poimirlem19

Description: Lemma for poimir establishing the vertices of the simplex in poimirlem20 . (Contributed by Brendan Leahy, 21-Aug-2020)

	Ref	Expression
Hypotheses	poimir.0	⊢ ( 𝜑 → 𝑁 ∈ ℕ )
	poimirlem22.s	⊢ 𝑆 = { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) × ( 0 ... 𝑁 ) ) ∣ 𝐹 = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑡 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑡 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) }
	poimirlem22.1	⊢ ( 𝜑 → 𝐹 : ( 0 ... ( 𝑁 − 1 ) ) ⟶ ( ( 0 ... 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) )
	poimirlem22.2	⊢ ( 𝜑 → 𝑇 ∈ 𝑆 )
	poimirlem22.3	⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ∃ 𝑝 ∈ ran 𝐹 ( 𝑝 ‘ 𝑛 ) ≠ 0 )
	poimirlem21.4	⊢ ( 𝜑 → ( 2^nd ‘ 𝑇 ) = 𝑁 )
Assertion	poimirlem19	⊢ ( 𝜑 → 𝐹 = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		poimir.0	⊢ ( 𝜑 → 𝑁 ∈ ℕ )
2		poimirlem22.s	⊢ 𝑆 = { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) × ( 0 ... 𝑁 ) ) ∣ 𝐹 = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑡 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑡 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) }
3		poimirlem22.1	⊢ ( 𝜑 → 𝐹 : ( 0 ... ( 𝑁 − 1 ) ) ⟶ ( ( 0 ... 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) )
4		poimirlem22.2	⊢ ( 𝜑 → 𝑇 ∈ 𝑆 )
5		poimirlem22.3	⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ∃ 𝑝 ∈ ran 𝐹 ( 𝑝 ‘ 𝑛 ) ≠ 0 )
6		poimirlem21.4	⊢ ( 𝜑 → ( 2^nd ‘ 𝑇 ) = 𝑁 )
7		fveq2	⊢ ( 𝑡 = 𝑇 → ( 2^nd ‘ 𝑡 ) = ( 2^nd ‘ 𝑇 ) )
8	7	breq2d	⊢ ( 𝑡 = 𝑇 → ( 𝑦 < ( 2^nd ‘ 𝑡 ) ↔ 𝑦 < ( 2^nd ‘ 𝑇 ) ) )
9	8	ifbid	⊢ ( 𝑡 = 𝑇 → if ( 𝑦 < ( 2^nd ‘ 𝑡 ) , 𝑦 , ( 𝑦 + 1 ) ) = if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) )
10		2fveq3	⊢ ( 𝑡 = 𝑇 → ( 1^st ‘ ( 1^st ‘ 𝑡 ) ) = ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) )
11		2fveq3	⊢ ( 𝑡 = 𝑇 → ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) = ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) )
12	11	imaeq1d	⊢ ( 𝑡 = 𝑇 → ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) )
13	12	xpeq1d	⊢ ( 𝑡 = 𝑇 → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) )
14	11	imaeq1d	⊢ ( 𝑡 = 𝑇 → ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) )
15	14	xpeq1d	⊢ ( 𝑡 = 𝑇 → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) )
16	13 15	uneq12d	⊢ ( 𝑡 = 𝑇 → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) )
17	10 16	oveq12d	⊢ ( 𝑡 = 𝑇 → ( ( 1^st ‘ ( 1^st ‘ 𝑡 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
18	9 17	csbeq12dv	⊢ ( 𝑡 = 𝑇 → ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑡 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑡 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
19	18	mpteq2dv	⊢ ( 𝑡 = 𝑇 → ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑡 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑡 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) )
20	19	eqeq2d	⊢ ( 𝑡 = 𝑇 → ( 𝐹 = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑡 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑡 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) ↔ 𝐹 = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) ) )
21	20 2	elrab2	⊢ ( 𝑇 ∈ 𝑆 ↔ ( 𝑇 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) × ( 0 ... 𝑁 ) ) ∧ 𝐹 = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) ) )
22	21	simprbi	⊢ ( 𝑇 ∈ 𝑆 → 𝐹 = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) )
23	4 22	syl	⊢ ( 𝜑 → 𝐹 = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) )
24		elrabi	⊢ ( 𝑇 ∈ { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) × ( 0 ... 𝑁 ) ) ∣ 𝐹 = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑡 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑡 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) } → 𝑇 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) × ( 0 ... 𝑁 ) ) )
25	24 2	eleq2s	⊢ ( 𝑇 ∈ 𝑆 → 𝑇 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) × ( 0 ... 𝑁 ) ) )
26	4 25	syl	⊢ ( 𝜑 → 𝑇 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) × ( 0 ... 𝑁 ) ) )
27		xp1st	⊢ ( 𝑇 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) × ( 0 ... 𝑁 ) ) → ( 1^st ‘ 𝑇 ) ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) )
28	26 27	syl	⊢ ( 𝜑 → ( 1^st ‘ 𝑇 ) ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) )
29		xp1st	⊢ ( ( 1^st ‘ 𝑇 ) ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) → ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∈ ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) )
30	28 29	syl	⊢ ( 𝜑 → ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∈ ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) )
31		elmapfn	⊢ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∈ ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) → ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) Fn ( 1 ... 𝑁 ) )
32	30 31	syl	⊢ ( 𝜑 → ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) Fn ( 1 ... 𝑁 ) )
33	32	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) Fn ( 1 ... 𝑁 ) )
34		1ex	⊢ 1 ∈ V
35		fnconstg	⊢ ( 1 ∈ V → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) Fn ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) )
36	34 35	ax-mp	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) Fn ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) )
37		c0ex	⊢ 0 ∈ V
38		fnconstg	⊢ ( 0 ∈ V → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) )
39	37 38	ax-mp	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) )
40	36 39	pm3.2i	⊢ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) Fn ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∧ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) )
41		xp2nd	⊢ ( ( 1^st ‘ 𝑇 ) ∈ ( ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) → ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∈ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } )
42	28 41	syl	⊢ ( 𝜑 → ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∈ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } )
43		fvex	⊢ ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∈ V
44		f1oeq1	⊢ ( 𝑓 = ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) → ( 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ↔ ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ) )
45	43 44	elab	⊢ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∈ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ↔ ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) )
46	42 45	sylib	⊢ ( 𝜑 → ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) )
47		dff1o3	⊢ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ↔ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –onto→ ( 1 ... 𝑁 ) ∧ Fun ^◡ ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ) )
