⊢ ( 0s +s 1s ) <s ( 1s +s 1s )
523	522 497 325	3brtr3i	⊢ 1s <s 2s
524	523	a1i	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ∧ 𝑤 = ( 𝑔 +s ( 1s /su 2s ) ) ) ) → 1s <s 2s )
525		simp-4r	⊢ ( ( ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ∧ 𝑤 = ( 𝑔 +s ( 1s /su 2s ) ) ) → ( 𝑔 +s 1s ) = 𝑁 )
526	525	adantl	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ∧ 𝑤 = ( 𝑔 +s ( 1s /su 2s ) ) ) ) → ( 𝑔 +s 1s ) = 𝑁 )
527	296	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ∧ 𝑤 = ( 𝑔 +s ( 1s /su 2s ) ) ) ) → 𝜑 )
528	527 32	syl	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ∧ 𝑤 = ( 𝑔 +s ( 1s /su 2s ) ) ) ) → 𝑁 ∈ No )
529	528	ltsp1d	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ∧ 𝑤 = ( 𝑔 +s ( 1s /su 2s ) ) ) ) → 𝑁 <s ( 𝑁 +s 1s ) )
530	526 529	eqbrtrd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ∧ 𝑤 = ( 𝑔 +s ( 1s /su 2s ) ) ) ) → ( 𝑔 +s 1s ) <s ( 𝑁 +s 1s ) )
531		oveq1	⊢ ( 𝑎 = 𝑔 → ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) = ( 𝑔 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) )
532	531	eqeq2d	⊢ ( 𝑎 = 𝑔 → ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ↔ 𝑤 = ( 𝑔 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ) )
533		oveq1	⊢ ( 𝑎 = 𝑔 → ( 𝑎 +s 𝑞 ) = ( 𝑔 +s 𝑞 ) )
534	533	breq1d	⊢ ( 𝑎 = 𝑔 → ( ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ↔ ( 𝑔 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) )
535	532 534	3anbi13d	⊢ ( 𝑎 = 𝑔 → ( ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ( 𝑤 = ( 𝑔 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑔 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
536		oveq1	⊢ ( 𝑏 = 1s → ( 𝑏 /su ( 2s ↑s 𝑞 ) ) = ( 1s /su ( 2s ↑s 𝑞 ) ) )
537	536	oveq2d	⊢ ( 𝑏 = 1s → ( 𝑔 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) = ( 𝑔 +s ( 1s /su ( 2s ↑s 𝑞 ) ) ) )
538	537	eqeq2d	⊢ ( 𝑏 = 1s → ( 𝑤 = ( 𝑔 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ↔ 𝑤 = ( 𝑔 +s ( 1s /su ( 2s ↑s 𝑞 ) ) ) ) )
539		breq1	⊢ ( 𝑏 = 1s → ( 𝑏 <s ( 2s ↑s 𝑞 ) ↔ 1s <s ( 2s ↑s 𝑞 ) ) )
540	538 539	3anbi12d	⊢ ( 𝑏 = 1s → ( ( 𝑤 = ( 𝑔 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑔 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ( 𝑤 = ( 𝑔 +s ( 1s /su ( 2s ↑s 𝑞 ) ) ) ∧ 1s <s ( 2s ↑s 𝑞 ) ∧ ( 𝑔 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
541		oveq2	⊢ ( 𝑞 = 1s → ( 2s ↑s 𝑞 ) = ( 2s ↑s 1s ) )
542	541 503	eqtrdi	⊢ ( 𝑞 = 1s → ( 2s ↑s 𝑞 ) = 2s )
543	542	oveq2d	⊢ ( 𝑞 = 1s → ( 1s /su ( 2s ↑s 𝑞 ) ) = ( 1s /su 2s ) )
544	543	oveq2d	⊢ ( 𝑞 = 1s → ( 𝑔 +s ( 1s /su ( 2s ↑s 𝑞 ) ) ) = ( 𝑔 +s ( 1s /su 2s ) ) )
545	544	eqeq2d	⊢ ( 𝑞 = 1s → ( 𝑤 = ( 𝑔 +s ( 1s /su ( 2s ↑s 𝑞 ) ) ) ↔ 𝑤 = ( 𝑔 +s ( 1s /su 2s ) ) ) )
546	542	breq2d	⊢ ( 𝑞 = 1s → ( 1s <s ( 2s ↑s 𝑞 ) ↔ 1s <s 2s ) )
547		oveq2	⊢ ( 𝑞 = 1s → ( 𝑔 +s 𝑞 ) = ( 𝑔 +s 1s ) )
548	547	breq1d	⊢ ( 𝑞 = 1s → ( ( 𝑔 +s 𝑞 ) <s ( 𝑁 +s 1s ) ↔ ( 𝑔 +s 1s ) <s ( 𝑁 +s 1s ) ) )
549	545 546 548	3anbi123d	⊢ ( 𝑞 = 1s → ( ( 𝑤 = ( 𝑔 +s ( 1s /su ( 2s ↑s 𝑞 ) ) ) ∧ 1s <s ( 2s ↑s 𝑞 ) ∧ ( 𝑔 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ( 𝑤 = ( 𝑔 +s ( 1s /su 2s ) ) ∧ 1s <s 2s ∧ ( 𝑔 +s 1s ) <s ( 𝑁 +s 1s ) ) ) )
550	535 540 549	rspc3ev	⊢ ( ( ( 𝑔 ∈ ℕ0s ∧ 1s ∈ ℕ0s ∧ 1s ∈ ℕ0s ) ∧ ( 𝑤 = ( 𝑔 +s ( 1s /su 2s ) ) ∧ 1s <s 2s ∧ ( 𝑔 +s 1s ) <s ( 𝑁 +s 1s ) ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) )
551	517 518 518 519 524 530 550	syl33anc	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ∧ 𝑤 = ( 𝑔 +s ( 1s /su 2s ) ) ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) )
552	551	expr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( 𝑤 = ( 𝑔 +s ( 1s /su 2s ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
553	516 552	sylbid	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ( 𝑤 = ( ( ( 2s ·s 𝑔 ) +s 1s ) /su ( 2s ↑s ( 0s +s 1s ) ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
554	492 553	mpd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ∧ 𝑐 = 𝑔 ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) )
555	554	expr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ) → ( 𝑐 = 𝑔 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
556	466 555	mpd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ∧ 𝑖 = 0s ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) )
557	556	expr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ) → ( 𝑖 = 0s → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
558	448 557	mpd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑑 = 𝑁 ∧ ( 𝑔 +s 1s ) = 𝑁 ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) )
559	558	expr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ( ( 𝑔 +s 1s ) = 𝑁 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
560	435 559	mpd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ 𝑑 = 𝑁 ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) )
561	560	ex	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → ( 𝑑 = 𝑁 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
562		simprr1	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) )
563		simprr2	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 𝑘 <s ( 2s ↑s 𝑙 ) )
564		simprl3	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 𝑙 ∈ ℕ0s )
565		expscl	⊢ ( ( 2s ∈ No ∧ 𝑙 ∈ ℕ0s ) → ( 2s ↑s 𝑙 ) ∈ No )
566	52 564 565	sylancr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → ( 2s ↑s 𝑙 ) ∈ No )
567	566	mulslidd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → ( 1s ·s ( 2s ↑s 𝑙 ) ) = ( 2s ↑s 𝑙 ) )
568	563 567	breqtrrd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 𝑘 <s ( 1s ·s ( 2s ↑s 𝑙 ) ) )
569		simprl2	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 𝑘 ∈ ℕ0s )
570	569	n0nod	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 𝑘 ∈ No )
571	322	a1i	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 1s ∈ No )
572	570 571 564	pw2ltdivmuls2d	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → ( ( 𝑘 /su ( 2s ↑s 𝑙 ) ) <s 1s ↔ 𝑘 <s ( 1s ·s ( 2s ↑s 𝑙 ) ) ) )
573	568 572	mpbird	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → ( 𝑘 /su ( 2s ↑s 𝑙 ) ) <s 1s )
574	570 564	pw2divscld	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ∈ No )
575		simprl1	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 𝑗 ∈ ℕ0s )
576	575	n0nod	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 𝑗 ∈ No )
577	574 571 576	ltadds2d	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → ( ( 𝑘 /su ( 2s ↑s 𝑙 ) ) <s 1s ↔ ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) <s ( 𝑗 +s 1s ) ) )
578	573 577	mpbid	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) <s ( 𝑗 +s 1s ) )
579	562 578	eqbrtrd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 𝑑 <s ( 𝑗 +s 1s ) )
580	288	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 𝑔 ∈ ℕ0s )
581	580	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ 𝑑 <s ( 𝑗 +s 1s ) ) → 𝑔 ∈ ℕ0s )
582	581	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ 𝑑 <s ( 𝑗 +s 1s ) ) → 𝑔 ∈ No )
583	582	addsridd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ 𝑑 <s ( 𝑗 +s 1s ) ) → ( 𝑔 +s 0s ) = 𝑔 )
584	354	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → ℎ ∈ ℕ0s )
585	584	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ 𝑑 <s ( 𝑗 +s 1s ) ) → ℎ ∈ ℕ0s )
586		n0sge0	⊢ ( ℎ ∈ ℕ0s → 0s ≤s ℎ )
587	585 586	syl	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ 𝑑 <s ( 𝑗 +s 1s ) ) → 0s ≤s ℎ )
588	585	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ 𝑑 <s ( 𝑗 +s 1s ) ) → ℎ ∈ No )
589	291	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 𝑖 ∈ ℕ0s )
590	589	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ 𝑑 <s ( 𝑗 +s 1s ) ) → 𝑖 ∈ ℕ0s )
591	588 590	pw2ge0divsd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ 𝑑 <s ( 𝑗 +s 1s ) ) → ( 0s ≤s ℎ ↔ 0s ≤s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) )
592	57	a1i	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ 𝑑 <s ( 𝑗 +s 1s ) ) → 0s ∈ No )
593	588 590	pw2divscld	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ 𝑑 <s ( 𝑗 +s 1s ) ) → ( ℎ /su ( 2s ↑s 𝑖 ) ) ∈ No )
594	592 593 582	leadds2d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ 𝑑 <s ( 𝑗 +s 1s ) ) → ( 0s ≤s ( ℎ /su ( 2s ↑s 𝑖 ) ) ↔ ( 𝑔 +s 0s ) ≤s ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ) )
595	591 594	bitrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ 𝑑 <s ( 𝑗 +s 1s ) ) → ( 0s ≤s ℎ ↔ ( 𝑔 +s 0s ) ≤s ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ) )
596	587 595	mpbid	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ 𝑑 <s ( 𝑗 +s 1s ) ) → ( 𝑔 +s 0s ) ≤s ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) )
597	583 596	eqbrtrrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ 𝑑 <s ( 𝑗 +s 1s ) ) → 𝑔 ≤s ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) )
598	345	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) )
599	598	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ 𝑑 <s ( 𝑗 +s 1s ) ) → 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) )
600	597 599	breqtrrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ 𝑑 <s ( 𝑗 +s 1s ) ) → 𝑔 ≤s 𝑐 )
601	580	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → 𝑔 ∈ ℕ0s )
602	601	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → 𝑔 ∈ No )
603	337	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 𝑐 ∈ No )
604	603	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → 𝑐 ∈ No )
605	385	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 𝑑 ∈ No )
606	605	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → 𝑑 ∈ No )
607		simprr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → 𝑔 ≤s 𝑐 )
608	241	3ad2ant1	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → 𝑐 <s 𝑑 )
609	608	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 𝑐 <s 𝑑 )
610	609	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → 𝑐 <s 𝑑 )
611	602 604 606 607 610	leltstrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → 𝑔 <s 𝑑 )
612	580	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑑 ) ) → 𝑔 ∈ ℕ0s )
613	612	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑑 ) ) → 𝑔 ∈ No )
614	605	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑑 ) ) → 𝑑 ∈ No )
615	575	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑑 ) ) → 𝑗 ∈ ℕ0s )
616	615	peano2n0sd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑑 ) ) → ( 𝑗 +s 1s ) ∈ ℕ0s )
617	616	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑑 ) ) → ( 𝑗 +s 1s ) ∈ No )
618		simprr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑑 ) ) → 𝑔 <s 𝑑 )
619		simprll	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑑 ) ) → 𝑑 <s ( 𝑗 +s 1s ) )
620	613 614 617 618 619	ltstrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑑 ) ) → 𝑔 <s ( 𝑗 +s 1s ) )
621		n0lesltp1	⊢ ( ( 𝑔 ∈ ℕ0s ∧ 𝑗 ∈ ℕ0s ) → ( 𝑔 ≤s 𝑗 ↔ 𝑔 <s ( 𝑗 +s 1s ) ) )
622	612 615 621	syl2anc	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑑 ) ) → ( 𝑔 ≤s 𝑗 ↔ 𝑔 <s ( 𝑗 +s 1s ) ) )
623	620 622	mpbird	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑑 ) ) → 𝑔 ≤s 𝑗 )
624	623	expr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → ( 𝑔 <s 𝑑 → 𝑔 ≤s 𝑗 ) )
625	611 624	mpd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → 𝑔 ≤s 𝑗 )
626	576	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → 𝑗 ∈ No )
627	602 626	lesloed	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → ( 𝑔 ≤s 𝑗 ↔ ( 𝑔 <s 𝑗 ∨ 𝑔 = 𝑗 ) ) )
628	575	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → 𝑗 ∈ ℕ0s )
629		n0ltsp1le	⊢ ( ( 𝑔 ∈ ℕ0s ∧ 𝑗 ∈ ℕ0s ) → ( 𝑔 <s 𝑗 ↔ ( 𝑔 +s 1s ) ≤s 𝑗 ) )
630	601 628 629	syl2anc	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → ( 𝑔 <s 𝑗 ↔ ( 𝑔 +s 1s ) ≤s 𝑗 ) )
631	630	biimpd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → ( 𝑔 <s 𝑗 → ( 𝑔 +s 1s ) ≤s 𝑗 ) )
632	631	impr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ) → ( 𝑔 +s 1s ) ≤s 𝑗 )
633	478	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → ( 𝑔 +s 1s ) ∈ ℕ0s )
634	633	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → ( 𝑔 +s 1s ) ∈ ℕ0s )
635	634	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → ( 𝑔 +s 1s ) ∈ No )
636	575	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → 𝑗 ∈ ℕ0s )
637	636	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → 𝑗 ∈ No )
638	605	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → 𝑑 ∈ No )
639		simprr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → ( 𝑔 +s 1s ) ≤s 𝑗 )
640	569	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → 𝑘 ∈ ℕ0s )
641		n0sge0	⊢ ( 𝑘 ∈ ℕ0s → 0s ≤s 𝑘 )
642	640 641	syl	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → 0s ≤s 𝑘 )
643	640	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → 𝑘 ∈ No )
644	564	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → 𝑙 ∈ ℕ0s )
645	643 644	pw2ge0divsd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → ( 0s ≤s 𝑘 ↔ 0s ≤s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) )
646	643 644	pw2divscld	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ∈ No )
647	637 646	addsge01d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → ( 0s ≤s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ↔ 𝑗 ≤s ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) )
648	645 647	bitrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → ( 0s ≤s 𝑘 ↔ 𝑗 ≤s ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) )
649	642 648	mpbid	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → 𝑗 ≤s ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) )
650	562	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) )
651	649 650	breqtrrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → 𝑗 ≤s 𝑑 )
652	635 637 638 639 651	lestrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → ( 𝑔 +s 1s ) ≤s 𝑑 )
653	575	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → 𝑗 ∈ ℕ0s )
654	564	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → 𝑙 ∈ ℕ0s )
655		n0addscl	⊢ ( ( 𝑗 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) → ( 𝑗 +s 𝑙 ) ∈ ℕ0s )
656	653 654 655	syl2anc	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → ( 𝑗 +s 𝑙 ) ∈ ℕ0s )
657	656	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → ( 𝑗 +s 𝑙 ) ∈ No )
658	296	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 𝜑 )
659	658	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → 𝜑 )
660	659 32	syl	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → 𝑁 ∈ No )
661	659 317	syl	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → ( 𝑁 +s 1s ) ∈ No )
662		simprr3	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → ( 𝑗 +s 𝑙 ) <s 𝑁 )
663	662	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → ( 𝑗 +s 𝑙 ) <s 𝑁 )
664	660	ltsp1d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → 𝑁 <s ( 𝑁 +s 1s ) )
665	657 660 661 663 664	ltstrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → ( 𝑗 +s 𝑙 ) <s ( 𝑁 +s 1s ) )
666	657 661	ltsnled	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → ( ( 𝑗 +s 𝑙 ) <s ( 𝑁 +s 1s ) ↔ ¬ ( 𝑁 +s 1s ) ≤s ( 𝑗 +s 𝑙 ) ) )
667	665 666	mpbid	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → ¬ ( 𝑁 +s 1s ) ≤s ( 𝑗 +s 𝑙 ) )
668	633	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → ( 𝑔 +s 1s ) ∈ ℕ0s )
669	668	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → ( 𝑔 +s 1s ) ∈ No )
670	605	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → 𝑑 ∈ No )
671	669 670	ltsnled	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → ( ( 𝑔 +s 1s ) <s 𝑑 ↔ ¬ 𝑑 ≤s ( 𝑔 +s 1s ) ) )
672	661	adantrr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( 𝑁 +s 1s ) ∈ No )
673	576	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → 𝑗 ∈ No )
674	657	adantrr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( 𝑗 +s 𝑙 ) ∈ No )
675	633	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( 𝑔 +s 1s ) ∈ ℕ0s )
676	675	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( 𝑔 +s 1s ) ∈ No )
677	333	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) )
678	677	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) )
679		simpll2	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → 𝑤 = ( { 𝑐 } |s { 𝑑 } ) )
680	603	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → 𝑐 ∈ No )
681	598	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) )
682	347	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → ℎ <s ( 2s ↑s 𝑖 ) )
683	682	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ℎ <s ( 2s ↑s 𝑖 ) )
684	589	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → 𝑖 ∈ ℕ0s )
685	52 684 350	sylancr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( 2s ↑s 𝑖 ) ∈ No )
686	685	mulslidd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( 1s ·s ( 2s ↑s 𝑖 ) ) = ( 2s ↑s 𝑖 ) )
687	683 686	breqtrrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ℎ <s ( 1s ·s ( 2s ↑s 𝑖 ) ) )
688	584	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ℎ ∈ ℕ0s )
689	688	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ℎ ∈ No )
690	322	a1i	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → 1s ∈ No )
691	689 690 684	pw2ltdivmuls2d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( ( ℎ /su ( 2s ↑s 𝑖 ) ) <s 1s ↔ ℎ <s ( 1s ·s ( 2s ↑s 𝑖 ) ) ) )
692	687 691	mpbird	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( ℎ /su ( 2s ↑s 𝑖 ) ) <s 1s )
693	689 684	pw2divscld	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( ℎ /su ( 2s ↑s 𝑖 ) ) ∈ No )
694	580	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → 𝑔 ∈ ℕ0s )
695	694	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → 𝑔 ∈ No )
696	693 690 695	ltadds2d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( ( ℎ /su ( 2s ↑s 𝑖 ) ) <s 1s ↔ ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) <s ( 𝑔 +s 1s ) ) )
697	692 696	mpbid	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) <s ( 𝑔 +s 1s ) )
698	681 697	eqbrtrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → 𝑐 <s ( 𝑔 +s 1s ) )
699	680 676 698	sltssn	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → { 𝑐 } <<s { ( 𝑔 +s 1s ) } )
700	605	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → 𝑑 ∈ No )
701		simprr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( 𝑔 +s 1s ) <s 𝑑 )
702	676 700 701	sltssn	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → { ( 𝑔 +s 1s ) } <<s { 𝑑 } )
703	679 676 699 702	sltsbday	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( bday ‘ 𝑤 ) ⊆ ( bday ‘ ( 𝑔 +s 1s ) ) )
704	678 703	eqsstrrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( bday ‘ ( 𝑁 +s 1s ) ) ⊆ ( bday ‘ ( 𝑔 +s 1s ) ) )
705	658	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → 𝜑 )
706	705 395	syl	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( 𝑁 +s 1s ) ∈ Ons )
707		n0on	⊢ ( ( 𝑔 +s 1s ) ∈ ℕ0s → ( 𝑔 +s 1s ) ∈ Ons )
708	675 707	syl	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( 𝑔 +s 1s ) ∈ Ons )
709	706 708	onlesd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( ( 𝑁 +s 1s ) ≤s ( 𝑔 +s 1s ) ↔ ( bday ‘ ( 𝑁 +s 1s ) ) ⊆ ( bday ‘ ( 𝑔 +s 1s ) ) ) )
710	704 709	mpbird	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( 𝑁 +s 1s ) ≤s ( 𝑔 +s 1s ) )
711		simpllr	⊢ ( ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) → ( 𝑔 +s 1s ) ≤s 𝑗 )
712	711	adantl	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( 𝑔 +s 1s ) ≤s 𝑗 )
713	672 676 673 710 712	lestrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( 𝑁 +s 1s ) ≤s 𝑗 )
714	564	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → 𝑙 ∈ ℕ0s )
715		n0sge0	⊢ ( 𝑙 ∈ ℕ0s → 0s ≤s 𝑙 )
716	714 715	syl	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → 0s ≤s 𝑙 )
717	714	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → 𝑙 ∈ No )
718	673 717	addsge01d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( 0s ≤s 𝑙 ↔ 𝑗 ≤s ( 𝑗 +s 𝑙 ) ) )
719	716 718	mpbid	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → 𝑗 ≤s ( 𝑗 +s 𝑙 ) )
720	672 673 674 713 719	lestrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ∧ ( 𝑔 +s 1s ) <s 𝑑 ) ) → ( 𝑁 +s 1s ) ≤s ( 𝑗 +s 𝑙 ) )
721	720	expr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → ( ( 𝑔 +s 1s ) <s 𝑑 → ( 𝑁 +s 1s ) ≤s ( 𝑗 +s 𝑙 ) ) )
722	671 721	sylbird	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → ( ¬ 𝑑 ≤s ( 𝑔 +s 1s ) → ( 𝑁 +s 1s ) ≤s ( 𝑗 +s 𝑙 ) ) )
723	667 722	mt3d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → 𝑑 ≤s ( 𝑔 +s 1s ) )
724		simprr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → ( 𝑔 +s 1s ) ≤s 𝑑 )
725	670 669	lestri3d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → ( 𝑑 = ( 𝑔 +s 1s ) ↔ ( 𝑑 ≤s ( 𝑔 +s 1s ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) )
726	723 724 725	mpbir2and	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → 𝑑 = ( 𝑔 +s 1s ) )
727	682	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → ℎ <s ( 2s ↑s 𝑖 ) )
728	584	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → ℎ ∈ ℕ0s )
729	589	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → 𝑖 ∈ ℕ0s )
730		n0expscl	⊢ ( ( 2s ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) → ( 2s ↑s 𝑖 ) ∈ ℕ0s )
731	412 729 730	sylancr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → ( 2s ↑s 𝑖 ) ∈ ℕ0s )
732		n0ltsp1le	⊢ ( ( ℎ ∈ ℕ0s ∧ ( 2s ↑s 𝑖 ) ∈ ℕ0s ) → ( ℎ <s ( 2s ↑s 𝑖 ) ↔ ( ℎ +s 1s ) ≤s ( 2s ↑s 𝑖 ) ) )
733	728 731 732	syl2anc	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → ( ℎ <s ( 2s ↑s 𝑖 ) ↔ ( ℎ +s 1s ) ≤s ( 2s ↑s 𝑖 ) ) )
734	727 733	mpbid	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → ( ℎ +s 1s ) ≤s ( 2s ↑s 𝑖 ) )
735	354	peano2n0sd	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → ( ℎ +s 1s ) ∈ ℕ0s )
736	735	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → ( ℎ +s 1s ) ∈ ℕ0s )
737	736	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → ( ℎ +s 1s ) ∈ ℕ0s )
738	737	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → ( ℎ +s 1s ) ∈ No )
739	731	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → ( 2s ↑s 𝑖 ) ∈ No )
740	738 739	lesloed	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → ( ( ℎ +s 1s ) ≤s ( 2s ↑s 𝑖 ) ↔ ( ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ∨ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) )
741	658	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → 𝜑 )
742	32 317	ltsnled	⊢ ( 𝜑 → ( 𝑁 <s ( 𝑁 +s 1s ) ↔ ¬ ( 𝑁 +s 1s ) ≤s 𝑁 ) )
743	38 742	mpbid	⊢ ( 𝜑 → ¬ ( 𝑁 +s 1s ) ≤s 𝑁 )
744	741 743	syl	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → ¬ ( 𝑁 +s 1s ) ≤s 𝑁 )
745	677	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) )
746		simpll2	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → 𝑤 = ( { 𝑐 } |s { 𝑑 } ) )
747	580	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → 𝑔 ∈ ℕ0s )
748	747	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → 𝑔 ∈ No )
749	736	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( ℎ +s 1s ) ∈ ℕ0s )
750	749	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( ℎ +s 1s ) ∈ No )
751	589	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → 𝑖 ∈ ℕ0s )
752	750 751	pw2divscld	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) ∈ No )
753	748 752	addscld	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( 𝑔 +s ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) ) ∈ No )
754	603	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → 𝑐 ∈ No )
755	598	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) )
756	584	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ℎ ∈ ℕ0s )
757	756	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ℎ ∈ No )
758	757	ltsp1d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ℎ <s ( ℎ +s 1s ) )
759	757 750 751	pw2ltsdiv1d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( ℎ <s ( ℎ +s 1s ) ↔ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) ) )
760	758 759	mpbid	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) )
761	757 751	pw2divscld	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( ℎ /su ( 2s ↑s 𝑖 ) ) ∈ No )
762	761 752 748	ltadds2d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) ↔ ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) <s ( 𝑔 +s ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) ) ) )
763	760 762	mpbid	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) <s ( 𝑔 +s ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) ) )
764	755 763	eqbrtrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → 𝑐 <s ( 𝑔 +s ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) ) )
765	754 753 764	sltssn	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → { 𝑐 } <<s { ( 𝑔 +s ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) ) } )
766	605	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → 𝑑 ∈ No )
767		simprr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) )
768	52 751 350	sylancr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( 2s ↑s 𝑖 ) ∈ No )
769	768	mulslidd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( 1s ·s ( 2s ↑s 𝑖 ) ) = ( 2s ↑s 𝑖 ) )
770	767 769	breqtrrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( ℎ +s 1s ) <s ( 1s ·s ( 2s ↑s 𝑖 ) ) )
771	322	a1i	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → 1s ∈ No )
772	750 771 751	pw2ltdivmuls2d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) <s 1s ↔ ( ℎ +s 1s ) <s ( 1s ·s ( 2s ↑s 𝑖 ) ) ) )
773	752 771 748	ltadds2d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) <s 1s ↔ ( 𝑔 +s ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) ) <s ( 𝑔 +s 1s ) ) )
774	772 773	bitr3d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( ( ℎ +s 1s ) <s ( 1s ·s ( 2s ↑s 𝑖 ) ) ↔ ( 𝑔 +s ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) ) <s ( 𝑔 +s 1s ) ) )
775	770 774	mpbid	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( 𝑔 +s ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) ) <s ( 𝑔 +s 1s ) )
776		simprlr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → 𝑑 = ( 𝑔 +s 1s ) )
777	775 776	breqtrrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( 𝑔 +s ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) ) <s 𝑑 )
778	753 766 777	sltssn	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → { ( 𝑔 +s ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) ) } <<s { 𝑑 } )
779	746 753 765 778	sltsbday	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( bday ‘ 𝑤 ) ⊆ ( bday ‘ ( 𝑔 +s ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) ) ) )
780	745 779	eqsstrrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( bday ‘ ( 𝑁 +s 1s ) ) ⊆ ( bday ‘ ( 𝑔 +s ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) ) ) )
781	658	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → 𝜑 )
782	781 1	syl	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → 𝑁 ∈ ℕ0s )
783	304	adantr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → ( 𝑔 +s 𝑖 ) <s 𝑁 )
784	783	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( 𝑔 +s 𝑖 ) <s 𝑁 )
785	782 747 749 751 767 784	bdaypw2bnd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( bday ‘ ( 𝑔 +s ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) ) ) ⊆ ( bday ‘ 𝑁 ) )
786	780 785	sstrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( bday ‘ ( 𝑁 +s 1s ) ) ⊆ ( bday ‘ 𝑁 ) )
787	219 395	onltsd	⊢ ( 𝜑 → ( 𝑁 <s ( 𝑁 +s 1s ) ↔ ( bday ‘ 𝑁 ) ∈ ( bday ‘ ( 𝑁 +s 1s ) ) ) )
788	781 787	syl	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( 𝑁 <s ( 𝑁 +s 1s ) ↔ ( bday ‘ 𝑁 ) ∈ ( bday ‘ ( 𝑁 +s 1s ) ) ) )
789	788	notbid	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( ¬ 𝑁 <s ( 𝑁 +s 1s ) ↔ ¬ ( bday ‘ 𝑁 ) ∈ ( bday ‘ ( 𝑁 +s 1s ) ) ) )
790	317 32	lesnltd	⊢ ( 𝜑 → ( ( 𝑁 +s 1s ) ≤s 𝑁 ↔ ¬ 𝑁 <s ( 𝑁 +s 1s ) ) )
791	781 790	syl	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( ( 𝑁 +s 1s ) ≤s 𝑁 ↔ ¬ 𝑁 <s ( 𝑁 +s 1s ) ) )
792		bdayon	⊢ ( bday ‘ ( 𝑁 +s 1s ) ) ∈ On
793		ontri1	⊢ ( ( ( bday ‘ ( 𝑁 +s 1s ) ) ∈ On ∧ ( bday ‘ 𝑁 ) ∈ On ) → ( ( bday ‘ ( 𝑁 +s 1s ) ) ⊆ ( bday ‘ 𝑁 ) ↔ ¬ ( bday ‘ 𝑁 ) ∈ ( bday ‘ ( 𝑁 +s 1s ) ) ) )
794	792 8 793	mp2an	⊢ ( ( bday ‘ ( 𝑁 +s 1s ) ) ⊆ ( bday ‘ 𝑁 ) ↔ ¬ ( bday ‘ 𝑁 ) ∈ ( bday ‘ ( 𝑁 +s 1s ) ) )
795	794	a1i	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( ( bday ‘ ( 𝑁 +s 1s ) ) ⊆ ( bday ‘ 𝑁 ) ↔ ¬ ( bday ‘ 𝑁 ) ∈ ( bday ‘ ( 𝑁 +s 1s ) ) ) )
796	789 791 795	3bitr4d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( ( 𝑁 +s 1s ) ≤s 𝑁 ↔ ( bday ‘ ( 𝑁 +s 1s ) ) ⊆ ( bday ‘ 𝑁 ) ) )
797	786 796	mpbird	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ) ) → ( 𝑁 +s 1s ) ≤s 𝑁 )
798	797	expr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → ( ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) → ( 𝑁 +s 1s ) ≤s 𝑁 ) )
799	744 798	mtod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → ¬ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) )
800		orel1	⊢ ( ¬ ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) → ( ( ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ∨ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) → ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) )
801	799 800	syl	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → ( ( ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ∨ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) → ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) )
802	580	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → 𝑔 ∈ ℕ0s )
803	584	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ℎ ∈ ℕ0s )