48	47	simprbi	⊢ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) → Fun ^◡ ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) )
49	46 48	syl	⊢ ( 𝜑 → Fun ^◡ ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) )
50		imain	⊢ ( Fun ^◡ ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑦 ) ∩ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∩ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) )
51	49 50	syl	⊢ ( 𝜑 → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑦 ) ∩ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∩ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) )
52		elfznn0	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 𝑦 ∈ ℕ₀ )
53	52	nn0red	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 𝑦 ∈ ℝ )
54	53	ltp1d	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 𝑦 < ( 𝑦 + 1 ) )
55		fzdisj	⊢ ( 𝑦 < ( 𝑦 + 1 ) → ( ( 1 ... 𝑦 ) ∩ ( ( 𝑦 + 1 ) ... 𝑁 ) ) = ∅ )
56	54 55	syl	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 1 ... 𝑦 ) ∩ ( ( 𝑦 + 1 ) ... 𝑁 ) ) = ∅ )
57	56	imaeq2d	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑦 ) ∩ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ∅ ) )
58		ima0	⊢ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ∅ ) = ∅
59	57 58	eqtrdi	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑦 ) ∩ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) = ∅ )
60	51 59	sylan9req	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∩ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) = ∅ )
61		fnun	⊢ ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) Fn ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∧ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) ∧ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∩ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) = ∅ ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∪ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) )
62	40 60 61	sylancr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∪ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) )
63		imaundi	⊢ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑦 ) ∪ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∪ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) )
64		nn0p1nn	⊢ ( 𝑦 ∈ ℕ₀ → ( 𝑦 + 1 ) ∈ ℕ )
65	52 64	syl	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 𝑦 + 1 ) ∈ ℕ )
66		nnuz	⊢ ℕ = ( ℤ_≥ ‘ 1 )
67	65 66	eleqtrdi	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 𝑦 + 1 ) ∈ ( ℤ_≥ ‘ 1 ) )
68	67	adantl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑦 + 1 ) ∈ ( ℤ_≥ ‘ 1 ) )
69	1	nncnd	⊢ ( 𝜑 → 𝑁 ∈ ℂ )
70		npcan1	⊢ ( 𝑁 ∈ ℂ → ( ( 𝑁 − 1 ) + 1 ) = 𝑁 )
71	69 70	syl	⊢ ( 𝜑 → ( ( 𝑁 − 1 ) + 1 ) = 𝑁 )
72	71	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑁 − 1 ) + 1 ) = 𝑁 )
73		elfzuz3	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 𝑁 − 1 ) ∈ ( ℤ_≥ ‘ 𝑦 ) )
74		peano2uz	⊢ ( ( 𝑁 − 1 ) ∈ ( ℤ_≥ ‘ 𝑦 ) → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ_≥ ‘ 𝑦 ) )
75	73 74	syl	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ_≥ ‘ 𝑦 ) )
76	75	adantl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ_≥ ‘ 𝑦 ) )
77	72 76	eqeltrrd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 𝑁 ∈ ( ℤ_≥ ‘ 𝑦 ) )
78		fzsplit2	⊢ ( ( ( 𝑦 + 1 ) ∈ ( ℤ_≥ ‘ 1 ) ∧ 𝑁 ∈ ( ℤ_≥ ‘ 𝑦 ) ) → ( 1 ... 𝑁 ) = ( ( 1 ... 𝑦 ) ∪ ( ( 𝑦 + 1 ) ... 𝑁 ) ) )
79	68 77 78	syl2anc	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 1 ... 𝑁 ) = ( ( 1 ... 𝑦 ) ∪ ( ( 𝑦 + 1 ) ... 𝑁 ) ) )
80	79	imaeq2d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑁 ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑦 ) ∪ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) )
81		f1ofo	⊢ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) → ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –onto→ ( 1 ... 𝑁 ) )
82		foima	⊢ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –onto→ ( 1 ... 𝑁 ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑁 ) ) = ( 1 ... 𝑁 ) )
83	46 81 82	3syl	⊢ ( 𝜑 → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑁 ) ) = ( 1 ... 𝑁 ) )
84	83	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑁 ) ) = ( 1 ... 𝑁 ) )
85	80 84	eqtr3d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑦 ) ∪ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) = ( 1 ... 𝑁 ) )
86	63 85	eqtr3id	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∪ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) = ( 1 ... 𝑁 ) )
87	86	fneq2d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∪ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) ↔ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( 1 ... 𝑁 ) ) )
88	62 87	mpbid	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( 1 ... 𝑁 ) )
89		ovexd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 1 ... 𝑁 ) ∈ V )
90		inidm	⊢ ( ( 1 ... 𝑁 ) ∩ ( 1 ... 𝑁 ) ) = ( 1 ... 𝑁 )
91		eqidd	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) = ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) )
92		eqidd	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) )
93	33 88 89 89 90 91 92	offval	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) )
94		elmapi	⊢ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∈ ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) → ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ..^ 𝐾 ) )
95	30 94	syl	⊢ ( 𝜑 → ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ..^ 𝐾 ) )
96	95	ffvelrnda	⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) ∈ ( 0 ..^ 𝐾 ) )
97		elfzonn0	⊢ ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) ∈ ( 0 ..^ 𝐾 ) → ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) ∈ ℕ₀ )
98	96 97	syl	⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) ∈ ℕ₀ )
99	98	nn0cnd	⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) ∈ ℂ )
100	99	adantlr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) ∈ ℂ )
101		ax-1cn	⊢ 1 ∈ ℂ
102		0cn	⊢ 0 ∈ ℂ
103	101 102	ifcli	⊢ if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ∈ ℂ
104	103	a1i	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ∈ ℂ )
105		snssi	⊢ ( 1 ∈ ℂ → { 1 } ⊆ ℂ )
106	101 105	ax-mp	⊢ { 1 } ⊆ ℂ
107		snssi	⊢ ( 0 ∈ ℂ → { 0 } ⊆ ℂ )
108	102 107	ax-mp	⊢ { 0 } ⊆ ℂ
109	106 108	unssi	⊢ ( { 1 } ∪ { 0 } ) ⊆ ℂ
110	34	fconst	⊢ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) : ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ⟶ { 1 }
111	37	fconst	⊢ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) : ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ⟶ { 0 }
112	110 111	pm3.2i	⊢ ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) : ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ⟶ { 1 } ∧ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) : ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ⟶ { 0 } )
113		simpr	⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ) → 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) )
114	1	nnzd	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
115		1z	⊢ 1 ∈ ℤ
116		peano2z	⊢ ( 1 ∈ ℤ → ( 1 + 1 ) ∈ ℤ )
117	115 116	ax-mp	⊢ ( 1 + 1 ) ∈ ℤ
118	114 117	jctil	⊢ ( 𝜑 → ( ( 1 + 1 ) ∈ ℤ ∧ 𝑁 ∈ ℤ ) )
119		elfzelz	⊢ ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) → 𝑛 ∈ ℤ )
120	119 115	jctir	⊢ ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) → ( 𝑛 ∈ ℤ ∧ 1 ∈ ℤ ) )
121		fzsubel	⊢ ( ( ( ( 1 + 1 ) ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑛 ∈ ℤ ∧ 1 ∈ ℤ ) ) → ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↔ ( 𝑛 − 1 ) ∈ ( ( ( 1 + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) )
122	118 120 121	syl2an	⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ) → ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↔ ( 𝑛 − 1 ) ∈ ( ( ( 1 + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) )
123	113 122	mpbid	⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ) → ( 𝑛 − 1 ) ∈ ( ( ( 1 + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) )
124	101 101	pncan3oi	⊢ ( ( 1 + 1 ) − 1 ) = 1
125	124	oveq1i	⊢ ( ( ( 1 + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) = ( 1 ... ( 𝑁 − 1 ) )
126	123 125	eleqtrdi	⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ) → ( 𝑛 − 1 ) ∈ ( 1 ... ( 𝑁 − 1 ) ) )
127	126	ralrimiva	⊢ ( 𝜑 → ∀ 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ( 𝑛 − 1 ) ∈ ( 1 ... ( 𝑁 − 1 ) ) )
128		simpr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 1 ... ( 𝑁 − 1 ) ) ) → 𝑦 ∈ ( 1 ... ( 𝑁 − 1 ) ) )
129		peano2zm	⊢ ( 𝑁 ∈ ℤ → ( 𝑁 − 1 ) ∈ ℤ )
130	114 129	syl	⊢ ( 𝜑 → ( 𝑁 − 1 ) ∈ ℤ )
131	130 115	jctil	⊢ ( 𝜑 → ( 1 ∈ ℤ ∧ ( 𝑁 − 1 ) ∈ ℤ ) )
132		elfzelz	⊢ ( 𝑦 ∈ ( 1 ... ( 𝑁 − 1 ) ) → 𝑦 ∈ ℤ )
133	132 115	jctir	⊢ ( 𝑦 ∈ ( 1 ... ( 𝑁 − 1 ) ) → ( 𝑦 ∈ ℤ ∧ 1 ∈ ℤ ) )
134		fzaddel	⊢ ( ( ( 1 ∈ ℤ ∧ ( 𝑁 − 1 ) ∈ ℤ ) ∧ ( 𝑦 ∈ ℤ ∧ 1 ∈ ℤ ) ) → ( 𝑦 ∈ ( 1 ... ( 𝑁 − 1 ) ) ↔ ( 𝑦 + 1 ) ∈ ( ( 1 + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) ) )
135	131 133 134	syl2an	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 1 ... ( 𝑁 − 1 ) ) ) → ( 𝑦 ∈ ( 1 ... ( 𝑁 − 1 ) ) ↔ ( 𝑦 + 1 ) ∈ ( ( 1 + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) ) )
136	128 135	mpbid	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 1 ... ( 𝑁 − 1 ) ) ) → ( 𝑦 + 1 ) ∈ ( ( 1 + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) )
137	71	oveq2d	⊢ ( 𝜑 → ( ( 1 + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) = ( ( 1 + 1 ) ... 𝑁 ) )
138	137	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 1 ... ( 𝑁 − 1 ) ) ) → ( ( 1 + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) = ( ( 1 + 1 ) ... 𝑁 ) )
139	136 138	eleqtrd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 1 ... ( 𝑁 − 1 ) ) ) → ( 𝑦 + 1 ) ∈ ( ( 1 + 1 ) ... 𝑁 ) )
140	119	zcnd	⊢ ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) → 𝑛 ∈ ℂ )
141	132	zcnd	⊢ ( 𝑦 ∈ ( 1 ... ( 𝑁 − 1 ) ) → 𝑦 ∈ ℂ )
142		subadd2	⊢ ( ( 𝑛 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( ( 𝑛 − 1 ) = 𝑦 ↔ ( 𝑦 + 1 ) = 𝑛 ) )
143	101 142	mp3an2	⊢ ( ( 𝑛 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( ( 𝑛 − 1 ) = 𝑦 ↔ ( 𝑦 + 1 ) = 𝑛 ) )
144		eqcom	⊢ ( 𝑦 = ( 𝑛 − 1 ) ↔ ( 𝑛 − 1 ) = 𝑦 )
145		eqcom	⊢ ( 𝑛 = ( 𝑦 + 1 ) ↔ ( 𝑦 + 1 ) = 𝑛 )
146	143 144 145	3bitr4g	⊢ ( ( 𝑛 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( 𝑦 = ( 𝑛 − 1 ) ↔ 𝑛 = ( 𝑦 + 1 ) ) )
147	140 141 146	syl2anr	⊢ ( ( 𝑦 ∈ ( 1 ... ( 𝑁 − 1 ) ) ∧ 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ) → ( 𝑦 = ( 𝑛 − 1 ) ↔ 𝑛 = ( 𝑦 + 1 ) ) )
148	147	ralrimiva	⊢ ( 𝑦 ∈ ( 1 ... ( 𝑁 − 1 ) ) → ∀ 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ( 𝑦 = ( 𝑛 − 1 ) ↔ 𝑛 = ( 𝑦 + 1 ) ) )
149	148	adantl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 1 ... ( 𝑁 − 1 ) ) ) → ∀ 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ( 𝑦 = ( 𝑛 − 1 ) ↔ 𝑛 = ( 𝑦 + 1 ) ) )
150		reu6i	⊢ ( ( ( 𝑦 + 1 ) ∈ ( ( 1 + 1 ) ... 𝑁 ) ∧ ∀ 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ( 𝑦 = ( 𝑛 − 1 ) ↔ 𝑛 = ( 𝑦 + 1 ) ) ) → ∃! 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) 𝑦 = ( 𝑛 − 1 ) )
151	139 149 150	syl2anc	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 1 ... ( 𝑁 − 1 ) ) ) → ∃! 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) 𝑦 = ( 𝑛 − 1 ) )
152	151	ralrimiva	⊢ ( 𝜑 → ∀ 𝑦 ∈ ( 1 ... ( 𝑁 − 1 ) ) ∃! 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) 𝑦 = ( 𝑛 − 1 ) )
153		eqid	⊢ ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) = ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) )
154	153	f1ompt	⊢ ( ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) : ( ( 1 + 1 ) ... 𝑁 ) –1-1-onto→ ( 1 ... ( 𝑁 − 1 ) ) ↔ ( ∀ 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ( 𝑛 − 1 ) ∈ ( 1 ... ( 𝑁 − 1 ) ) ∧ ∀ 𝑦 ∈ ( 1 ... ( 𝑁 − 1 ) ) ∃! 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) 𝑦 = ( 𝑛 − 1 ) ) )
155	127 152 154	sylanbrc	⊢ ( 𝜑 → ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) : ( ( 1 + 1 ) ... 𝑁 ) –1-1-onto→ ( 1 ... ( 𝑁 − 1 ) ) )
156		f1osng	⊢ ( ( 1 ∈ V ∧ 𝑁 ∈ ℕ ) → { ⟨ 1 , 𝑁 ⟩ } : { 1 } –1-1-onto→ { 𝑁 } )
157	34 1 156	sylancr	⊢ ( 𝜑 → { ⟨ 1 , 𝑁 ⟩ } : { 1 } –1-1-onto→ { 𝑁 } )
158	1	nnred	⊢ ( 𝜑 → 𝑁 ∈ ℝ )
159	158	ltm1d	⊢ ( 𝜑 → ( 𝑁 − 1 ) < 𝑁 )
160	130	zred	⊢ ( 𝜑 → ( 𝑁 − 1 ) ∈ ℝ )
161	160 158	ltnled	⊢ ( 𝜑 → ( ( 𝑁 − 1 ) < 𝑁 ↔ ¬ 𝑁 ≤ ( 𝑁 − 1 ) ) )
162	159 161	mpbid	⊢ ( 𝜑 → ¬ 𝑁 ≤ ( 𝑁 − 1 ) )
163		elfzle2	⊢ ( 𝑁 ∈ ( 1 ... ( 𝑁 − 1 ) ) → 𝑁 ≤ ( 𝑁 − 1 ) )
164	162 163	nsyl	⊢ ( 𝜑 → ¬ 𝑁 ∈ ( 1 ... ( 𝑁 − 1 ) ) )
165		disjsn	⊢ ( ( ( 1 ... ( 𝑁 − 1 ) ) ∩ { 𝑁 } ) = ∅ ↔ ¬ 𝑁 ∈ ( 1 ... ( 𝑁 − 1 ) ) )
166	164 165	sylibr	⊢ ( 𝜑 → ( ( 1 ... ( 𝑁 − 1 ) ) ∩ { 𝑁 } ) = ∅ )
167		1re	⊢ 1 ∈ ℝ
168	167	ltp1i	⊢ 1 < ( 1 + 1 )
169	117	zrei	⊢ ( 1 + 1 ) ∈ ℝ
170	167 169	ltnlei	⊢ ( 1 < ( 1 + 1 ) ↔ ¬ ( 1 + 1 ) ≤ 1 )
171	168 170	mpbi	⊢ ¬ ( 1 + 1 ) ≤ 1
172		elfzle1	⊢ ( 1 ∈ ( ( 1 + 1 ) ... 𝑁 ) → ( 1 + 1 ) ≤ 1 )
173	171 172	mto	⊢ ¬ 1 ∈ ( ( 1 + 1 ) ... 𝑁 )
174		disjsn	⊢ ( ( ( ( 1 + 1 ) ... 𝑁 ) ∩ { 1 } ) = ∅ ↔ ¬ 1 ∈ ( ( 1 + 1 ) ... 𝑁 ) )
175	173 174	mpbir	⊢ ( ( ( 1 + 1 ) ... 𝑁 ) ∩ { 1 } ) = ∅