804		n0mulscl	⊢ ( ( 2s ∈ ℕ0s ∧ ℎ ∈ ℕ0s ) → ( 2s ·s ℎ ) ∈ ℕ0s )
805	412 803 804	sylancr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 2s ·s ℎ ) ∈ ℕ0s )
806	805	peano2n0sd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( 2s ·s ℎ ) +s 1s ) ∈ ℕ0s )
807	589	peano2n0sd	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → ( 𝑖 +s 1s ) ∈ ℕ0s )
808	807	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 𝑖 +s 1s ) ∈ ℕ0s )
809		simpll2	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → 𝑤 = ( { 𝑐 } |s { 𝑑 } ) )
810	802	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → 𝑔 ∈ No )
811	589	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → 𝑖 ∈ ℕ0s )
812	810 811	pw2divscan3d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) /su ( 2s ↑s 𝑖 ) ) = 𝑔 )
813	812	oveq1d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) /su ( 2s ↑s 𝑖 ) ) +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) )
814	52 589 350	sylancr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → ( 2s ↑s 𝑖 ) ∈ No )
815	814	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 2s ↑s 𝑖 ) ∈ No )
816	815 810	mulscld	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) ∈ No )
817	584	n0nod	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → ℎ ∈ No )
818	817	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ℎ ∈ No )
819	816 818 811	pw2divsdird	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) /su ( 2s ↑s 𝑖 ) ) = ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) /su ( 2s ↑s 𝑖 ) ) +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) )
820	598	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) )
821	813 819 820	3eqtr4rd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → 𝑐 = ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) /su ( 2s ↑s 𝑖 ) ) )
822	821	sneqd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → { 𝑐 } = { ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) /su ( 2s ↑s 𝑖 ) ) } )
823		simprr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) )
824	823 815	eqeltrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ℎ +s 1s ) ∈ No )
825	816 824 811	pw2divsdird	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ( ℎ +s 1s ) ) /su ( 2s ↑s 𝑖 ) ) = ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) /su ( 2s ↑s 𝑖 ) ) +s ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) ) )
826	823	oveq1d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) = ( ( 2s ↑s 𝑖 ) /su ( 2s ↑s 𝑖 ) ) )
827	811	pw2divsidd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( 2s ↑s 𝑖 ) /su ( 2s ↑s 𝑖 ) ) = 1s )
828	826 827	eqtrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) = 1s )
829	812 828	oveq12d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) /su ( 2s ↑s 𝑖 ) ) +s ( ( ℎ +s 1s ) /su ( 2s ↑s 𝑖 ) ) ) = ( 𝑔 +s 1s ) )
830	825 829	eqtrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ( ℎ +s 1s ) ) /su ( 2s ↑s 𝑖 ) ) = ( 𝑔 +s 1s ) )
831	322	a1i	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → 1s ∈ No )
832	816 818 831	addsassd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) +s 1s ) = ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ( ℎ +s 1s ) ) )
833	832	oveq1d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) +s 1s ) /su ( 2s ↑s 𝑖 ) ) = ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ( ℎ +s 1s ) ) /su ( 2s ↑s 𝑖 ) ) )
834		simprlr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → 𝑑 = ( 𝑔 +s 1s ) )
835	830 833 834	3eqtr4rd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → 𝑑 = ( ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) +s 1s ) /su ( 2s ↑s 𝑖 ) ) )
836	835	sneqd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → { 𝑑 } = { ( ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) +s 1s ) /su ( 2s ↑s 𝑖 ) ) } )
837	822 836	oveq12d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( { 𝑐 } |s { 𝑑 } ) = ( { ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) /su ( 2s ↑s 𝑖 ) ) } |s { ( ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) +s 1s ) /su ( 2s ↑s 𝑖 ) ) } ) )
838	412 811 730	sylancr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 2s ↑s 𝑖 ) ∈ ℕ0s )
839		n0mulscl	⊢ ( ( ( 2s ↑s 𝑖 ) ∈ ℕ0s ∧ 𝑔 ∈ ℕ0s ) → ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) ∈ ℕ0s )
840	838 802 839	syl2anc	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) ∈ ℕ0s )
841		n0addscl	⊢ ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) ∈ ℕ0s ∧ ℎ ∈ ℕ0s ) → ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) ∈ ℕ0s )
842	840 803 841	syl2anc	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) ∈ ℕ0s )
843	842	n0zsd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) ∈ ℤs )
844	843 811	pw2cutp1	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( { ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) /su ( 2s ↑s 𝑖 ) ) } |s { ( ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) +s 1s ) /su ( 2s ↑s 𝑖 ) ) } ) = ( ( ( 2s ·s ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) ) +s 1s ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) )
845	52	a1i	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → 2s ∈ No )
846	845 816 818	addsdid	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 2s ·s ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) ) = ( ( 2s ·s ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) ) +s ( 2s ·s ℎ ) ) )
847		expsp1	⊢ ( ( 2s ∈ No ∧ 𝑖 ∈ ℕ0s ) → ( 2s ↑s ( 𝑖 +s 1s ) ) = ( ( 2s ↑s 𝑖 ) ·s 2s ) )
848	52 811 847	sylancr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 2s ↑s ( 𝑖 +s 1s ) ) = ( ( 2s ↑s 𝑖 ) ·s 2s ) )
849	815 845	mulscomd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( 2s ↑s 𝑖 ) ·s 2s ) = ( 2s ·s ( 2s ↑s 𝑖 ) ) )
850	848 849	eqtrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 2s ↑s ( 𝑖 +s 1s ) ) = ( 2s ·s ( 2s ↑s 𝑖 ) ) )
851	850	oveq1d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( 2s ↑s ( 𝑖 +s 1s ) ) ·s 𝑔 ) = ( ( 2s ·s ( 2s ↑s 𝑖 ) ) ·s 𝑔 ) )
852	845 815 810	mulsassd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( 2s ·s ( 2s ↑s 𝑖 ) ) ·s 𝑔 ) = ( 2s ·s ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) ) )
853	851 852	eqtr2d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 2s ·s ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) ) = ( ( 2s ↑s ( 𝑖 +s 1s ) ) ·s 𝑔 ) )
854	853	oveq1d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( 2s ·s ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) ) +s ( 2s ·s ℎ ) ) = ( ( ( 2s ↑s ( 𝑖 +s 1s ) ) ·s 𝑔 ) +s ( 2s ·s ℎ ) ) )
855	846 854	eqtrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 2s ·s ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) ) = ( ( ( 2s ↑s ( 𝑖 +s 1s ) ) ·s 𝑔 ) +s ( 2s ·s ℎ ) ) )
856	855	oveq1d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( 2s ·s ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) ) +s 1s ) = ( ( ( ( 2s ↑s ( 𝑖 +s 1s ) ) ·s 𝑔 ) +s ( 2s ·s ℎ ) ) +s 1s ) )
857	856	oveq1d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ( 2s ·s ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) ) +s 1s ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) = ( ( ( ( ( 2s ↑s ( 𝑖 +s 1s ) ) ·s 𝑔 ) +s ( 2s ·s ℎ ) ) +s 1s ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) )
858	844 857	eqtrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( { ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) /su ( 2s ↑s 𝑖 ) ) } |s { ( ( ( ( ( 2s ↑s 𝑖 ) ·s 𝑔 ) +s ℎ ) +s 1s ) /su ( 2s ↑s 𝑖 ) ) } ) = ( ( ( ( ( 2s ↑s ( 𝑖 +s 1s ) ) ·s 𝑔 ) +s ( 2s ·s ℎ ) ) +s 1s ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) )
859	809 837 858	3eqtrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → 𝑤 = ( ( ( ( ( 2s ↑s ( 𝑖 +s 1s ) ) ·s 𝑔 ) +s ( 2s ·s ℎ ) ) +s 1s ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) )
860		expscl	⊢ ( ( 2s ∈ No ∧ ( 𝑖 +s 1s ) ∈ ℕ0s ) → ( 2s ↑s ( 𝑖 +s 1s ) ) ∈ No )
861	52 808 860	sylancr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 2s ↑s ( 𝑖 +s 1s ) ) ∈ No )
862	861 810	mulscld	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( 2s ↑s ( 𝑖 +s 1s ) ) ·s 𝑔 ) ∈ No )
863	805	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 2s ·s ℎ ) ∈ No )
864	862 863 831	addsassd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ( ( 2s ↑s ( 𝑖 +s 1s ) ) ·s 𝑔 ) +s ( 2s ·s ℎ ) ) +s 1s ) = ( ( ( 2s ↑s ( 𝑖 +s 1s ) ) ·s 𝑔 ) +s ( ( 2s ·s ℎ ) +s 1s ) ) )
865	864	oveq1d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ( ( ( 2s ↑s ( 𝑖 +s 1s ) ) ·s 𝑔 ) +s ( 2s ·s ℎ ) ) +s 1s ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) = ( ( ( ( 2s ↑s ( 𝑖 +s 1s ) ) ·s 𝑔 ) +s ( ( 2s ·s ℎ ) +s 1s ) ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) )
866	806	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( 2s ·s ℎ ) +s 1s ) ∈ No )
867	862 866 808	pw2divsdird	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ( ( 2s ↑s ( 𝑖 +s 1s ) ) ·s 𝑔 ) +s ( ( 2s ·s ℎ ) +s 1s ) ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) = ( ( ( ( 2s ↑s ( 𝑖 +s 1s ) ) ·s 𝑔 ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) +s ( ( ( 2s ·s ℎ ) +s 1s ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) ) )
868	810 808	pw2divscan3d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ( 2s ↑s ( 𝑖 +s 1s ) ) ·s 𝑔 ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) = 𝑔 )
869	868	oveq1d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ( ( 2s ↑s ( 𝑖 +s 1s ) ) ·s 𝑔 ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) +s ( ( ( 2s ·s ℎ ) +s 1s ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) ) = ( 𝑔 +s ( ( ( 2s ·s ℎ ) +s 1s ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) ) )
870	867 869	eqtrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( ( ( 2s ↑s ( 𝑖 +s 1s ) ) ·s 𝑔 ) +s ( ( 2s ·s ℎ ) +s 1s ) ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) = ( 𝑔 +s ( ( ( 2s ·s ℎ ) +s 1s ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) ) )
871	859 865 870	3eqtrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → 𝑤 = ( 𝑔 +s ( ( ( 2s ·s ℎ ) +s 1s ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) ) )
872	831 845 863	ltadds2d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 1s <s 2s ↔ ( ( 2s ·s ℎ ) +s 1s ) <s ( ( 2s ·s ℎ ) +s 2s ) ) )
873	523 872	mpbii	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( 2s ·s ℎ ) +s 1s ) <s ( ( 2s ·s ℎ ) +s 2s ) )
874	823	oveq2d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 2s ·s ( ℎ +s 1s ) ) = ( 2s ·s ( 2s ↑s 𝑖 ) ) )
875	845 818 831	addsdid	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 2s ·s ( ℎ +s 1s ) ) = ( ( 2s ·s ℎ ) +s ( 2s ·s 1s ) ) )
876		mulsrid	⊢ ( 2s ∈ No → ( 2s ·s 1s ) = 2s )
877	52 876	ax-mp	⊢ ( 2s ·s 1s ) = 2s
878	877	oveq2i	⊢ ( ( 2s ·s ℎ ) +s ( 2s ·s 1s ) ) = ( ( 2s ·s ℎ ) +s 2s )
879	875 878	eqtrdi	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 2s ·s ( ℎ +s 1s ) ) = ( ( 2s ·s ℎ ) +s 2s ) )
880	849 874 879	3eqtr2d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( 2s ↑s 𝑖 ) ·s 2s ) = ( ( 2s ·s ℎ ) +s 2s ) )
881	848 880	eqtrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 2s ↑s ( 𝑖 +s 1s ) ) = ( ( 2s ·s ℎ ) +s 2s ) )
882	873 881	breqtrrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( 2s ·s ℎ ) +s 1s ) <s ( 2s ↑s ( 𝑖 +s 1s ) ) )
883	811	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → 𝑖 ∈ No )
884	810 883 831	addsassd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( 𝑔 +s 𝑖 ) +s 1s ) = ( 𝑔 +s ( 𝑖 +s 1s ) ) )
885	783	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 𝑔 +s 𝑖 ) <s 𝑁 )
886	810 883	addscld	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 𝑔 +s 𝑖 ) ∈ No )
887	658	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → 𝜑 )
888	887 32	syl	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → 𝑁 ∈ No )
889	886 888 831	ltadds1d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( 𝑔 +s 𝑖 ) <s 𝑁 ↔ ( ( 𝑔 +s 𝑖 ) +s 1s ) <s ( 𝑁 +s 1s ) ) )
890	885 889	mpbid	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( ( 𝑔 +s 𝑖 ) +s 1s ) <s ( 𝑁 +s 1s ) )
891	884 890	eqbrtrrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ( 𝑔 +s ( 𝑖 +s 1s ) ) <s ( 𝑁 +s 1s ) )
892		oveq1	⊢ ( 𝑏 = ( ( 2s ·s ℎ ) +s 1s ) → ( 𝑏 /su ( 2s ↑s 𝑞 ) ) = ( ( ( 2s ·s ℎ ) +s 1s ) /su ( 2s ↑s 𝑞 ) ) )
893	892	oveq2d	⊢ ( 𝑏 = ( ( 2s ·s ℎ ) +s 1s ) → ( 𝑔 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) = ( 𝑔 +s ( ( ( 2s ·s ℎ ) +s 1s ) /su ( 2s ↑s 𝑞 ) ) ) )
894	893	eqeq2d	⊢ ( 𝑏 = ( ( 2s ·s ℎ ) +s 1s ) → ( 𝑤 = ( 𝑔 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ↔ 𝑤 = ( 𝑔 +s ( ( ( 2s ·s ℎ ) +s 1s ) /su ( 2s ↑s 𝑞 ) ) ) ) )
895		breq1	⊢ ( 𝑏 = ( ( 2s ·s ℎ ) +s 1s ) → ( 𝑏 <s ( 2s ↑s 𝑞 ) ↔ ( ( 2s ·s ℎ ) +s 1s ) <s ( 2s ↑s 𝑞 ) ) )
896	894 895	3anbi12d	⊢ ( 𝑏 = ( ( 2s ·s ℎ ) +s 1s ) → ( ( 𝑤 = ( 𝑔 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑔 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ( 𝑤 = ( 𝑔 +s ( ( ( 2s ·s ℎ ) +s 1s ) /su ( 2s ↑s 𝑞 ) ) ) ∧ ( ( 2s ·s ℎ ) +s 1s ) <s ( 2s ↑s 𝑞 ) ∧ ( 𝑔 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
897		oveq2	⊢ ( 𝑞 = ( 𝑖 +s 1s ) → ( 2s ↑s 𝑞 ) = ( 2s ↑s ( 𝑖 +s 1s ) ) )
898	897	oveq2d	⊢ ( 𝑞 = ( 𝑖 +s 1s ) → ( ( ( 2s ·s ℎ ) +s 1s ) /su ( 2s ↑s 𝑞 ) ) = ( ( ( 2s ·s ℎ ) +s 1s ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) )
899	898	oveq2d	⊢ ( 𝑞 = ( 𝑖 +s 1s ) → ( 𝑔 +s ( ( ( 2s ·s ℎ ) +s 1s ) /su ( 2s ↑s 𝑞 ) ) ) = ( 𝑔 +s ( ( ( 2s ·s ℎ ) +s 1s ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) ) )
900	899	eqeq2d	⊢ ( 𝑞 = ( 𝑖 +s 1s ) → ( 𝑤 = ( 𝑔 +s ( ( ( 2s ·s ℎ ) +s 1s ) /su ( 2s ↑s 𝑞 ) ) ) ↔ 𝑤 = ( 𝑔 +s ( ( ( 2s ·s ℎ ) +s 1s ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) ) ) )
901	897	breq2d	⊢ ( 𝑞 = ( 𝑖 +s 1s ) → ( ( ( 2s ·s ℎ ) +s 1s ) <s ( 2s ↑s 𝑞 ) ↔ ( ( 2s ·s ℎ ) +s 1s ) <s ( 2s ↑s ( 𝑖 +s 1s ) ) ) )
902		oveq2	⊢ ( 𝑞 = ( 𝑖 +s 1s ) → ( 𝑔 +s 𝑞 ) = ( 𝑔 +s ( 𝑖 +s 1s ) ) )
903	902	breq1d	⊢ ( 𝑞 = ( 𝑖 +s 1s ) → ( ( 𝑔 +s 𝑞 ) <s ( 𝑁 +s 1s ) ↔ ( 𝑔 +s ( 𝑖 +s 1s ) ) <s ( 𝑁 +s 1s ) ) )
904	900 901 903	3anbi123d	⊢ ( 𝑞 = ( 𝑖 +s 1s ) → ( ( 𝑤 = ( 𝑔 +s ( ( ( 2s ·s ℎ ) +s 1s ) /su ( 2s ↑s 𝑞 ) ) ) ∧ ( ( 2s ·s ℎ ) +s 1s ) <s ( 2s ↑s 𝑞 ) ∧ ( 𝑔 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ( 𝑤 = ( 𝑔 +s ( ( ( 2s ·s ℎ ) +s 1s ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) ) ∧ ( ( 2s ·s ℎ ) +s 1s ) <s ( 2s ↑s ( 𝑖 +s 1s ) ) ∧ ( 𝑔 +s ( 𝑖 +s 1s ) ) <s ( 𝑁 +s 1s ) ) ) )
905	535 896 904	rspc3ev	⊢ ( ( ( 𝑔 ∈ ℕ0s ∧ ( ( 2s ·s ℎ ) +s 1s ) ∈ ℕ0s ∧ ( 𝑖 +s 1s ) ∈ ℕ0s ) ∧ ( 𝑤 = ( 𝑔 +s ( ( ( 2s ·s ℎ ) +s 1s ) /su ( 2s ↑s ( 𝑖 +s 1s ) ) ) ) ∧ ( ( 2s ·s ℎ ) +s 1s ) <s ( 2s ↑s ( 𝑖 +s 1s ) ) ∧ ( 𝑔 +s ( 𝑖 +s 1s ) ) <s ( 𝑁 +s 1s ) ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) )