176		f1oun	⊢ ( ( ( ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) : ( ( 1 + 1 ) ... 𝑁 ) –1-1-onto→ ( 1 ... ( 𝑁 − 1 ) ) ∧ { ⟨ 1 , 𝑁 ⟩ } : { 1 } –1-1-onto→ { 𝑁 } ) ∧ ( ( ( ( 1 + 1 ) ... 𝑁 ) ∩ { 1 } ) = ∅ ∧ ( ( 1 ... ( 𝑁 − 1 ) ) ∩ { 𝑁 } ) = ∅ ) ) → ( ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) ∪ { ⟨ 1 , 𝑁 ⟩ } ) : ( ( ( 1 + 1 ) ... 𝑁 ) ∪ { 1 } ) –1-1-onto→ ( ( 1 ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) )
177	175 176	mpanr1	⊢ ( ( ( ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) : ( ( 1 + 1 ) ... 𝑁 ) –1-1-onto→ ( 1 ... ( 𝑁 − 1 ) ) ∧ { ⟨ 1 , 𝑁 ⟩ } : { 1 } –1-1-onto→ { 𝑁 } ) ∧ ( ( 1 ... ( 𝑁 − 1 ) ) ∩ { 𝑁 } ) = ∅ ) → ( ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) ∪ { ⟨ 1 , 𝑁 ⟩ } ) : ( ( ( 1 + 1 ) ... 𝑁 ) ∪ { 1 } ) –1-1-onto→ ( ( 1 ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) )
178	155 157 166 177	syl21anc	⊢ ( 𝜑 → ( ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) ∪ { ⟨ 1 , 𝑁 ⟩ } ) : ( ( ( 1 + 1 ) ... 𝑁 ) ∪ { 1 } ) –1-1-onto→ ( ( 1 ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) )
179		eleq1	⊢ ( 𝑛 = 1 → ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↔ 1 ∈ ( ( 1 + 1 ) ... 𝑁 ) ) )
180	173 179	mtbiri	⊢ ( 𝑛 = 1 → ¬ 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) )
181	180	necon2ai	⊢ ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) → 𝑛 ≠ 1 )
182		ifnefalse	⊢ ( 𝑛 ≠ 1 → if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) = ( 𝑛 − 1 ) )
183	181 182	syl	⊢ ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) → if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) = ( 𝑛 − 1 ) )
184	183	mpteq2ia	⊢ ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) = ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) )
185	184	uneq1i	⊢ ( ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ∪ { ⟨ 1 , 𝑁 ⟩ } ) = ( ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) ∪ { ⟨ 1 , 𝑁 ⟩ } )
186	34	a1i	⊢ ( 𝜑 → 1 ∈ V )
187	1 66	eleqtrdi	⊢ ( 𝜑 → 𝑁 ∈ ( ℤ_≥ ‘ 1 ) )
188		fzpred	⊢ ( 𝑁 ∈ ( ℤ_≥ ‘ 1 ) → ( 1 ... 𝑁 ) = ( { 1 } ∪ ( ( 1 + 1 ) ... 𝑁 ) ) )
189	187 188	syl	⊢ ( 𝜑 → ( 1 ... 𝑁 ) = ( { 1 } ∪ ( ( 1 + 1 ) ... 𝑁 ) ) )
190		uncom	⊢ ( { 1 } ∪ ( ( 1 + 1 ) ... 𝑁 ) ) = ( ( ( 1 + 1 ) ... 𝑁 ) ∪ { 1 } )
191	189 190	eqtr2di	⊢ ( 𝜑 → ( ( ( 1 + 1 ) ... 𝑁 ) ∪ { 1 } ) = ( 1 ... 𝑁 ) )
192		iftrue	⊢ ( 𝑛 = 1 → if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) = 𝑁 )
193	192	adantl	⊢ ( ( 𝜑 ∧ 𝑛 = 1 ) → if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) = 𝑁 )
194	186 1 191 193	fmptapd	⊢ ( 𝜑 → ( ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ∪ { ⟨ 1 , 𝑁 ⟩ } ) = ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) )
195	185 194	eqtr3id	⊢ ( 𝜑 → ( ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) ∪ { ⟨ 1 , 𝑁 ⟩ } ) = ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) )
196	71 187	eqeltrd	⊢ ( 𝜑 → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ_≥ ‘ 1 ) )
197		uzid	⊢ ( ( 𝑁 − 1 ) ∈ ℤ → ( 𝑁 − 1 ) ∈ ( ℤ_≥ ‘ ( 𝑁 − 1 ) ) )
198		peano2uz	⊢ ( ( 𝑁 − 1 ) ∈ ( ℤ_≥ ‘ ( 𝑁 − 1 ) ) → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ_≥ ‘ ( 𝑁 − 1 ) ) )
199	130 197 198	3syl	⊢ ( 𝜑 → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ_≥ ‘ ( 𝑁 − 1 ) ) )
200	71 199	eqeltrrd	⊢ ( 𝜑 → 𝑁 ∈ ( ℤ_≥ ‘ ( 𝑁 − 1 ) ) )
201		fzsplit2	⊢ ( ( ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ_≥ ‘ 1 ) ∧ 𝑁 ∈ ( ℤ_≥ ‘ ( 𝑁 − 1 ) ) ) → ( 1 ... 𝑁 ) = ( ( 1 ... ( 𝑁 − 1 ) ) ∪ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) )
202	196 200 201	syl2anc	⊢ ( 𝜑 → ( 1 ... 𝑁 ) = ( ( 1 ... ( 𝑁 − 1 ) ) ∪ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) )
203	71	oveq1d	⊢ ( 𝜑 → ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) = ( 𝑁 ... 𝑁 ) )
204		fzsn	⊢ ( 𝑁 ∈ ℤ → ( 𝑁 ... 𝑁 ) = { 𝑁 } )
205	114 204	syl	⊢ ( 𝜑 → ( 𝑁 ... 𝑁 ) = { 𝑁 } )
206	203 205	eqtrd	⊢ ( 𝜑 → ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) = { 𝑁 } )
207	206	uneq2d	⊢ ( 𝜑 → ( ( 1 ... ( 𝑁 − 1 ) ) ∪ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) = ( ( 1 ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) )
208	202 207	eqtr2d	⊢ ( 𝜑 → ( ( 1 ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) = ( 1 ... 𝑁 ) )
209	195 191 208	f1oeq123d	⊢ ( 𝜑 → ( ( ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) ∪ { ⟨ 1 , 𝑁 ⟩ } ) : ( ( ( 1 + 1 ) ... 𝑁 ) ∪ { 1 } ) –1-1-onto→ ( ( 1 ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) ↔ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ) )
210	178 209	mpbid	⊢ ( 𝜑 → ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) )
211		f1oco	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) )
212	46 210 211	syl2anc	⊢ ( 𝜑 → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) )
213		dff1o3	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ↔ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) : ( 1 ... 𝑁 ) –onto→ ( 1 ... 𝑁 ) ∧ Fun ^◡ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) ) )
214	213	simprbi	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) → Fun ^◡ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) )
215		imain	⊢ ( Fun ^◡ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( 1 ... ( 𝑦 + 1 ) ) ∩ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∩ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) )
216	212 214 215	3syl	⊢ ( 𝜑 → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( 1 ... ( 𝑦 + 1 ) ) ∩ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∩ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) )
217	65	nnred	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 𝑦 + 1 ) ∈ ℝ )
218	217	ltp1d	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 𝑦 + 1 ) < ( ( 𝑦 + 1 ) + 1 ) )
219		fzdisj	⊢ ( ( 𝑦 + 1 ) < ( ( 𝑦 + 1 ) + 1 ) → ( ( 1 ... ( 𝑦 + 1 ) ) ∩ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) = ∅ )
220	218 219	syl	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 1 ... ( 𝑦 + 1 ) ) ∩ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) = ∅ )
221	220	imaeq2d	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( 1 ... ( 𝑦 + 1 ) ) ∩ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ∅ ) )
222		ima0	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ∅ ) = ∅
223	221 222	eqtrdi	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( 1 ... ( 𝑦 + 1 ) ) ∩ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ∅ )
224	216 223	sylan9req	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∩ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ∅ )
225		fun	⊢ ( ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) : ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ⟶ { 1 } ∧ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) : ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ⟶ { 0 } ) ∧ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∩ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ∅ ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) : ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) ⟶ ( { 1 } ∪ { 0 } ) )
226	112 224 225	sylancr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) : ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) ⟶ ( { 1 } ∪ { 0 } ) )
227		imaundi	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( 1 ... ( 𝑦 + 1 ) ) ∪ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) )
228	65	peano2nnd	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 𝑦 + 1 ) + 1 ) ∈ ℕ )
229	228 66	eleqtrdi	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 𝑦 + 1 ) + 1 ) ∈ ( ℤ_≥ ‘ 1 ) )
230	229	adantl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑦 + 1 ) + 1 ) ∈ ( ℤ_≥ ‘ 1 ) )
231		eluzp1p1	⊢ ( ( 𝑁 − 1 ) ∈ ( ℤ_≥ ‘ 𝑦 ) → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ_≥ ‘ ( 𝑦 + 1 ) ) )
232	73 231	syl	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ_≥ ‘ ( 𝑦 + 1 ) ) )
233	232	adantl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ_≥ ‘ ( 𝑦 + 1 ) ) )
234	72 233	eqeltrrd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 𝑁 ∈ ( ℤ_≥ ‘ ( 𝑦 + 1 ) ) )
235		fzsplit2	⊢ ( ( ( ( 𝑦 + 1 ) + 1 ) ∈ ( ℤ_≥ ‘ 1 ) ∧ 𝑁 ∈ ( ℤ_≥ ‘ ( 𝑦 + 1 ) ) ) → ( 1 ... 𝑁 ) = ( ( 1 ... ( 𝑦 + 1 ) ) ∪ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) )
236	230 234 235	syl2anc	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 1 ... 𝑁 ) = ( ( 1 ... ( 𝑦 + 1 ) ) ∪ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) )
237	236	imaeq2d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... 𝑁 ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( 1 ... ( 𝑦 + 1 ) ) ∪ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) )
238		f1ofo	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) : ( 1 ... 𝑁 ) –onto→ ( 1 ... 𝑁 ) )
239		foima	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) : ( 1 ... 𝑁 ) –onto→ ( 1 ... 𝑁 ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... 𝑁 ) ) = ( 1 ... 𝑁 ) )
240	212 238 239	3syl	⊢ ( 𝜑 → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... 𝑁 ) ) = ( 1 ... 𝑁 ) )
241	240	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... 𝑁 ) ) = ( 1 ... 𝑁 ) )
242	237 241	eqtr3d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( 1 ... ( 𝑦 + 1 ) ) ∪ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ( 1 ... 𝑁 ) )
243	227 242	eqtr3id	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ( 1 ... 𝑁 ) )
244	243	feq2d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) : ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) ⟶ ( { 1 } ∪ { 0 } ) ↔ ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( { 1 } ∪ { 0 } ) ) )
245	226 244	mpbid	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( { 1 } ∪ { 0 } ) )
246	245	ffvelrnda	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ∈ ( { 1 } ∪ { 0 } ) )
247	109 246	sseldi	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ∈ ℂ )
248	100 104 247	subadd23d	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) + ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) = ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) + ( ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) ) )
249		oveq2	⊢ ( 1 = if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) → ( ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − 1 ) = ( ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) )
250	249	eqeq1d	⊢ ( 1 = if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) → ( ( ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − 1 ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ↔ ( ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) )
251		oveq2	⊢ ( 0 = if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) → ( ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − 0 ) = ( ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) )
252	251	eqeq1d	⊢ ( 0 = if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) → ( ( ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − 0 ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ↔ ( ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) )
253		1m1e0	⊢ ( 1 − 1 ) = 0
254		f1ofn	⊢ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) → ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) Fn ( 1 ... 𝑁 ) )
255	46 254	syl	⊢ ( 𝜑 → ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) Fn ( 1 ... 𝑁 ) )
256	255	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) Fn ( 1 ... 𝑁 ) )
257		imassrn	⊢ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ⊆ ran ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) )
258		f1of	⊢ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) → ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) : ( 1 ... 𝑁 ) ⟶ ( 1 ... 𝑁 ) )
259	210 258	syl	⊢ ( 𝜑 → ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) : ( 1 ... 𝑁 ) ⟶ ( 1 ... 𝑁 ) )
260	259	frnd	⊢ ( 𝜑 → ran ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ⊆ ( 1 ... 𝑁 ) )
261	257 260	sstrid	⊢ ( 𝜑 → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ⊆ ( 1 ... 𝑁 ) )
262	261	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ⊆ ( 1 ... 𝑁 ) )
263		eqidd	⊢ ( 𝜑 → ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) = ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) )
264		eluzfz1	⊢ ( 𝑁 ∈ ( ℤ_≥ ‘ 1 ) → 1 ∈ ( 1 ... 𝑁 ) )
265	187 264	syl	⊢ ( 𝜑 → 1 ∈ ( 1 ... 𝑁 ) )
266	263 193 265 1	fvmptd	⊢ ( 𝜑 → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ‘ 1 ) = 𝑁 )
267	266	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ‘ 1 ) = 𝑁 )
268		f1ofn	⊢ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) → ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) Fn ( 1 ... 𝑁 ) )
269	210 268	syl	⊢ ( 𝜑 → ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) Fn ( 1 ... 𝑁 ) )
270	269	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) Fn ( 1 ... 𝑁 ) )
271		fzss2	⊢ ( 𝑁 ∈ ( ℤ_≥ ‘ ( 𝑦 + 1 ) ) → ( 1 ... ( 𝑦 + 1 ) ) ⊆ ( 1 ... 𝑁 ) )
272	234 271	syl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 1 ... ( 𝑦 + 1 ) ) ⊆ ( 1 ... 𝑁 ) )
273		eluzfz1	⊢ ( ( 𝑦 + 1 ) ∈ ( ℤ_≥ ‘ 1 ) → 1 ∈ ( 1 ... ( 𝑦 + 1 ) ) )
274	67 273	syl	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 1 ∈ ( 1 ... ( 𝑦 + 1 ) ) )
275	274	adantl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 1 ∈ ( 1 ... ( 𝑦 + 1 ) ) )
276		fnfvima	⊢ ( ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) Fn ( 1 ... 𝑁 ) ∧ ( 1 ... ( 𝑦 + 1 ) ) ⊆ ( 1 ... 𝑁 ) ∧ 1 ∈ ( 1 ... ( 𝑦 + 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ‘ 1 ) ∈ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) )