906	802 806 808 871 882 891 905	syl33anc	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ∧ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) )
907	906	expr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → ( ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
908	801 907	syld	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → ( ( ( ℎ +s 1s ) <s ( 2s ↑s 𝑖 ) ∨ ( ℎ +s 1s ) = ( 2s ↑s 𝑖 ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
909	740 908	sylbid	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → ( ( ℎ +s 1s ) ≤s ( 2s ↑s 𝑖 ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
910	734 909	mpd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ 𝑑 = ( 𝑔 +s 1s ) ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) )
911	910	expr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → ( 𝑑 = ( 𝑔 +s 1s ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
912	911	adantrr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → ( 𝑑 = ( 𝑔 +s 1s ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
913	726 912	mpd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑑 ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) )
914	913	expr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → ( ( 𝑔 +s 1s ) ≤s 𝑑 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
915	652 914	mpd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ∧ ( 𝑔 +s 1s ) ≤s 𝑗 ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) )
916	915	expr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ) → ( ( 𝑔 +s 1s ) ≤s 𝑗 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
917	632 916	mpd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 <s 𝑗 ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) )
918	917	expr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → ( 𝑔 <s 𝑗 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
919	609	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ) → 𝑐 <s 𝑑 )
920	598	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ) → 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) )
921	562	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ) → 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) )
922		simprr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ) → 𝑔 = 𝑗 )
923	922	oveq1d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ) → ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) )
924	921 923	eqtr4d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ) → 𝑑 = ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) )
925	919 920 924	3brtr3d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ) → ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) <s ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) )
926	817	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ) → ℎ ∈ No )
927	589	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ) → 𝑖 ∈ ℕ0s )
928	926 927	pw2divscld	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ) → ( ℎ /su ( 2s ↑s 𝑖 ) ) ∈ No )
929	570	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ) → 𝑘 ∈ No )
930	564	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ) → 𝑙 ∈ ℕ0s )
931	929 930	pw2divscld	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ) → ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ∈ No )
932	580	n0nod	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → 𝑔 ∈ No )
933	932	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ) → 𝑔 ∈ No )
934	928 931 933	ltadds2d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ) → ( ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ↔ ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) <s ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) )
935	925 934	mpbird	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ) → ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) )
936	584	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ℎ ∈ ℕ0s )
937	564	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → 𝑙 ∈ ℕ0s )
938	589	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → 𝑖 ∈ ℕ0s )
939		n0subs	⊢ ( ( 𝑙 ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) → ( 𝑙 ≤s 𝑖 ↔ ( 𝑖 -s 𝑙 ) ∈ ℕ0s ) )
940	937 938 939	syl2anc	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → ( 𝑙 ≤s 𝑖 ↔ ( 𝑖 -s 𝑙 ) ∈ ℕ0s ) )
941	940	biimpd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → ( 𝑙 ≤s 𝑖 → ( 𝑖 -s 𝑙 ) ∈ ℕ0s ) )
942	941	impr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( 𝑖 -s 𝑙 ) ∈ ℕ0s )
943		n0expscl	⊢ ( ( 2s ∈ ℕ0s ∧ ( 𝑖 -s 𝑙 ) ∈ ℕ0s ) → ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ∈ ℕ0s )
944	412 942 943	sylancr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ∈ ℕ0s )
945	569	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → 𝑘 ∈ ℕ0s )
946		n0mulscl	⊢ ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ) → ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) ∈ ℕ0s )
947	944 945 946	syl2anc	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) ∈ ℕ0s )
948	589	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → 𝑖 ∈ ℕ0s )
949		simprlr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) )
950	945	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → 𝑘 ∈ No )
951	564	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → 𝑙 ∈ ℕ0s )
952	950 951 942	pw2divscan4d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( 𝑘 /su ( 2s ↑s 𝑙 ) ) = ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s ( 𝑙 +s ( 𝑖 -s 𝑙 ) ) ) ) )
953	951	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → 𝑙 ∈ No )
954	942	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( 𝑖 -s 𝑙 ) ∈ No )
955	953 954	addscomd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( 𝑙 +s ( 𝑖 -s 𝑙 ) ) = ( ( 𝑖 -s 𝑙 ) +s 𝑙 ) )
956	948	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → 𝑖 ∈ No )
957		npcans	⊢ ( ( 𝑖 ∈ No ∧ 𝑙 ∈ No ) → ( ( 𝑖 -s 𝑙 ) +s 𝑙 ) = 𝑖 )
958	956 953 957	syl2anc	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( ( 𝑖 -s 𝑙 ) +s 𝑙 ) = 𝑖 )
959	955 958	eqtrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( 𝑙 +s ( 𝑖 -s 𝑙 ) ) = 𝑖 )
960	959	oveq2d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( 2s ↑s ( 𝑙 +s ( 𝑖 -s 𝑙 ) ) ) = ( 2s ↑s 𝑖 ) )
961	960	oveq2d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s ( 𝑙 +s ( 𝑖 -s 𝑙 ) ) ) ) = ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑖 ) ) )
962	952 961	eqtr2d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑖 ) ) = ( 𝑘 /su ( 2s ↑s 𝑙 ) ) )
963	949 962	breqtrrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑖 ) ) )
964	936	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ℎ ∈ No )
965	947	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) ∈ No )
966	964 965 948	pw2ltsdiv1d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( ℎ <s ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) ↔ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑖 ) ) ) )
967	963 966	mpbird	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ℎ <s ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) )
968	682	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ℎ <s ( 2s ↑s 𝑖 ) )
969	563	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → 𝑘 <s ( 2s ↑s 𝑙 ) )
970		n0expscl	⊢ ( ( 2s ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) → ( 2s ↑s 𝑙 ) ∈ ℕ0s )
971	412 951 970	sylancr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( 2s ↑s 𝑙 ) ∈ ℕ0s )
972	971	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( 2s ↑s 𝑙 ) ∈ No )
973	944	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ∈ No )
974		nnsgt0	⊢ ( 2s ∈ ℕs → 0s <s 2s )
975	410 974	ax-mp	⊢ 0s <s 2s
976		expsgt0	⊢ ( ( 2s ∈ No ∧ ( 𝑖 -s 𝑙 ) ∈ ℕ0s ∧ 0s <s 2s ) → 0s <s ( 2s ↑s ( 𝑖 -s 𝑙 ) ) )
977	52 975 976	mp3an13	⊢ ( ( 𝑖 -s 𝑙 ) ∈ ℕ0s → 0s <s ( 2s ↑s ( 𝑖 -s 𝑙 ) ) )
978	942 977	syl	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → 0s <s ( 2s ↑s ( 𝑖 -s 𝑙 ) ) )
979	950 972 973 978	ltmuls2d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( 𝑘 <s ( 2s ↑s 𝑙 ) ↔ ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) <s ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s ( 2s ↑s 𝑙 ) ) ) )
980	969 979	mpbid	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) <s ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s ( 2s ↑s 𝑙 ) ) )
981		expadds	⊢ ( ( 2s ∈ No ∧ ( 𝑖 -s 𝑙 ) ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) → ( 2s ↑s ( ( 𝑖 -s 𝑙 ) +s 𝑙 ) ) = ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s ( 2s ↑s 𝑙 ) ) )
982	52 942 951 981	mp3an2i	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( 2s ↑s ( ( 𝑖 -s 𝑙 ) +s 𝑙 ) ) = ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s ( 2s ↑s 𝑙 ) ) )
983	958	oveq2d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( 2s ↑s ( ( 𝑖 -s 𝑙 ) +s 𝑙 ) ) = ( 2s ↑s 𝑖 ) )
984	982 983	eqtr3d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s ( 2s ↑s 𝑙 ) ) = ( 2s ↑s 𝑖 ) )
985	980 984	breqtrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) <s ( 2s ↑s 𝑖 ) )
986	967 968 985	3jca	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( ℎ <s ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) <s ( 2s ↑s 𝑖 ) ) )
987	598	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) )
988	562	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) )
989		simpllr	⊢ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) → 𝑔 = 𝑗 )
990	989	adantl	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → 𝑔 = 𝑗 )
991	990 962	oveq12d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( 𝑔 +s ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑖 ) ) ) = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) )
992	988 991	eqtr4d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → 𝑑 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑖 ) ) ) )
993	783	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( 𝑔 +s 𝑖 ) <s 𝑁 )
994	987 992 993	3jca	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑖 ) ) ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) )
995		breq1	⊢ ( 𝑚 = ℎ → ( 𝑚 <s 𝑛 ↔ ℎ <s 𝑛 ) )
996		breq1	⊢ ( 𝑚 = ℎ → ( 𝑚 <s ( 2s ↑s 𝑜 ) ↔ ℎ <s ( 2s ↑s 𝑜 ) ) )
997	995 996	3anbi12d	⊢ ( 𝑚 = ℎ → ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ↔ ( ℎ <s 𝑛 ∧ ℎ <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ) )
998		oveq1	⊢ ( 𝑚 = ℎ → ( 𝑚 /su ( 2s ↑s 𝑜 ) ) = ( ℎ /su ( 2s ↑s 𝑜 ) ) )
999	998	oveq2d	⊢ ( 𝑚 = ℎ → ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑜 ) ) ) )
1000	999	eqeq2d	⊢ ( 𝑚 = ℎ → ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ↔ 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑜 ) ) ) ) )
1001	1000	3anbi1d	⊢ ( 𝑚 = ℎ → ( ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ↔ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) )
1002	997 1001	anbi12d	⊢ ( 𝑚 = ℎ → ( ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ↔ ( ( ℎ <s 𝑛 ∧ ℎ <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) )
1003		breq2	⊢ ( 𝑛 = ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) → ( ℎ <s 𝑛 ↔ ℎ <s ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) ) )
1004		breq1	⊢ ( 𝑛 = ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) → ( 𝑛 <s ( 2s ↑s 𝑜 ) ↔ ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) <s ( 2s ↑s 𝑜 ) ) )
1005	1003 1004	3anbi13d	⊢ ( 𝑛 = ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) → ( ( ℎ <s 𝑛 ∧ ℎ <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ↔ ( ℎ <s ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) ∧ ℎ <s ( 2s ↑s 𝑜 ) ∧ ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) <s ( 2s ↑s 𝑜 ) ) ) )
1006		oveq1	⊢ ( 𝑛 = ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) → ( 𝑛 /su ( 2s ↑s 𝑜 ) ) = ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑜 ) ) )
1007	1006	oveq2d	⊢ ( 𝑛 = ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) → ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑜 ) ) ) )
1008	1007	eqeq2d	⊢ ( 𝑛 = ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) → ( 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ↔ 𝑑 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑜 ) ) ) ) )
1009	1008	3anbi2d	⊢ ( 𝑛 = ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) → ( ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ↔ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) )
1010	1005 1009	anbi12d	⊢ ( 𝑛 = ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) → ( ( ( ℎ <s 𝑛 ∧ ℎ <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ↔ ( ( ℎ <s ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) ∧ ℎ <s ( 2s ↑s 𝑜 ) ∧ ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) )
1011		oveq2	⊢ ( 𝑜 = 𝑖 → ( 2s ↑s 𝑜 ) = ( 2s ↑s 𝑖 ) )
1012	1011	breq2d	⊢ ( 𝑜 = 𝑖 → ( ℎ <s ( 2s ↑s 𝑜 ) ↔ ℎ <s ( 2s ↑s 𝑖 ) ) )
1013	1011	breq2d	⊢ ( 𝑜 = 𝑖 → ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) <s ( 2s ↑s 𝑜 ) ↔ ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) <s ( 2s ↑s 𝑖 ) ) )
1014	1012 1013	3anbi23d	⊢ ( 𝑜 = 𝑖 → ( ( ℎ <s ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) ∧ ℎ <s ( 2s ↑s 𝑜 ) ∧ ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) <s ( 2s ↑s 𝑜 ) ) ↔ ( ℎ <s ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) <s ( 2s ↑s 𝑖 ) ) ) )
1015	1011	oveq2d	⊢ ( 𝑜 = 𝑖 → ( ℎ /su ( 2s ↑s 𝑜 ) ) = ( ℎ /su ( 2s ↑s 𝑖 ) ) )
1016	1015	oveq2d	⊢ ( 𝑜 = 𝑖 → ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑜 ) ) ) = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) )
1017	1016	eqeq2d	⊢ ( 𝑜 = 𝑖 → ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑜 ) ) ) ↔ 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ) )
1018	1011	oveq2d	⊢ ( 𝑜 = 𝑖 → ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑜 ) ) = ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑖 ) ) )
1019	1018	oveq2d	⊢ ( 𝑜 = 𝑖 → ( 𝑔 +s ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑜 ) ) ) = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑖 ) ) ) )
1020	1019	eqeq2d	⊢ ( 𝑜 = 𝑖 → ( 𝑑 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑜 ) ) ) ↔ 𝑑 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑖 ) ) ) ) )
1021		oveq2	⊢ ( 𝑜 = 𝑖 → ( 𝑔 +s 𝑜 ) = ( 𝑔 +s 𝑖 ) )
1022	1021	breq1d	⊢ ( 𝑜 = 𝑖 → ( ( 𝑔 +s 𝑜 ) <s 𝑁 ↔ ( 𝑔 +s 𝑖 ) <s 𝑁 ) )
1023	1017 1020 1022	3anbi123d	⊢ ( 𝑜 = 𝑖 → ( ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ↔ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑖 ) ) ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) )