277	270 272 275 276	syl3anc	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ‘ 1 ) ∈ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) )
278	267 277	eqeltrrd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 𝑁 ∈ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) )
279		fnfvima	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) Fn ( 1 ... 𝑁 ) ∧ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ⊆ ( 1 ... 𝑁 ) ∧ 𝑁 ∈ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ∈ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ) )
280	256 262 278 279	syl3anc	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ∈ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ) )
281		imaco	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) )
282	280 281	eleqtrrdi	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ∈ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) )
283		fnconstg	⊢ ( 1 ∈ V → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) Fn ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) )
284	34 283	ax-mp	⊢ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) Fn ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) )
285		fnconstg	⊢ ( 0 ∈ V → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) )
286	37 285	ax-mp	⊢ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) )
287		fvun1	⊢ ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) Fn ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∧ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ∧ ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∩ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ∅ ∧ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ∈ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) )
288	284 286 287	mp3an12	⊢ ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∩ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ∅ ∧ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ∈ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) )
289	224 282 288	syl2anc	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) )
290	34	fvconst2	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ∈ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) = 1 )
291	282 290	syl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) = 1 )
292	289 291	eqtrd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) = 1 )
293	292	oveq1d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) − 1 ) = ( 1 − 1 ) )
294		fzss1	⊢ ( ( 𝑦 + 1 ) ∈ ( ℤ_≥ ‘ 1 ) → ( ( 𝑦 + 1 ) ... 𝑁 ) ⊆ ( 1 ... 𝑁 ) )
295	67 294	syl	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 𝑦 + 1 ) ... 𝑁 ) ⊆ ( 1 ... 𝑁 ) )
296	295	adantl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑦 + 1 ) ... 𝑁 ) ⊆ ( 1 ... 𝑁 ) )
297		eluzfz2	⊢ ( 𝑁 ∈ ( ℤ_≥ ‘ ( 𝑦 + 1 ) ) → 𝑁 ∈ ( ( 𝑦 + 1 ) ... 𝑁 ) )
298	234 297	syl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 𝑁 ∈ ( ( 𝑦 + 1 ) ... 𝑁 ) )
299		fnfvima	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) Fn ( 1 ... 𝑁 ) ∧ ( ( 𝑦 + 1 ) ... 𝑁 ) ⊆ ( 1 ... 𝑁 ) ∧ 𝑁 ∈ ( ( 𝑦 + 1 ) ... 𝑁 ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ∈ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) )
300	256 296 298 299	syl3anc	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ∈ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) )
301		fvun2	⊢ ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) Fn ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∧ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ∧ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∩ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) = ∅ ∧ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ∈ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) )
302	36 39 301	mp3an12	⊢ ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∩ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) = ∅ ∧ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ∈ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) )
303	60 300 302	syl2anc	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) )
304	37	fvconst2	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ∈ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) = 0 )
305	300 304	syl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) = 0 )
306	303 305	eqtrd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) = 0 )
307	253 293 306	3eqtr4a	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) − 1 ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) )
308		fveq2	⊢ ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) )
309	308	oveq1d	⊢ ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) → ( ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − 1 ) = ( ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) − 1 ) )
310		fveq2	⊢ ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) )
311	309 310	eqeq12d	⊢ ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) → ( ( ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − 1 ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ↔ ( ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) − 1 ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) ) )
312	307 311	syl5ibrcom	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) → ( ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − 1 ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) )
313	312	imp	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) → ( ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − 1 ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) )
314	313	adantlr	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) → ( ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − 1 ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) )
315	247	subid1d	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − 0 ) = ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) )
316	315	adantr	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) ∧ ¬ 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) → ( ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − 0 ) = ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) )
317		eldifsn	⊢ ( 𝑛 ∈ ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ↔ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑛 ≠ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) )
318		df-ne	⊢ ( 𝑛 ≠ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ↔ ¬ 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) )
319	318	anbi2i	⊢ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑛 ≠ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) ↔ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ ¬ 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) )
320	317 319	bitri	⊢ ( 𝑛 ∈ ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ↔ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ ¬ 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) )
321		fnconstg	⊢ ( 0 ∈ V → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) Fn ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) )
322	37 321	ax-mp	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) Fn ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) )
323	36 322	pm3.2i	⊢ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) Fn ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∧ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) Fn ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) )
324		imain	⊢ ( Fun ^◡ ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑦 ) ∩ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∩ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) )
325	49 324	syl	⊢ ( 𝜑 → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑦 ) ∩ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∩ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) )
326		fzdisj	⊢ ( 𝑦 < ( 𝑦 + 1 ) → ( ( 1 ... 𝑦 ) ∩ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) = ∅ )
327	54 326	syl	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 1 ... 𝑦 ) ∩ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) = ∅ )
328	327	imaeq2d	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑦 ) ∩ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ∅ ) )
329	328 58	eqtrdi	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑦 ) ∩ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) = ∅ )
330	325 329	sylan9req	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∩ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) = ∅ )
331		fnun	⊢ ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) Fn ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∧ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) Fn ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) ∧ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∩ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) = ∅ ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) Fn ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∪ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) )
332	323 330 331	sylancr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) Fn ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∪ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) )
333		imaundi	⊢ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑦 ) ∪ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∪ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) )
334	202 207	eqtrd	⊢ ( 𝜑 → ( 1 ... 𝑁 ) = ( ( 1 ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) )
335	334	difeq1d	⊢ ( 𝜑 → ( ( 1 ... 𝑁 ) ∖ { 𝑁 } ) = ( ( ( 1 ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) ∖ { 𝑁 } ) )
336		difun2	⊢ ( ( ( 1 ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) ∖ { 𝑁 } ) = ( ( 1 ... ( 𝑁 − 1 ) ) ∖ { 𝑁 } )
337	335 336	eqtrdi	⊢ ( 𝜑 → ( ( 1 ... 𝑁 ) ∖ { 𝑁 } ) = ( ( 1 ... ( 𝑁 − 1 ) ) ∖ { 𝑁 } ) )
338		difsn	⊢ ( ¬ 𝑁 ∈ ( 1 ... ( 𝑁 − 1 ) ) → ( ( 1 ... ( 𝑁 − 1 ) ) ∖ { 𝑁 } ) = ( 1 ... ( 𝑁 − 1 ) ) )
339	164 338	syl	⊢ ( 𝜑 → ( ( 1 ... ( 𝑁 − 1 ) ) ∖ { 𝑁 } ) = ( 1 ... ( 𝑁 − 1 ) ) )
340	337 339	eqtrd	⊢ ( 𝜑 → ( ( 1 ... 𝑁 ) ∖ { 𝑁 } ) = ( 1 ... ( 𝑁 − 1 ) ) )
341	340	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 1 ... 𝑁 ) ∖ { 𝑁 } ) = ( 1 ... ( 𝑁 − 1 ) ) )
342	73	adantl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑁 − 1 ) ∈ ( ℤ_≥ ‘ 𝑦 ) )
343		fzsplit2	⊢ ( ( ( 𝑦 + 1 ) ∈ ( ℤ_≥ ‘ 1 ) ∧ ( 𝑁 − 1 ) ∈ ( ℤ_≥ ‘ 𝑦 ) ) → ( 1 ... ( 𝑁 − 1 ) ) = ( ( 1 ... 𝑦 ) ∪ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) )
344	68 342 343	syl2anc	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 1 ... ( 𝑁 − 1 ) ) = ( ( 1 ... 𝑦 ) ∪ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) )
345	341 344	eqtrd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 1 ... 𝑁 ) ∖ { 𝑁 } ) = ( ( 1 ... 𝑦 ) ∪ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) )
346	345	imaeq2d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑁 ) ∖ { 𝑁 } ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑦 ) ∪ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) )
347		imadif	⊢ ( Fun ^◡ ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑁 ) ∖ { 𝑁 } ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑁 ) ) ∖ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ { 𝑁 } ) ) )
348	49 347	syl	⊢ ( 𝜑 → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑁 ) ∖ { 𝑁 } ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑁 ) ) ∖ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ { 𝑁 } ) ) )
349		elfz1end	⊢ ( 𝑁 ∈ ℕ ↔ 𝑁 ∈ ( 1 ... 𝑁 ) )
350	1 349	sylib	⊢ ( 𝜑 → 𝑁 ∈ ( 1 ... 𝑁 ) )
351		fnsnfv	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) Fn ( 1 ... 𝑁 ) ∧ 𝑁 ∈ ( 1 ... 𝑁 ) ) → { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ { 𝑁 } ) )
352	255 350 351	syl2anc	⊢ ( 𝜑 → { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ { 𝑁 } ) )
353	352	eqcomd	⊢ ( 𝜑 → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ { 𝑁 } ) = { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } )
354	83 353	difeq12d	⊢ ( 𝜑 → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑁 ) ) ∖ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ { 𝑁 } ) ) = ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) )
355	348 354	eqtrd	⊢ ( 𝜑 → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑁 ) ∖ { 𝑁 } ) ) = ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) )
356	355	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑁 ) ∖ { 𝑁 } ) ) = ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) )
357	346 356	eqtr3d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑦 ) ∪ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) = ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) )
358	333 357	eqtr3id	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∪ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) = ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) )
359	358	fneq2d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) Fn ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∪ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) ↔ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) Fn ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ) )
360	332 359	mpbid	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) Fn ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) )