1024	1014 1023	anbi12d	⊢ ( 𝑜 = 𝑖 → ( ( ( ℎ <s ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) ∧ ℎ <s ( 2s ↑s 𝑜 ) ∧ ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ↔ ( ( ℎ <s ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) <s ( 2s ↑s 𝑖 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑖 ) ) ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) )
1025	1002 1010 1024	rspc3ev	⊢ ( ( ( ℎ ∈ ℕ0s ∧ ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( ( ℎ <s ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) <s ( 2s ↑s 𝑖 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑖 -s 𝑙 ) ) ·s 𝑘 ) /su ( 2s ↑s 𝑖 ) ) ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → ∃ 𝑚 ∈ ℕ0s ∃ 𝑛 ∈ ℕ0s ∃ 𝑜 ∈ ℕ0s ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) )
1026	936 947 948 986 994 1025	syl32anc	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑙 ≤s 𝑖 ) ) → ∃ 𝑚 ∈ ℕ0s ∃ 𝑛 ∈ ℕ0s ∃ 𝑜 ∈ ℕ0s ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) )
1027	1026	expr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → ( 𝑙 ≤s 𝑖 → ∃ 𝑚 ∈ ℕ0s ∃ 𝑛 ∈ ℕ0s ∃ 𝑜 ∈ ℕ0s ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) )
1028		n0subs	⊢ ( ( 𝑖 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) → ( 𝑖 ≤s 𝑙 ↔ ( 𝑙 -s 𝑖 ) ∈ ℕ0s ) )
1029	938 937 1028	syl2anc	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → ( 𝑖 ≤s 𝑙 ↔ ( 𝑙 -s 𝑖 ) ∈ ℕ0s ) )
1030	1029	biimpd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → ( 𝑖 ≤s 𝑙 → ( 𝑙 -s 𝑖 ) ∈ ℕ0s ) )
1031	1030	impr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( 𝑙 -s 𝑖 ) ∈ ℕ0s )
1032		n0expscl	⊢ ( ( 2s ∈ ℕ0s ∧ ( 𝑙 -s 𝑖 ) ∈ ℕ0s ) → ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ∈ ℕ0s )
1033	412 1031 1032	sylancr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ∈ ℕ0s )
1034	584	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ℎ ∈ ℕ0s )
1035		n0mulscl	⊢ ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ∈ ℕ0s ∧ ℎ ∈ ℕ0s ) → ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) ∈ ℕ0s )
1036	1033 1034 1035	syl2anc	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) ∈ ℕ0s )
1037	569	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → 𝑘 ∈ ℕ0s )
1038	564	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → 𝑙 ∈ ℕ0s )
1039	1034	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ℎ ∈ No )
1040	589	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → 𝑖 ∈ ℕ0s )
1041	1039 1040 1031	pw2divscan4d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( ℎ /su ( 2s ↑s 𝑖 ) ) = ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s ( 𝑖 +s ( 𝑙 -s 𝑖 ) ) ) ) )
1042	1040	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → 𝑖 ∈ No )
1043	1031	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( 𝑙 -s 𝑖 ) ∈ No )
1044	1042 1043	addscomd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( 𝑖 +s ( 𝑙 -s 𝑖 ) ) = ( ( 𝑙 -s 𝑖 ) +s 𝑖 ) )
1045	1044	oveq2d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( 2s ↑s ( 𝑖 +s ( 𝑙 -s 𝑖 ) ) ) = ( 2s ↑s ( ( 𝑙 -s 𝑖 ) +s 𝑖 ) ) )
1046	1045	oveq2d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s ( 𝑖 +s ( 𝑙 -s 𝑖 ) ) ) ) = ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s ( ( 𝑙 -s 𝑖 ) +s 𝑖 ) ) ) )
1047	1038	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → 𝑙 ∈ No )
1048		npcans	⊢ ( ( 𝑙 ∈ No ∧ 𝑖 ∈ No ) → ( ( 𝑙 -s 𝑖 ) +s 𝑖 ) = 𝑙 )
1049	1047 1042 1048	syl2anc	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( ( 𝑙 -s 𝑖 ) +s 𝑖 ) = 𝑙 )
1050	1049	oveq2d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( 2s ↑s ( ( 𝑙 -s 𝑖 ) +s 𝑖 ) ) = ( 2s ↑s 𝑙 ) )
1051	1050	oveq2d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s ( ( 𝑙 -s 𝑖 ) +s 𝑖 ) ) ) = ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑙 ) ) )
1052	1041 1046 1051	3eqtrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( ℎ /su ( 2s ↑s 𝑖 ) ) = ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑙 ) ) )
1053		simprlr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) )
1054	1052 1053	eqbrtrrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑙 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) )
1055		expscl	⊢ ( ( 2s ∈ No ∧ ( 𝑙 -s 𝑖 ) ∈ ℕ0s ) → ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ∈ No )
1056	52 1031 1055	sylancr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ∈ No )
1057	1056 1039	mulscld	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) ∈ No )
1058	1037	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → 𝑘 ∈ No )
1059	1057 1058 1038	pw2ltsdiv1d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s 𝑘 ↔ ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑙 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) )
1060	1054 1059	mpbird	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s 𝑘 )
1061	682	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ℎ <s ( 2s ↑s 𝑖 ) )
1062	52 1040 350	sylancr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( 2s ↑s 𝑖 ) ∈ No )
1063		expsgt0	⊢ ( ( 2s ∈ No ∧ ( 𝑙 -s 𝑖 ) ∈ ℕ0s ∧ 0s <s 2s ) → 0s <s ( 2s ↑s ( 𝑙 -s 𝑖 ) ) )
1064	52 975 1063	mp3an13	⊢ ( ( 𝑙 -s 𝑖 ) ∈ ℕ0s → 0s <s ( 2s ↑s ( 𝑙 -s 𝑖 ) ) )
1065	1031 1064	syl	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → 0s <s ( 2s ↑s ( 𝑙 -s 𝑖 ) ) )
1066	1039 1062 1056 1065	ltmuls2d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( ℎ <s ( 2s ↑s 𝑖 ) ↔ ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ( 2s ↑s 𝑖 ) ) ) )
1067	1061 1066	mpbid	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ( 2s ↑s 𝑖 ) ) )
1068		expadds	⊢ ( ( 2s ∈ No ∧ ( 𝑙 -s 𝑖 ) ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) → ( 2s ↑s ( ( 𝑙 -s 𝑖 ) +s 𝑖 ) ) = ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ( 2s ↑s 𝑖 ) ) )
1069	52 1031 1040 1068	mp3an2i	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( 2s ↑s ( ( 𝑙 -s 𝑖 ) +s 𝑖 ) ) = ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ( 2s ↑s 𝑖 ) ) )
1070	1069 1050	eqtr3d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ( 2s ↑s 𝑖 ) ) = ( 2s ↑s 𝑙 ) )
1071	1067 1070	breqtrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s ( 2s ↑s 𝑙 ) )
1072	563	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → 𝑘 <s ( 2s ↑s 𝑙 ) )
1073	1060 1071 1072	3jca	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s 𝑘 ∧ ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s ( 2s ↑s 𝑙 ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ) )
1074	598	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) )
1075	1052	oveq2d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑙 ) ) ) )
1076	1074 1075	eqtrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → 𝑐 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑙 ) ) ) )
1077	562	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) )
1078		simpllr	⊢ ( ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) → 𝑔 = 𝑗 )
1079	1078	adantl	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → 𝑔 = 𝑗 )
1080	1079	oveq1d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) )
1081	1077 1080	eqtr4d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → 𝑑 = ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) )
1082	1079	oveq1d	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( 𝑔 +s 𝑙 ) = ( 𝑗 +s 𝑙 ) )
1083	662	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( 𝑗 +s 𝑙 ) <s 𝑁 )
1084	1082 1083	eqbrtrd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( 𝑔 +s 𝑙 ) <s 𝑁 )
1085	1076 1081 1084	3jca	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ( 𝑐 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ ( 𝑔 +s 𝑙 ) <s 𝑁 ) )
1086		breq1	⊢ ( 𝑚 = ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) → ( 𝑚 <s 𝑛 ↔ ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s 𝑛 ) )
1087		breq1	⊢ ( 𝑚 = ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) → ( 𝑚 <s ( 2s ↑s 𝑜 ) ↔ ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s ( 2s ↑s 𝑜 ) ) )
1088	1086 1087	3anbi12d	⊢ ( 𝑚 = ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) → ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ↔ ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s 𝑛 ∧ ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ) )
1089		oveq1	⊢ ( 𝑚 = ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) → ( 𝑚 /su ( 2s ↑s 𝑜 ) ) = ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑜 ) ) )
1090	1089	oveq2d	⊢ ( 𝑚 = ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) → ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑜 ) ) ) )
1091	1090	eqeq2d	⊢ ( 𝑚 = ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) → ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ↔ 𝑐 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑜 ) ) ) ) )
1092	1091	3anbi1d	⊢ ( 𝑚 = ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) → ( ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ↔ ( 𝑐 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) )
1093	1088 1092	anbi12d	⊢ ( 𝑚 = ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) → ( ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ↔ ( ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s 𝑛 ∧ ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) )
1094		breq2	⊢ ( 𝑛 = 𝑘 → ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s 𝑛 ↔ ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s 𝑘 ) )
1095		breq1	⊢ ( 𝑛 = 𝑘 → ( 𝑛 <s ( 2s ↑s 𝑜 ) ↔ 𝑘 <s ( 2s ↑s 𝑜 ) ) )
1096	1094 1095	3anbi13d	⊢ ( 𝑛 = 𝑘 → ( ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s 𝑛 ∧ ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ↔ ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s 𝑘 ∧ ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s ( 2s ↑s 𝑜 ) ∧ 𝑘 <s ( 2s ↑s 𝑜 ) ) ) )
1097		oveq1	⊢ ( 𝑛 = 𝑘 → ( 𝑛 /su ( 2s ↑s 𝑜 ) ) = ( 𝑘 /su ( 2s ↑s 𝑜 ) ) )
1098	1097	oveq2d	⊢ ( 𝑛 = 𝑘 → ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) = ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑜 ) ) ) )
1099	1098	eqeq2d	⊢ ( 𝑛 = 𝑘 → ( 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ↔ 𝑑 = ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑜 ) ) ) ) )
1100	1099	3anbi2d	⊢ ( 𝑛 = 𝑘 → ( ( 𝑐 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ↔ ( 𝑐 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) )
1101	1096 1100	anbi12d	⊢ ( 𝑛 = 𝑘 → ( ( ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s 𝑛 ∧ ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ↔ ( ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s 𝑘 ∧ ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s ( 2s ↑s 𝑜 ) ∧ 𝑘 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) )
1102		oveq2	⊢ ( 𝑜 = 𝑙 → ( 2s ↑s 𝑜 ) = ( 2s ↑s 𝑙 ) )
1103	1102	breq2d	⊢ ( 𝑜 = 𝑙 → ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s ( 2s ↑s 𝑜 ) ↔ ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s ( 2s ↑s 𝑙 ) ) )
1104	1102	breq2d	⊢ ( 𝑜 = 𝑙 → ( 𝑘 <s ( 2s ↑s 𝑜 ) ↔ 𝑘 <s ( 2s ↑s 𝑙 ) ) )
1105	1103 1104	3anbi23d	⊢ ( 𝑜 = 𝑙 → ( ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s 𝑘 ∧ ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s ( 2s ↑s 𝑜 ) ∧ 𝑘 <s ( 2s ↑s 𝑜 ) ) ↔ ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s 𝑘 ∧ ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s ( 2s ↑s 𝑙 ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ) ) )
1106	1102	oveq2d	⊢ ( 𝑜 = 𝑙 → ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑜 ) ) = ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑙 ) ) )
1107	1106	oveq2d	⊢ ( 𝑜 = 𝑙 → ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑜 ) ) ) = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑙 ) ) ) )
1108	1107	eqeq2d	⊢ ( 𝑜 = 𝑙 → ( 𝑐 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑜 ) ) ) ↔ 𝑐 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑙 ) ) ) ) )
1109	1102	oveq2d	⊢ ( 𝑜 = 𝑙 → ( 𝑘 /su ( 2s ↑s 𝑜 ) ) = ( 𝑘 /su ( 2s ↑s 𝑙 ) ) )
1110	1109	oveq2d	⊢ ( 𝑜 = 𝑙 → ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑜 ) ) ) = ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) )
1111	1110	eqeq2d	⊢ ( 𝑜 = 𝑙 → ( 𝑑 = ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑜 ) ) ) ↔ 𝑑 = ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) )
1112		oveq2	⊢ ( 𝑜 = 𝑙 → ( 𝑔 +s 𝑜 ) = ( 𝑔 +s 𝑙 ) )
1113	1112	breq1d	⊢ ( 𝑜 = 𝑙 → ( ( 𝑔 +s 𝑜 ) <s 𝑁 ↔ ( 𝑔 +s 𝑙 ) <s 𝑁 ) )
1114	1108 1111 1113	3anbi123d	⊢ ( 𝑜 = 𝑙 → ( ( 𝑐 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ↔ ( 𝑐 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ ( 𝑔 +s 𝑙 ) <s 𝑁 ) ) )
1115	1105 1114	anbi12d	⊢ ( 𝑜 = 𝑙 → ( ( ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s 𝑘 ∧ ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s ( 2s ↑s 𝑜 ) ∧ 𝑘 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ↔ ( ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s 𝑘 ∧ ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s ( 2s ↑s 𝑙 ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ ( 𝑔 +s 𝑙 ) <s 𝑁 ) ) ) )
1116	1093 1101 1115	rspc3ev	⊢ ( ( ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s 𝑘 ∧ ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) <s ( 2s ↑s 𝑙 ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( ( ( 2s ↑s ( 𝑙 -s 𝑖 ) ) ·s ℎ ) /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ ( 𝑔 +s 𝑙 ) <s 𝑁 ) ) ) → ∃ 𝑚 ∈ ℕ0s ∃ 𝑛 ∈ ℕ0s ∃ 𝑜 ∈ ℕ0s ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) )
1117	1036 1037 1038 1073 1085 1116	syl32anc	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑖 ≤s 𝑙 ) ) → ∃ 𝑚 ∈ ℕ0s ∃ 𝑛 ∈ ℕ0s ∃ 𝑜 ∈ ℕ0s ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) )
1118	1117	expr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → ( 𝑖 ≤s 𝑙 → ∃ 𝑚 ∈ ℕ0s ∃ 𝑛 ∈ ℕ0s ∃ 𝑜 ∈ ℕ0s ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) )
1119	937	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → 𝑙 ∈ No )
1120	938	n0nod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → 𝑖 ∈ No )
1121		lestric	⊢ ( ( 𝑙 ∈ No ∧ 𝑖 ∈ No ) → ( 𝑙 ≤s 𝑖 ∨ 𝑖 ≤s 𝑙 ) )
1122	1119 1120 1121	syl2anc	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → ( 𝑙 ≤s 𝑖 ∨ 𝑖 ≤s 𝑙 ) )
1123	1027 1118 1122	mpjaod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → ∃ 𝑚 ∈ ℕ0s ∃ 𝑛 ∈ ℕ0s ∃ 𝑜 ∈ ℕ0s ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) )
1124	580	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → 𝑔 ∈ ℕ0s )
1125	1124	adantr	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑔 ∈ ℕ0s )
1126		simprl1	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑚 ∈ ℕ0s )
1127		n0mulscl	⊢ ( ( 2s ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s ) → ( 2s ·s 𝑚 ) ∈ ℕ0s )
1128	412 1126 1127	sylancr	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 2s ·s 𝑚 ) ∈ ℕ0s )
1129	1128	peano2n0sd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 2s ·s 𝑚 ) +s 1s ) ∈ ℕ0s )
1130		simprl3	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑜 ∈ ℕ0s )
1131	1130	peano2n0sd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑜 +s 1s ) ∈ ℕ0s )
1132		simpll2	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → 𝑤 = ( { 𝑐 } |s { 𝑑 } ) )
1133	1132	adantr	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑤 = ( { 𝑐 } |s { 𝑑 } ) )
1134	1125	n0nod	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑔 ∈ No )
1135	1134 1130	pw2divscan3d	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) /su ( 2s ↑s 𝑜 ) ) = 𝑔 )
1136	1135	oveq1d	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) /su ( 2s ↑s 𝑜 ) ) +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) )
1137		n0expscl	⊢ ( ( 2s ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) → ( 2s ↑s 𝑜 ) ∈ ℕ0s )
1138	412 1130 1137	sylancr	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 2s ↑s 𝑜 ) ∈ ℕ0s )
1139		n0mulscl	⊢ ( ( ( 2s ↑s 𝑜 ) ∈ ℕ0s ∧ 𝑔 ∈ ℕ0s ) → ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) ∈ ℕ0s )
1140	1138 1125 1139	syl2anc	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) ∈ ℕ0s )
1141	1140	n0nod	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) ∈ No )
1142	1126	n0nod	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑚 ∈ No )
1143	1141 1142 1130	pw2divsdird	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) /su ( 2s ↑s 𝑜 ) ) = ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) /su ( 2s ↑s 𝑜 ) ) +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) )
1144		simprr1	⊢ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) → 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) )
1145	1144	adantl	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) )
1146	1136 1143 1145	3eqtr4rd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑐 = ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) /su ( 2s ↑s 𝑜 ) ) )
1147	1146	sneqd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → { 𝑐 } = { ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) /su ( 2s ↑s 𝑜 ) ) } )
1148	1126	peano2n0sd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑚 +s 1s ) ∈ ℕ0s )
1149	1148	n0nod	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑚 +s 1s ) ∈ No )
1150	1141 1149 1130	pw2divsdird	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s ( 𝑚 +s 1s ) ) /su ( 2s ↑s 𝑜 ) ) = ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) /su ( 2s ↑s 𝑜 ) ) +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) )
1151	1135	oveq1d	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) /su ( 2s ↑s 𝑜 ) ) +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) = ( 𝑔 +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) )
1152	1150 1151	eqtr2d	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑔 +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) = ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s ( 𝑚 +s 1s ) ) /su ( 2s ↑s 𝑜 ) ) )
1153		simprr2	⊢ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) → 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) )
1154	1153	adantl	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) )
1155	658	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → 𝜑 )
1156	1155	adantr	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝜑 )
1157	1156 743	syl	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ¬ ( 𝑁 +s 1s ) ≤s 𝑁 )
1158	322	a1i	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 1s ∈ No )
1159		simprl2	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑛 ∈ ℕ0s )
1160	1159	n0nod	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑛 ∈ No )
1161	1160 1142	subscld	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑛 -s 𝑚 ) ∈ No )
1162	1158 1161	ltsnled	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 1s <s ( 𝑛 -s 𝑚 ) ↔ ¬ ( 𝑛 -s 𝑚 ) ≤s 1s ) )
1163	677	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) )
1164	1163	adantr	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) )
1165	1132	adantr	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → 𝑤 = ( { 𝑐 } |s { 𝑑 } ) )
1166	1124	adantr	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → 𝑔 ∈ ℕ0s )
1167	1166	n0nod	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → 𝑔 ∈ No )
1168	1126	adantrr	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → 𝑚 ∈ ℕ0s )
1169	1168	peano2n0sd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( 𝑚 +s 1s ) ∈ ℕ0s )
1170	1169	n0nod	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( 𝑚 +s 1s ) ∈ No )
1171	1130	adantrr	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → 𝑜 ∈ ℕ0s )
1172	1170 1171	pw2divscld	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ∈ No )
1173	1167 1172	addscld	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( 𝑔 +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) ∈ No )
1174	603	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → 𝑐 ∈ No )
1175	1174	adantr	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → 𝑐 ∈ No )
1176	1144	ad2antrl	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) )
1177	1142	adantrr	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → 𝑚 ∈ No )
1178	1177	ltsp1d	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → 𝑚 <s ( 𝑚 +s 1s ) )
1179	1177 1170 1171	pw2ltsdiv1d	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( 𝑚 <s ( 𝑚 +s 1s ) ↔ ( 𝑚 /su ( 2s ↑s 𝑜 ) ) <s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) )
1180	1177 1171	pw2divscld	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ∈ No )
1181	1180 1172 1167	ltadds2d	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( ( 𝑚 /su ( 2s ↑s 𝑜 ) ) <s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ↔ ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) <s ( 𝑔 +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) ) )
1182	1179 1181	bitrd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( 𝑚 <s ( 𝑚 +s 1s ) ↔ ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) <s ( 𝑔 +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) ) )
1183	1178 1182	mpbid	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) <s ( 𝑔 +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) )
1184	1176 1183	eqbrtrd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → 𝑐 <s ( 𝑔 +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) )
1185	1175 1173 1184	sltssn	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → { 𝑐 } <<s { ( 𝑔 +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) } )
1186	605	adantr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → 𝑑 ∈ No )
1187	1186	adantr	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → 𝑑 ∈ No )
1188	1142 1158 1160	ltaddsubs2d	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 𝑚 +s 1s ) <s 𝑛 ↔ 1s <s ( 𝑛 -s 𝑚 ) ) )
1189	1188	biimprd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 1s <s ( 𝑛 -s 𝑚 ) → ( 𝑚 +s 1s ) <s 𝑛 ) )
1190	1189	impr	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( 𝑚 +s 1s ) <s 𝑛 )
1191	1159	adantrr	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → 𝑛 ∈ ℕ0s )
1192	1191	n0nod	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → 𝑛 ∈ No )
1193	1170 1192 1171	pw2ltsdiv1d	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( ( 𝑚 +s 1s ) <s 𝑛 ↔ ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) <s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) )
1194	1192 1171	pw2divscld	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ∈ No )
1195	1172 1194 1167	ltadds2d	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) <s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ↔ ( 𝑔 +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) <s ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ) )
1196	1193 1195	bitrd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( ( 𝑚 +s 1s ) <s 𝑛 ↔ ( 𝑔 +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) <s ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ) )
1197	1190 1196	mpbid	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( 𝑔 +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) <s ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) )
1198	1153	ad2antrl	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) )
1199	1197 1198	breqtrrd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( 𝑔 +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) <s 𝑑 )
1200	1173 1187 1199	sltssn	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → { ( 𝑔 +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) } <<s { 𝑑 } )
1201	1165 1173 1185 1200	sltsbday	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( bday ‘ 𝑤 ) ⊆ ( bday ‘ ( 𝑔 +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) ) )
1202	1164 1201	eqsstrrd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( bday ‘ ( 𝑁 +s 1s ) ) ⊆ ( bday ‘ ( 𝑔 +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) ) )
1203	1155	adantr	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → 𝜑 )
1204	1203 1	syl	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → 𝑁 ∈ ℕ0s )
1205		expscl	⊢ ( ( 2s ∈ No ∧ 𝑜 ∈ ℕ0s ) → ( 2s ↑s 𝑜 ) ∈ No )
1206	52 1130 1205	sylancr	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 2s ↑s 𝑜 ) ∈ No )
1207	1206	adantrr	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( 2s ↑s 𝑜 ) ∈ No )
1208		simprl1	⊢ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) → 𝑚 <s 𝑛 )
1209	1208	ad2antrl	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → 𝑚 <s 𝑛 )
1210		n0ltsp1le	⊢ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ) → ( 𝑚 <s 𝑛 ↔ ( 𝑚 +s 1s ) ≤s 𝑛 ) )
1211	1168 1191 1210	syl2anc	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( 𝑚 <s 𝑛 ↔ ( 𝑚 +s 1s ) ≤s 𝑛 ) )
1212	1209 1211	mpbid	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( 𝑚 +s 1s ) ≤s 𝑛 )
1213		simprl3	⊢ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) → 𝑛 <s ( 2s ↑s 𝑜 ) )
1214	1213	ad2antrl	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → 𝑛 <s ( 2s ↑s 𝑜 ) )
1215	1170 1192 1207 1212 1214	leltstrd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( 𝑚 +s 1s ) <s ( 2s ↑s 𝑜 ) )
1216		simprr3	⊢ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) → ( 𝑔 +s 𝑜 ) <s 𝑁 )
1217	1216	ad2antrl	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( 𝑔 +s 𝑜 ) <s 𝑁 )
1218	1204 1166 1169 1171 1215 1217	bdaypw2bnd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( bday ‘ ( 𝑔 +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) ) ⊆ ( bday ‘ 𝑁 ) )
1219	1202 1218	sstrd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( bday ‘ ( 𝑁 +s 1s ) ) ⊆ ( bday ‘ 𝑁 ) )
1220	395 219	onlesd	⊢ ( 𝜑 → ( ( 𝑁 +s 1s ) ≤s 𝑁 ↔ ( bday ‘ ( 𝑁 +s 1s ) ) ⊆ ( bday ‘ 𝑁 ) ) )
1221	1203 1220	syl	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( ( 𝑁 +s 1s ) ≤s 𝑁 ↔ ( bday ‘ ( 𝑁 +s 1s ) ) ⊆ ( bday ‘ 𝑁 ) ) )
1222	1219 1221	mpbird	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ∧ 1s <s ( 𝑛 -s 𝑚 ) ) ) → ( 𝑁 +s 1s ) ≤s 𝑁 )
1223	1222	expr	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 1s <s ( 𝑛 -s 𝑚 ) → ( 𝑁 +s 1s ) ≤s 𝑁 ) )
1224	1162 1223	sylbird	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ¬ ( 𝑛 -s 𝑚 ) ≤s 1s → ( 𝑁 +s 1s ) ≤s 𝑁 ) )
1225	1157 1224	mt3d	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑛 -s 𝑚 ) ≤s 1s )
1226	1208	adantl	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑚 <s 𝑛 )
1227		npcans	⊢ ( ( 𝑛 ∈ No ∧ 1s ∈ No ) → ( ( 𝑛 -s 1s ) +s 1s ) = 𝑛 )
1228	1160 322 1227	sylancl	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 𝑛 -s 1s ) +s 1s ) = 𝑛 )
1229	1228	breq2d	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 𝑚 +s 1s ) ≤s ( ( 𝑛 -s 1s ) +s 1s ) ↔ ( 𝑚 +s 1s ) ≤s 𝑛 ) )
1230	1160 1158	subscld	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑛 -s 1s ) ∈ No )
1231	1142 1230 1158	leadds1d	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑚 ≤s ( 𝑛 -s 1s ) ↔ ( 𝑚 +s 1s ) ≤s ( ( 𝑛 -s 1s ) +s 1s ) ) )
1232	1126 1159 1210	syl2anc	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑚 <s 𝑛 ↔ ( 𝑚 +s 1s ) ≤s 𝑛 ) )
1233	1229 1231 1232	3bitr4rd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑚 <s 𝑛 ↔ 𝑚 ≤s ( 𝑛 -s 1s ) ) )
1234	1142 1160 1158	lesubsd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑚 ≤s ( 𝑛 -s 1s ) ↔ 1s ≤s ( 𝑛 -s 𝑚 ) ) )
1235	1233 1234	bitrd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑚 <s 𝑛 ↔ 1s ≤s ( 𝑛 -s 𝑚 ) ) )
1236	1226 1235	mpbid	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 1s ≤s ( 𝑛 -s 𝑚 ) )
1237	1161 1158	lestri3d	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 𝑛 -s 𝑚 ) = 1s ↔ ( ( 𝑛 -s 𝑚 ) ≤s 1s ∧ 1s ≤s ( 𝑛 -s 𝑚 ) ) ) )
1238	1225 1236 1237	mpbir2and	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑛 -s 𝑚 ) = 1s )
1239	1160 1142 1158	subaddsd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 𝑛 -s 𝑚 ) = 1s ↔ ( 𝑚 +s 1s ) = 𝑛 ) )
1240	1238 1239	mpbid	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑚 +s 1s ) = 𝑛 )
1241	1240	eqcomd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑛 = ( 𝑚 +s 1s ) )
1242	1241	oveq1d	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑛 /su ( 2s ↑s 𝑜 ) ) = ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) )
1243	1242	oveq2d	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) = ( 𝑔 +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) )
1244	1154 1243	eqtrd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑑 = ( 𝑔 +s ( ( 𝑚 +s 1s ) /su ( 2s ↑s 𝑜 ) ) ) )
1245	1141 1142 1158	addsassd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) +s 1s ) = ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s ( 𝑚 +s 1s ) ) )
1246	1245	oveq1d	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) +s 1s ) /su ( 2s ↑s 𝑜 ) ) = ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s ( 𝑚 +s 1s ) ) /su ( 2s ↑s 𝑜 ) ) )
1247	1152 1244 1246	3eqtr4d	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑑 = ( ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) +s 1s ) /su ( 2s ↑s 𝑜 ) ) )
1248	1247	sneqd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → { 𝑑 } = { ( ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) +s 1s ) /su ( 2s ↑s 𝑜 ) ) } )
1249	1147 1248	oveq12d	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( { 𝑐 } |s { 𝑑 } ) = ( { ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) /su ( 2s ↑s 𝑜 ) ) } |s { ( ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) +s 1s ) /su ( 2s ↑s 𝑜 ) ) } ) )
1250		n0addscl	⊢ ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) ∈ ℕ0s ∧ 𝑚 ∈ ℕ0s ) → ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) ∈ ℕ0s )
1251	1140 1126 1250	syl2anc	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) ∈ ℕ0s )
1252	1251	n0zsd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) ∈ ℤs )
1253	1252 1130	pw2cutp1	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( { ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) /su ( 2s ↑s 𝑜 ) ) } |s { ( ( ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) +s 1s ) /su ( 2s ↑s 𝑜 ) ) } ) = ( ( ( 2s ·s ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) ) +s 1s ) /su ( 2s ↑s ( 𝑜 +s 1s ) ) ) )
1254	1133 1249 1253	3eqtrd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑤 = ( ( ( 2s ·s ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) ) +s 1s ) /su ( 2s ↑s ( 𝑜 +s 1s ) ) ) )
1255	52	a1i	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 2s ∈ No )
1256	1255 1141 1142	addsdid	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 2s ·s ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) ) = ( ( 2s ·s ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) ) +s ( 2s ·s 𝑚 ) ) )
1257		expsp1	⊢ ( ( 2s ∈ No ∧ 𝑜 ∈ ℕ0s ) → ( 2s ↑s ( 𝑜 +s 1s ) ) = ( ( 2s ↑s 𝑜 ) ·s 2s ) )
1258	52 1130 1257	sylancr	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 2s ↑s ( 𝑜 +s 1s ) ) = ( ( 2s ↑s 𝑜 ) ·s 2s ) )
1259	1206 1255	mulscomd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 2s ↑s 𝑜 ) ·s 2s ) = ( 2s ·s ( 2s ↑s 𝑜 ) ) )
1260	1258 1259	eqtrd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 2s ↑s ( 𝑜 +s 1s ) ) = ( 2s ·s ( 2s ↑s 𝑜 ) ) )
1261	1260	oveq1d	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 2s ↑s ( 𝑜 +s 1s ) ) ·s 𝑔 ) = ( ( 2s ·s ( 2s ↑s 𝑜 ) ) ·s 𝑔 ) )
1262	1255 1206 1134	mulsassd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 2s ·s ( 2s ↑s 𝑜 ) ) ·s 𝑔 ) = ( 2s ·s ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) ) )
1263	1261 1262	eqtrd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 2s ↑s ( 𝑜 +s 1s ) ) ·s 𝑔 ) = ( 2s ·s ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) ) )
1264	1263	oveq1d	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( ( 2s ↑s ( 𝑜 +s 1s ) ) ·s 𝑔 ) +s ( 2s ·s 𝑚 ) ) = ( ( 2s ·s ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) ) +s ( 2s ·s 𝑚 ) ) )
1265	1256 1264	eqtr4d	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 2s ·s ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) ) = ( ( ( 2s ↑s ( 𝑜 +s 1s ) ) ·s 𝑔 ) +s ( 2s ·s 𝑚 ) ) )
1266	1265	oveq1d	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 2s ·s ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) ) +s 1s ) = ( ( ( ( 2s ↑s ( 𝑜 +s 1s ) ) ·s 𝑔 ) +s ( 2s ·s 𝑚 ) ) +s 1s ) )
1267		n0expscl	⊢ ( ( 2s ∈ ℕ0s ∧ ( 𝑜 +s 1s ) ∈ ℕ0s ) → ( 2s ↑s ( 𝑜 +s 1s ) ) ∈ ℕ0s )
1268	412 1131 1267	sylancr	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 2s ↑s ( 𝑜 +s 1s ) ) ∈ ℕ0s )
1269		n0mulscl	⊢ ( ( ( 2s ↑s ( 𝑜 +s 1s ) ) ∈ ℕ0s ∧ 𝑔 ∈ ℕ0s ) → ( ( 2s ↑s ( 𝑜 +s 1s ) ) ·s 𝑔 ) ∈ ℕ0s )
1270	1268 1125 1269	syl2anc	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 2s ↑s ( 𝑜 +s 1s ) ) ·s 𝑔 ) ∈ ℕ0s )
1271	1270	n0nod	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 2s ↑s ( 𝑜 +s 1s ) ) ·s 𝑔 ) ∈ No )
1272	1128	n0nod	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 2s ·s 𝑚 ) ∈ No )
1273	1271 1272 1158	addsassd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( ( ( 2s ↑s ( 𝑜 +s 1s ) ) ·s 𝑔 ) +s ( 2s ·s 𝑚 ) ) +s 1s ) = ( ( ( 2s ↑s ( 𝑜 +s 1s ) ) ·s 𝑔 ) +s ( ( 2s ·s 𝑚 ) +s 1s ) ) )
1274	1266 1273	eqtrd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 2s ·s ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) ) +s 1s ) = ( ( ( 2s ↑s ( 𝑜 +s 1s ) ) ·s 𝑔 ) +s ( ( 2s ·s 𝑚 ) +s 1s ) ) )
1275	1274	oveq1d	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( ( 2s ·s ( ( ( 2s ↑s 𝑜 ) ·s 𝑔 ) +s 𝑚 ) ) +s 1s ) /su ( 2s ↑s ( 𝑜 +s 1s ) ) ) = ( ( ( ( 2s ↑s ( 𝑜 +s 1s ) ) ·s 𝑔 ) +s ( ( 2s ·s 𝑚 ) +s 1s ) ) /su ( 2s ↑s ( 𝑜 +s 1s ) ) ) )
1276	1254 1275	eqtrd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑤 = ( ( ( ( 2s ↑s ( 𝑜 +s 1s ) ) ·s 𝑔 ) +s ( ( 2s ·s 𝑚 ) +s 1s ) ) /su ( 2s ↑s ( 𝑜 +s 1s ) ) ) )
1277	1129	n0nod	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 2s ·s 𝑚 ) +s 1s ) ∈ No )
1278	1271 1277 1131	pw2divsdird	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( ( ( 2s ↑s ( 𝑜 +s 1s ) ) ·s 𝑔 ) +s ( ( 2s ·s 𝑚 ) +s 1s ) ) /su ( 2s ↑s ( 𝑜 +s 1s ) ) ) = ( ( ( ( 2s ↑s ( 𝑜 +s 1s ) ) ·s 𝑔 ) /su ( 2s ↑s ( 𝑜 +s 1s ) ) ) +s ( ( ( 2s ·s 𝑚 ) +s 1s ) /su ( 2s ↑s ( 𝑜 +s 1s ) ) ) ) )
1279	1134 1131	pw2divscan3d	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( ( 2s ↑s ( 𝑜 +s 1s ) ) ·s 𝑔 ) /su ( 2s ↑s ( 𝑜 +s 1s ) ) ) = 𝑔 )
1280	1279	oveq1d	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( ( ( 2s ↑s ( 𝑜 +s 1s ) ) ·s 𝑔 ) /su ( 2s ↑s ( 𝑜 +s 1s ) ) ) +s ( ( ( 2s ·s 𝑚 ) +s 1s ) /su ( 2s ↑s ( 𝑜 +s 1s ) ) ) ) = ( 𝑔 +s ( ( ( 2s ·s 𝑚 ) +s 1s ) /su ( 2s ↑s ( 𝑜 +s 1s ) ) ) ) )
1281	1276 1278 1280	3eqtrd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑤 = ( 𝑔 +s ( ( ( 2s ·s 𝑚 ) +s 1s ) /su ( 2s ↑s ( 𝑜 +s 1s ) ) ) ) )
1282		n0mulscl	⊢ ( ( 2s ∈ ℕ0s ∧ ( 𝑚 +s 1s ) ∈ ℕ0s ) → ( 2s ·s ( 𝑚 +s 1s ) ) ∈ ℕ0s )
1283	412 1148 1282	sylancr	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 2s ·s ( 𝑚 +s 1s ) ) ∈ ℕ0s )
1284	1283	n0nod	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 2s ·s ( 𝑚 +s 1s ) ) ∈ No )
1285	1268	n0nod	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 2s ↑s ( 𝑜 +s 1s ) ) ∈ No )
1286	1158 1255 1272	ltadds2d	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 1s <s 2s ↔ ( ( 2s ·s 𝑚 ) +s 1s ) <s ( ( 2s ·s 𝑚 ) +s 2s ) ) )
1287	523 1286	mpbii	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 2s ·s 𝑚 ) +s 1s ) <s ( ( 2s ·s 𝑚 ) +s 2s ) )
1288	1255 1142 1158	addsdid	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 2s ·s ( 𝑚 +s 1s ) ) = ( ( 2s ·s 𝑚 ) +s ( 2s ·s 1s ) ) )
1289	877	oveq2i	⊢ ( ( 2s ·s 𝑚 ) +s ( 2s ·s 1s ) ) = ( ( 2s ·s 𝑚 ) +s 2s )
1290	1288 1289	eqtrdi	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 2s ·s ( 𝑚 +s 1s ) ) = ( ( 2s ·s 𝑚 ) +s 2s ) )
1291	1287 1290	breqtrrd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 2s ·s 𝑚 ) +s 1s ) <s ( 2s ·s ( 𝑚 +s 1s ) ) )
1292		simprl2	⊢ ( ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) → 𝑚 <s ( 2s ↑s 𝑜 ) )
1293	1292	adantl	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑚 <s ( 2s ↑s 𝑜 ) )
1294		n0ltsp1le	⊢ ( ( 𝑚 ∈ ℕ0s ∧ ( 2s ↑s 𝑜 ) ∈ ℕ0s ) → ( 𝑚 <s ( 2s ↑s 𝑜 ) ↔ ( 𝑚 +s 1s ) ≤s ( 2s ↑s 𝑜 ) ) )
1295	1126 1138 1294	syl2anc	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑚 <s ( 2s ↑s 𝑜 ) ↔ ( 𝑚 +s 1s ) ≤s ( 2s ↑s 𝑜 ) ) )
1296	1293 1295	mpbid	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑚 +s 1s ) ≤s ( 2s ↑s 𝑜 ) )
1297	975	a1i	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 0s <s 2s )
1298	1149 1206 1255 1297	lemuls2d	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 𝑚 +s 1s ) ≤s ( 2s ↑s 𝑜 ) ↔ ( 2s ·s ( 𝑚 +s 1s ) ) ≤s ( 2s ·s ( 2s ↑s 𝑜 ) ) ) )
1299	1296 1298	mpbid	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 2s ·s ( 𝑚 +s 1s ) ) ≤s ( 2s ·s ( 2s ↑s 𝑜 ) ) )
1300	1299 1260	breqtrrd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 2s ·s ( 𝑚 +s 1s ) ) ≤s ( 2s ↑s ( 𝑜 +s 1s ) ) )
1301	1277 1284 1285 1291 1300	ltlestrd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 2s ·s 𝑚 ) +s 1s ) <s ( 2s ↑s ( 𝑜 +s 1s ) ) )
1302	1130	n0nod	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑜 ∈ No )
1303	1134 1302 1158	addsassd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 𝑔 +s 𝑜 ) +s 1s ) = ( 𝑔 +s ( 𝑜 +s 1s ) ) )
1304	1216	adantl	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑔 +s 𝑜 ) <s 𝑁 )
1305		n0addscl	⊢ ( ( 𝑔 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) → ( 𝑔 +s 𝑜 ) ∈ ℕ0s )
1306	1125 1130 1305	syl2anc	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑔 +s 𝑜 ) ∈ ℕ0s )
1307	1306	n0nod	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑔 +s 𝑜 ) ∈ No )
1308	1156 32	syl	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → 𝑁 ∈ No )
1309	1307 1308 1158	ltadds1d	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 𝑔 +s 𝑜 ) <s 𝑁 ↔ ( ( 𝑔 +s 𝑜 ) +s 1s ) <s ( 𝑁 +s 1s ) ) )
1310	1304 1309	mpbid	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( ( 𝑔 +s 𝑜 ) +s 1s ) <s ( 𝑁 +s 1s ) )
1311	1303 1310	eqbrtrrd	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ( 𝑔 +s ( 𝑜 +s 1s ) ) <s ( 𝑁 +s 1s ) )
1312		oveq1	⊢ ( 𝑏 = ( ( 2s ·s 𝑚 ) +s 1s ) → ( 𝑏 /su ( 2s ↑s 𝑞 ) ) = ( ( ( 2s ·s 𝑚 ) +s 1s ) /su ( 2s ↑s 𝑞 ) ) )
1313	1312	oveq2d	⊢ ( 𝑏 = ( ( 2s ·s 𝑚 ) +s 1s ) → ( 𝑔 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) = ( 𝑔 +s ( ( ( 2s ·s 𝑚 ) +s 1s ) /su ( 2s ↑s 𝑞 ) ) ) )
1314	1313	eqeq2d	⊢ ( 𝑏 = ( ( 2s ·s 𝑚 ) +s 1s ) → ( 𝑤 = ( 𝑔 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ↔ 𝑤 = ( 𝑔 +s ( ( ( 2s ·s 𝑚 ) +s 1s ) /su ( 2s ↑s 𝑞 ) ) ) ) )
1315		breq1	⊢ ( 𝑏 = ( ( 2s ·s 𝑚 ) +s 1s ) → ( 𝑏 <s ( 2s ↑s 𝑞 ) ↔ ( ( 2s ·s 𝑚 ) +s 1s ) <s ( 2s ↑s 𝑞 ) ) )
1316	1314 1315	3anbi12d	⊢ ( 𝑏 = ( ( 2s ·s 𝑚 ) +s 1s ) → ( ( 𝑤 = ( 𝑔 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑔 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ( 𝑤 = ( 𝑔 +s ( ( ( 2s ·s 𝑚 ) +s 1s ) /su ( 2s ↑s 𝑞 ) ) ) ∧ ( ( 2s ·s 𝑚 ) +s 1s ) <s ( 2s ↑s 𝑞 ) ∧ ( 𝑔 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
1317		oveq2	⊢ ( 𝑞 = ( 𝑜 +s 1s ) → ( 2s ↑s 𝑞 ) = ( 2s ↑s ( 𝑜 +s 1s ) ) )
1318	1317	oveq2d	⊢ ( 𝑞 = ( 𝑜 +s 1s ) → ( ( ( 2s ·s 𝑚 ) +s 1s ) /su ( 2s ↑s 𝑞 ) ) = ( ( ( 2s ·s 𝑚 ) +s 1s ) /su ( 2s ↑s ( 𝑜 +s 1s ) ) ) )
1319	1318	oveq2d	⊢ ( 𝑞 = ( 𝑜 +s 1s ) → ( 𝑔 +s ( ( ( 2s ·s 𝑚 ) +s 1s ) /su ( 2s ↑s 𝑞 ) ) ) = ( 𝑔 +s ( ( ( 2s ·s 𝑚 ) +s 1s ) /su ( 2s ↑s ( 𝑜 +s 1s ) ) ) ) )
1320	1319	eqeq2d	⊢ ( 𝑞 = ( 𝑜 +s 1s ) → ( 𝑤 = ( 𝑔 +s ( ( ( 2s ·s 𝑚 ) +s 1s ) /su ( 2s ↑s 𝑞 ) ) ) ↔ 𝑤 = ( 𝑔 +s ( ( ( 2s ·s 𝑚 ) +s 1s ) /su ( 2s ↑s ( 𝑜 +s 1s ) ) ) ) ) )
1321	1317	breq2d	⊢ ( 𝑞 = ( 𝑜 +s 1s ) → ( ( ( 2s ·s 𝑚 ) +s 1s ) <s ( 2s ↑s 𝑞 ) ↔ ( ( 2s ·s 𝑚 ) +s 1s ) <s ( 2s ↑s ( 𝑜 +s 1s ) ) ) )
1322		oveq2	⊢ ( 𝑞 = ( 𝑜 +s 1s ) → ( 𝑔 +s 𝑞 ) = ( 𝑔 +s ( 𝑜 +s 1s ) ) )
1323	1322	breq1d	⊢ ( 𝑞 = ( 𝑜 +s 1s ) → ( ( 𝑔 +s 𝑞 ) <s ( 𝑁 +s 1s ) ↔ ( 𝑔 +s ( 𝑜 +s 1s ) ) <s ( 𝑁 +s 1s ) ) )
1324	1320 1321 1323	3anbi123d	⊢ ( 𝑞 = ( 𝑜 +s 1s ) → ( ( 𝑤 = ( 𝑔 +s ( ( ( 2s ·s 𝑚 ) +s 1s ) /su ( 2s ↑s 𝑞 ) ) ) ∧ ( ( 2s ·s 𝑚 ) +s 1s ) <s ( 2s ↑s 𝑞 ) ∧ ( 𝑔 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ( 𝑤 = ( 𝑔 +s ( ( ( 2s ·s 𝑚 ) +s 1s ) /su ( 2s ↑s ( 𝑜 +s 1s ) ) ) ) ∧ ( ( 2s ·s 𝑚 ) +s 1s ) <s ( 2s ↑s ( 𝑜 +s 1s ) ) ∧ ( 𝑔 +s ( 𝑜 +s 1s ) ) <s ( 𝑁 +s 1s ) ) ) )
1325	535 1316 1324	rspc3ev	⊢ ( ( ( 𝑔 ∈ ℕ0s ∧ ( ( 2s ·s 𝑚 ) +s 1s ) ∈ ℕ0s ∧ ( 𝑜 +s 1s ) ∈ ℕ0s ) ∧ ( 𝑤 = ( 𝑔 +s ( ( ( 2s ·s 𝑚 ) +s 1s ) /su ( 2s ↑s ( 𝑜 +s 1s ) ) ) ) ∧ ( ( 2s ·s 𝑚 ) +s 1s ) <s ( 2s ↑s ( 𝑜 +s 1s ) ) ∧ ( 𝑔 +s ( 𝑜 +s 1s ) ) <s ( 𝑁 +s 1s ) ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) )
1326	1125 1129 1131 1281 1301 1311 1325	syl33anc	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ∧ ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) )
1327	1326	expr	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) ∧ ( 𝑚 ∈ ℕ0s ∧ 𝑛 ∈ ℕ0s ∧ 𝑜 ∈ ℕ0s ) ) → ( ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
1328	1327	rexlimdvvva	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → ( ∃ 𝑚 ∈ ℕ0s ∃ 𝑛 ∈ ℕ0s ∃ 𝑜 ∈ ℕ0s ( ( 𝑚 <s 𝑛 ∧ 𝑚 <s ( 2s ↑s 𝑜 ) ∧ 𝑛 <s ( 2s ↑s 𝑜 ) ) ∧ ( 𝑐 = ( 𝑔 +s ( 𝑚 /su ( 2s ↑s 𝑜 ) ) ) ∧ 𝑑 = ( 𝑔 +s ( 𝑛 /su ( 2s ↑s 𝑜 ) ) ) ∧ ( 𝑔 +s 𝑜 ) <s 𝑁 ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
1329	1123 1328	mpd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ∧ ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) )
1330	1329	expr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ) → ( ( ℎ /su ( 2s ↑s 𝑖 ) ) <s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
1331	935 1330	mpd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ∧ 𝑔 = 𝑗 ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) )
1332	1331	expr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → ( 𝑔 = 𝑗 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
1333	918 1332	jaod	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → ( ( 𝑔 <s 𝑗 ∨ 𝑔 = 𝑗 ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
1334	627 1333	sylbid	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → ( 𝑔 ≤s 𝑗 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
1335	625 1334	mpd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ ( 𝑑 <s ( 𝑗 +s 1s ) ∧ 𝑔 ≤s 𝑐 ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) )
1336	1335	expr	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ 𝑑 <s ( 𝑗 +s 1s ) ) → ( 𝑔 ≤s 𝑐 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
1337	600 1336	mpd	⊢ ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ∧ 𝑑 <s ( 𝑗 +s 1s ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) )
1338	579 1337	mpdan	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ∧ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) )
1339	1338	expr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) ∧ ( 𝑗 ∈ ℕ0s ∧ 𝑘 ∈ ℕ0s ∧ 𝑙 ∈ ℕ0s ) ) → ( ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
1340	1339	rexlimdvvva	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → ( ∃ 𝑗 ∈ ℕ0s ∃ 𝑘 ∈ ℕ0s ∃ 𝑙 ∈ ℕ0s ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
1341	229	3adant3	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → ( bday ‘ 𝑑 ) ⊆ ( bday ‘ 𝑁 ) )
1342	57	a1i	⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → 0s ∈ No )
1343	135	3ad2ant1	⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → 𝑤 ∈ No )
1344		simp1r	⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → 0s <s 𝑤 )
1345	1343 1344	0elleft	⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → 0s ∈ ( L ‘ 𝑤 ) )
1346	240 1345 189	sltssepcd	⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → 0s <s 𝑑 )
1347	1342 384 1346	ltlesd	⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → 0s ≤s 𝑑 )
1348	1347	3ad2ant1	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → 0s ≤s 𝑑 )
1349		fveq2	⊢ ( 𝑧 = 𝑑 → ( bday ‘ 𝑧 ) = ( bday ‘ 𝑑 ) )
1350	1349	sseq1d	⊢ ( 𝑧 = 𝑑 → ( ( bday ‘ 𝑧 ) ⊆ ( bday ‘ 𝑁 ) ↔ ( bday ‘ 𝑑 ) ⊆ ( bday ‘ 𝑁 ) ) )
1351		breq2	⊢ ( 𝑧 = 𝑑 → ( 0s ≤s 𝑧 ↔ 0s ≤s 𝑑 ) )
1352	1350 1351	anbi12d	⊢ ( 𝑧 = 𝑑 → ( ( ( bday ‘ 𝑧 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑧 ) ↔ ( ( bday ‘ 𝑑 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑑 ) ) )
1353		eqeq1	⊢ ( 𝑧 = 𝑑 → ( 𝑧 = 𝑁 ↔ 𝑑 = 𝑁 ) )
1354		eqeq1	⊢ ( 𝑧 = 𝑑 → ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ↔ 𝑑 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ) )
1355	1354	3anbi1d	⊢ ( 𝑧 = 𝑑 → ( ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ↔ ( 𝑑 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) )
1356	1355	rexbidv	⊢ ( 𝑧 = 𝑑 → ( ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ↔ ∃ 𝑝 ∈ ℕ0s ( 𝑑 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) )
1357	1356	2rexbidv	⊢ ( 𝑧 = 𝑑 → ( ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ↔ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑑 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) )
1358		oveq1	⊢ ( 𝑥 = 𝑗 → ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) = ( 𝑗 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) )
1359	1358	eqeq2d	⊢ ( 𝑥 = 𝑗 → ( 𝑑 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ↔ 𝑑 = ( 𝑗 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ) )
1360		oveq1	⊢ ( 𝑥 = 𝑗 → ( 𝑥 +s 𝑝 ) = ( 𝑗 +s 𝑝 ) )
1361	1360	breq1d	⊢ ( 𝑥 = 𝑗 → ( ( 𝑥 +s 𝑝 ) <s 𝑁 ↔ ( 𝑗 +s 𝑝 ) <s 𝑁 ) )
1362	1359 1361	3anbi13d	⊢ ( 𝑥 = 𝑗 → ( ( 𝑑 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ↔ ( 𝑑 = ( 𝑗 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑗 +s 𝑝 ) <s 𝑁 ) ) )
1363	1362	rexbidv	⊢ ( 𝑥 = 𝑗 → ( ∃ 𝑝 ∈ ℕ0s ( 𝑑 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ↔ ∃ 𝑝 ∈ ℕ0s ( 𝑑 = ( 𝑗 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑗 +s 𝑝 ) <s 𝑁 ) ) )
1364		oveq1	⊢ ( 𝑦 = 𝑘 → ( 𝑦 /su ( 2s ↑s 𝑝 ) ) = ( 𝑘 /su ( 2s ↑s 𝑝 ) ) )
1365	1364	oveq2d	⊢ ( 𝑦 = 𝑘 → ( 𝑗 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑝 ) ) ) )
1366	1365	eqeq2d	⊢ ( 𝑦 = 𝑘 → ( 𝑑 = ( 𝑗 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ↔ 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑝 ) ) ) ) )
1367		breq1	⊢ ( 𝑦 = 𝑘 → ( 𝑦 <s ( 2s ↑s 𝑝 ) ↔ 𝑘 <s ( 2s ↑s 𝑝 ) ) )
1368	1366 1367	3anbi12d	⊢ ( 𝑦 = 𝑘 → ( ( 𝑑 = ( 𝑗 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑗 +s 𝑝 ) <s 𝑁 ) ↔ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑗 +s 𝑝 ) <s 𝑁 ) ) )
1369	1368	rexbidv	⊢ ( 𝑦 = 𝑘 → ( ∃ 𝑝 ∈ ℕ0s ( 𝑑 = ( 𝑗 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑗 +s 𝑝 ) <s 𝑁 ) ↔ ∃ 𝑝 ∈ ℕ0s ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑗 +s 𝑝 ) <s 𝑁 ) ) )
1370		oveq2	⊢ ( 𝑝 = 𝑙 → ( 2s ↑s 𝑝 ) = ( 2s ↑s 𝑙 ) )
1371	1370	oveq2d	⊢ ( 𝑝 = 𝑙 → ( 𝑘 /su ( 2s ↑s 𝑝 ) ) = ( 𝑘 /su ( 2s ↑s 𝑙 ) ) )
1372	1371	oveq2d	⊢ ( 𝑝 = 𝑙 → ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑝 ) ) ) = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) )
1373	1372	eqeq2d	⊢ ( 𝑝 = 𝑙 → ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑝 ) ) ) ↔ 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ) )
1374	1370	breq2d	⊢ ( 𝑝 = 𝑙 → ( 𝑘 <s ( 2s ↑s 𝑝 ) ↔ 𝑘 <s ( 2s ↑s 𝑙 ) ) )
1375		oveq2	⊢ ( 𝑝 = 𝑙 → ( 𝑗 +s 𝑝 ) = ( 𝑗 +s 𝑙 ) )
1376	1375	breq1d	⊢ ( 𝑝 = 𝑙 → ( ( 𝑗 +s 𝑝 ) <s 𝑁 ↔ ( 𝑗 +s 𝑙 ) <s 𝑁 ) )
1377	1373 1374 1376	3anbi123d	⊢ ( 𝑝 = 𝑙 → ( ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑗 +s 𝑝 ) <s 𝑁 ) ↔ ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) )
1378	1377	cbvrexvw	⊢ ( ∃ 𝑝 ∈ ℕ0s ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑗 +s 𝑝 ) <s 𝑁 ) ↔ ∃ 𝑙 ∈ ℕ0s ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) )
1379	1369 1378	bitrdi	⊢ ( 𝑦 = 𝑘 → ( ∃ 𝑝 ∈ ℕ0s ( 𝑑 = ( 𝑗 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑗 +s 𝑝 ) <s 𝑁 ) ↔ ∃ 𝑙 ∈ ℕ0s ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) )
1380	1363 1379	cbvrex2vw	⊢ ( ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑑 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ↔ ∃ 𝑗 ∈ ℕ0s ∃ 𝑘 ∈ ℕ0s ∃ 𝑙 ∈ ℕ0s ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) )
1381	1357 1380	bitrdi	⊢ ( 𝑧 = 𝑑 → ( ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ↔ ∃ 𝑗 ∈ ℕ0s ∃ 𝑘 ∈ ℕ0s ∃ 𝑙 ∈ ℕ0s ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) )
1382	1353 1381	orbi12d	⊢ ( 𝑧 = 𝑑 → ( ( 𝑧 = 𝑁 ∨ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ↔ ( 𝑑 = 𝑁 ∨ ∃ 𝑗 ∈ ℕ0s ∃ 𝑘 ∈ ℕ0s ∃ 𝑙 ∈ ℕ0s ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) )
1383	1352 1382	imbi12d	⊢ ( 𝑧 = 𝑑 → ( ( ( ( bday ‘ 𝑧 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑧 ) → ( 𝑧 = 𝑁 ∨ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) ↔ ( ( ( bday ‘ 𝑑 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑑 ) → ( 𝑑 = 𝑁 ∨ ∃ 𝑗 ∈ ℕ0s ∃ 𝑘 ∈ ℕ0s ∃ 𝑙 ∈ ℕ0s ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) ) )
1384	296 2	syl	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → ∀ 𝑧 ∈ No ( ( ( bday ‘ 𝑧 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑧 ) → ( 𝑧 = 𝑁 ∨ ∃ 𝑥 ∈ ℕ0s ∃ 𝑦 ∈ ℕ0s ∃ 𝑝 ∈ ℕ0s ( 𝑧 = ( 𝑥 +s ( 𝑦 /su ( 2s ↑s 𝑝 ) ) ) ∧ 𝑦 <s ( 2s ↑s 𝑝 ) ∧ ( 𝑥 +s 𝑝 ) <s 𝑁 ) ) ) )
1385	1383 1384 385	rspcdva	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → ( ( ( bday ‘ 𝑑 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑑 ) → ( 𝑑 = 𝑁 ∨ ∃ 𝑗 ∈ ℕ0s ∃ 𝑘 ∈ ℕ0s ∃ 𝑙 ∈ ℕ0s ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) ) )
1386	1341 1348 1385	mp2and	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → ( 𝑑 = 𝑁 ∨ ∃ 𝑗 ∈ ℕ0s ∃ 𝑘 ∈ ℕ0s ∃ 𝑙 ∈ ℕ0s ( 𝑑 = ( 𝑗 +s ( 𝑘 /su ( 2s ↑s 𝑙 ) ) ) ∧ 𝑘 <s ( 2s ↑s 𝑙 ) ∧ ( 𝑗 +s 𝑙 ) <s 𝑁 ) ) )
1387	561 1340 1386	mpjaod	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) )
1388	1387	3expa	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) ∧ ( ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ∧ ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) )
1389	1388	expr	⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) ∧ ( 𝑔 ∈ ℕ0s ∧ ℎ ∈ ℕ0s ∧ 𝑖 ∈ ℕ0s ) ) → ( ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
1390	1389	rexlimdvvva	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ( ∃ 𝑔 ∈ ℕ0s ∃ ℎ ∈ ℕ0s ∃ 𝑖 ∈ ℕ0s ( 𝑐 = ( 𝑔 +s ( ℎ /su ( 2s ↑s 𝑖 ) ) ) ∧ ℎ <s ( 2s ↑s 𝑖 ) ∧ ( 𝑔 +s 𝑖 ) <s 𝑁 ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
1391	287 1390	syld	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ( ( ( bday ‘ 𝑐 ) ⊆ ( bday ‘ 𝑁 ) ∧ 0s ≤s 𝑐 ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
1392	210 215 1391	mp2and	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) ∧ 𝑤 = ( { 𝑐 } |s { 𝑑 } ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) )
1393	191 1392	mpdan	⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) )
1394	1393	3expa	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) ∧ ( 𝑑 ∈ ( R ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) )
1395	1394	expr	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) ∧ 𝑑 ∈ ( R ‘ 𝑤 ) ) → ( ∀ 𝑓 ∈ ( R ‘ 𝑤 ) 𝑑 ≤s 𝑓 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
1396	186 1395	sylbird	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) ∧ 𝑑 ∈ ( R ‘ 𝑤 ) ) → ( ∀ 𝑓 ∈ ( R ‘ 𝑤 ) ¬ 𝑓 <s 𝑑 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
1397	1396	rexlimdva	⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) → ( ∃ 𝑑 ∈ ( R ‘ 𝑤 ) ∀ 𝑓 ∈ ( R ‘ 𝑤 ) ¬ 𝑓 <s 𝑑 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
1398	180 1397	syl5	⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) → ( ( ( R ‘ 𝑤 ) ∈ Fin ∧ ( R ‘ 𝑤 ) ≠ ∅ ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
1399	154 175 1398	mp2and	⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ ( 𝑐 ∈ ( L ‘ 𝑤 ) ∧ ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) )
1400	1399	expr	⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ 𝑐 ∈ ( L ‘ 𝑤 ) ) → ( ∀ 𝑒 ∈ ( L ‘ 𝑤 ) 𝑒 ≤s 𝑐 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
1401	150 1400	sylbird	⊢ ( ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) ∧ 𝑐 ∈ ( L ‘ 𝑤 ) ) → ( ∀ 𝑒 ∈ ( L ‘ 𝑤 ) ¬ 𝑐 <s 𝑒 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
1402	1401	rexlimdva	⊢ ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) → ( ∃ 𝑐 ∈ ( L ‘ 𝑤 ) ∀ 𝑒 ∈ ( L ‘ 𝑤 ) ¬ 𝑐 <s 𝑒 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
1403	144 1402	syl5	⊢ ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) → ( ( ( L ‘ 𝑤 ) ∈ Fin ∧ ( L ‘ 𝑤 ) ≠ ∅ ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
1404	134 138 1403	mp2and	⊢ ( ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) ∧ 0s <s 𝑤 ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) )
1405	1404	ex	⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) → ( 0s <s 𝑤 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
1406		addslid	⊢ ( 0s ∈ No → ( 0s +s 0s ) = 0s )
1407	57 1406	ax-mp	⊢ ( 0s +s 0s ) = 0s
1408	1407	eqcomi	⊢ 0s = ( 0s +s 0s )
1409		n0p1nns	⊢ ( 𝑁 ∈ ℕ0s → ( 𝑁 +s 1s ) ∈ ℕs )
1410	1 1409	syl	⊢ ( 𝜑 → ( 𝑁 +s 1s ) ∈ ℕs )
1411		nnsgt0	⊢ ( ( 𝑁 +s 1s ) ∈ ℕs → 0s <s ( 𝑁 +s 1s ) )
1412	1410 1411	syl	⊢ ( 𝜑 → 0s <s ( 𝑁 +s 1s ) )
1413	29 29 29	3pm3.2i	⊢ ( 0s ∈ ℕ0s ∧ 0s ∈ ℕ0s ∧ 0s ∈ ℕ0s )
1414		oveq1	⊢ ( 𝑎 = 0s → ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) = ( 0s +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) )
1415	1414	eqeq2d	⊢ ( 𝑎 = 0s → ( 0s = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ↔ 0s = ( 0s +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ) )
1416		oveq1	⊢ ( 𝑎 = 0s → ( 𝑎 +s 𝑞 ) = ( 0s +s 𝑞 ) )
1417	1416	breq1d	⊢ ( 𝑎 = 0s → ( ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ↔ ( 0s +s 𝑞 ) <s ( 𝑁 +s 1s ) ) )
1418	1415 1417	3anbi13d	⊢ ( 𝑎 = 0s → ( ( 0s = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ( 0s = ( 0s +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 0s +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
1419	46	oveq2d	⊢ ( 𝑏 = 0s → ( 0s +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) = ( 0s +s ( 0s /su ( 2s ↑s 𝑞 ) ) ) )
1420	1419	eqeq2d	⊢ ( 𝑏 = 0s → ( 0s = ( 0s +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ↔ 0s = ( 0s +s ( 0s /su ( 2s ↑s 𝑞 ) ) ) ) )
1421	1420 49	3anbi12d	⊢ ( 𝑏 = 0s → ( ( 0s = ( 0s +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 0s +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ( 0s = ( 0s +s ( 0s /su ( 2s ↑s 𝑞 ) ) ) ∧ 0s <s ( 2s ↑s 𝑞 ) ∧ ( 0s +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
1422	60	oveq2d	⊢ ( 𝑞 = 0s → ( 0s +s ( 0s /su ( 2s ↑s 𝑞 ) ) ) = ( 0s +s 0s ) )
1423	1422	eqeq2d	⊢ ( 𝑞 = 0s → ( 0s = ( 0s +s ( 0s /su ( 2s ↑s 𝑞 ) ) ) ↔ 0s = ( 0s +s 0s ) ) )
1424		oveq2	⊢ ( 𝑞 = 0s → ( 0s +s 𝑞 ) = ( 0s +s 0s ) )
1425	1424 1407	eqtrdi	⊢ ( 𝑞 = 0s → ( 0s +s 𝑞 ) = 0s )
1426	1425	breq1d	⊢ ( 𝑞 = 0s → ( ( 0s +s 𝑞 ) <s ( 𝑁 +s 1s ) ↔ 0s <s ( 𝑁 +s 1s ) ) )
1427	1423 63 1426	3anbi123d	⊢ ( 𝑞 = 0s → ( ( 0s = ( 0s +s ( 0s /su ( 2s ↑s 𝑞 ) ) ) ∧ 0s <s ( 2s ↑s 𝑞 ) ∧ ( 0s +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ( 0s = ( 0s +s 0s ) ∧ 0s <s 1s ∧ 0s <s ( 𝑁 +s 1s ) ) ) )
1428	1418 1421 1427	rspc3ev	⊢ ( ( ( 0s ∈ ℕ0s ∧ 0s ∈ ℕ0s ∧ 0s ∈ ℕ0s ) ∧ ( 0s = ( 0s +s 0s ) ∧ 0s <s 1s ∧ 0s <s ( 𝑁 +s 1s ) ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 0s = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) )
1429	1413 1428	mpan	⊢ ( ( 0s = ( 0s +s 0s ) ∧ 0s <s 1s ∧ 0s <s ( 𝑁 +s 1s ) ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 0s = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) )
1430	1408 36 1412 1429	mp3an12i	⊢ ( 𝜑 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 0s = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) )
1431		eqeq1	⊢ ( 0s = 𝑤 → ( 0s = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ↔ 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ) )
1432	1431	3anbi1d	⊢ ( 0s = 𝑤 → ( ( 0s = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
1433	1432	rexbidv	⊢ ( 0s = 𝑤 → ( ∃ 𝑞 ∈ ℕ0s ( 0s = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
1434	1433	2rexbidv	⊢ ( 0s = 𝑤 → ( ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 0s = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ↔ ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
1435	1430 1434	syl5ibcom	⊢ ( 𝜑 → ( 0s = 𝑤 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
1436	1435	adantr	⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) → ( 0s = 𝑤 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
1437	1405 1436	jaod	⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) → ( ( 0s <s 𝑤 ∨ 0s = 𝑤 ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
1438	121 1437	sylbid	⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) ) ) → ( 0s ≤s 𝑤 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
1439	1438	expr	⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 𝑤 ≠ ( 𝑁 +s 1s ) ) → ( 0s ≤s 𝑤 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) )
1440	1439	expd	⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) → ( 𝑤 ≠ ( 𝑁 +s 1s ) → ( 0s ≤s 𝑤 → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) ) )
1441	1440	com34	⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) → ( 0s ≤s 𝑤 → ( 𝑤 ≠ ( 𝑁 +s 1s ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) ) )
1442	1441	impd	⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 0s ≤s 𝑤 ) → ( 𝑤 ≠ ( 𝑁 +s 1s ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) )
1443	1442	impr	⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 0s ≤s 𝑤 ) ) ) → ( 𝑤 ≠ ( 𝑁 +s 1s ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
1444	118 1443	biimtrrid	⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 0s ≤s 𝑤 ) ) ) → ( ¬ 𝑤 = ( 𝑁 +s 1s ) → ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
1445	1444	orrd	⊢ ( ( 𝜑 ∧ ( 𝑤 ∈ No ∧ ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 0s ≤s 𝑤 ) ) ) → ( 𝑤 = ( 𝑁 +s 1s ) ∨ ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) )
1446	1445	expr	⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 0s ≤s 𝑤 ) → ( 𝑤 = ( 𝑁 +s 1s ) ∨ ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) )
1447	1446	expd	⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ( bday ‘ 𝑤 ) = ( bday ‘ ( 𝑁 +s 1s ) ) → ( 0s ≤s 𝑤 → ( 𝑤 = ( 𝑁 +s 1s ) ∨ ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) ) )
1448	117 1447	sylbird	⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ( bday ‘ 𝑤 ) = suc ( bday ‘ 𝑁 ) → ( 0s ≤s 𝑤 → ( 𝑤 = ( 𝑁 +s 1s ) ∨ ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) ) )
1449	116 1448	jaod	⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ 𝑁 ) ∨ ( bday ‘ 𝑤 ) = suc ( bday ‘ 𝑁 ) ) → ( 0s ≤s 𝑤 → ( 𝑤 = ( 𝑁 +s 1s ) ∨ ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) ) )
1450	14 1449	biimtrid	⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ( bday ‘ 𝑤 ) ⊆ suc ( bday ‘ 𝑁 ) → ( 0s ≤s 𝑤 → ( 𝑤 = ( 𝑁 +s 1s ) ∨ ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) ) )
1451	6 1450	sylbid	⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ ( 𝑁 +s 1s ) ) → ( 0s ≤s 𝑤 → ( 𝑤 = ( 𝑁 +s 1s ) ∨ ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) ) )
1452	1451	impd	⊢ ( ( 𝜑 ∧ 𝑤 ∈ No ) → ( ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 0s ≤s 𝑤 ) → ( 𝑤 = ( 𝑁 +s 1s ) ∨ ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) )
1453	1452	ralrimiva	⊢ ( 𝜑 → ∀ 𝑤 ∈ No ( ( ( bday ‘ 𝑤 ) ⊆ ( bday ‘ ( 𝑁 +s 1s ) ) ∧ 0s ≤s 𝑤 ) → ( 𝑤 = ( 𝑁 +s 1s ) ∨ ∃ 𝑎 ∈ ℕ0s ∃ 𝑏 ∈ ℕ0s ∃ 𝑞 ∈ ℕ0s ( 𝑤 = ( 𝑎 +s ( 𝑏 /su ( 2s ↑s 𝑞 ) ) ) ∧ 𝑏 <s ( 2s ↑s 𝑞 ) ∧ ( 𝑎 +s 𝑞 ) <s ( 𝑁 +s 1s ) ) ) ) )