361		disjdifr	⊢ ( ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ∩ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) = ∅
362		fnconstg	⊢ ( 1 ∈ V → ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 1 } ) Fn { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } )
363	34 362	ax-mp	⊢ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 1 } ) Fn { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) }
364		fvun1	⊢ ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) Fn ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ∧ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 1 } ) Fn { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ∧ ( ( ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ∩ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) = ∅ ∧ 𝑛 ∈ ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 1 } ) ) ‘ 𝑛 ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) )
365	363 364	mp3an2	⊢ ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) Fn ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ∧ ( ( ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ∩ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) = ∅ ∧ 𝑛 ∈ ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 1 } ) ) ‘ 𝑛 ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) )
366		fnconstg	⊢ ( 0 ∈ V → ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 0 } ) Fn { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } )
367	37 366	ax-mp	⊢ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 0 } ) Fn { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) }
368		fvun1	⊢ ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) Fn ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ∧ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 0 } ) Fn { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ∧ ( ( ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ∩ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) = ∅ ∧ 𝑛 ∈ ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) )
369	367 368	mp3an2	⊢ ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) Fn ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ∧ ( ( ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ∩ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) = ∅ ∧ 𝑛 ∈ ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) )
370	365 369	eqtr4d	⊢ ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) Fn ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ∧ ( ( ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ∩ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) = ∅ ∧ 𝑛 ∈ ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 1 } ) ) ‘ 𝑛 ) = ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 0 } ) ) ‘ 𝑛 ) )
371	361 370	mpanr1	⊢ ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) Fn ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ∧ 𝑛 ∈ ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 1 } ) ) ‘ 𝑛 ) = ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 0 } ) ) ‘ 𝑛 ) )
372	360 371	sylan	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 1 } ) ) ‘ 𝑛 ) = ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 0 } ) ) ‘ 𝑛 ) )
373	320 372	sylan2br	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ ¬ 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 1 } ) ) ‘ 𝑛 ) = ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 0 } ) ) ‘ 𝑛 ) )
374	373	anassrs	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) ∧ ¬ 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 1 } ) ) ‘ 𝑛 ) = ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 0 } ) ) ‘ 𝑛 ) )
375		imaundi	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ∪ { 1 } ) ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ { 1 } ) )
376		imaco	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) )
377		imaco	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ { 1 } ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ { 1 } ) )
378	376 377	uneq12i	⊢ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ { 1 } ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ) ∪ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ { 1 } ) ) )
379	375 378	eqtri	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ∪ { 1 } ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ) ∪ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ { 1 } ) ) )
380		fzpred	⊢ ( ( 𝑦 + 1 ) ∈ ( ℤ_≥ ‘ 1 ) → ( 1 ... ( 𝑦 + 1 ) ) = ( { 1 } ∪ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) )
381	67 380	syl	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 1 ... ( 𝑦 + 1 ) ) = ( { 1 } ∪ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) )
382		uncom	⊢ ( { 1 } ∪ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) = ( ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ∪ { 1 } )
383	381 382	eqtrdi	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 1 ... ( 𝑦 + 1 ) ) = ( ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ∪ { 1 } ) )
384	383	imaeq2d	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ∪ { 1 } ) ) )
385	384	adantl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ∪ { 1 } ) ) )
386		elfzelz	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 𝑦 ∈ ℤ )
387	124	a1i	⊢ ( 𝑦 ∈ ℤ → ( ( 1 + 1 ) − 1 ) = 1 )
388		zcn	⊢ ( 𝑦 ∈ ℤ → 𝑦 ∈ ℂ )
389		pncan1	⊢ ( 𝑦 ∈ ℂ → ( ( 𝑦 + 1 ) − 1 ) = 𝑦 )
390	388 389	syl	⊢ ( 𝑦 ∈ ℤ → ( ( 𝑦 + 1 ) − 1 ) = 𝑦 )
391	387 390	oveq12d	⊢ ( 𝑦 ∈ ℤ → ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) = ( 1 ... 𝑦 ) )
392		elfzelz	⊢ ( 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) → 𝑗 ∈ ℤ )
393	392	zcnd	⊢ ( 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) → 𝑗 ∈ ℂ )
394		pncan1	⊢ ( 𝑗 ∈ ℂ → ( ( 𝑗 + 1 ) − 1 ) = 𝑗 )
395	393 394	syl	⊢ ( 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) → ( ( 𝑗 + 1 ) − 1 ) = 𝑗 )
396	395	eleq1d	⊢ ( 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) → ( ( ( 𝑗 + 1 ) − 1 ) ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) ↔ 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) ) )
397	396	ibir	⊢ ( 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) → ( ( 𝑗 + 1 ) − 1 ) ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) )
398	397	adantl	⊢ ( ( 𝑦 ∈ ℤ ∧ 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) ) → ( ( 𝑗 + 1 ) − 1 ) ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) )
399		peano2z	⊢ ( 𝑦 ∈ ℤ → ( 𝑦 + 1 ) ∈ ℤ )
400	399 117	jctil	⊢ ( 𝑦 ∈ ℤ → ( ( 1 + 1 ) ∈ ℤ ∧ ( 𝑦 + 1 ) ∈ ℤ ) )
401	392	peano2zd	⊢ ( 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) → ( 𝑗 + 1 ) ∈ ℤ )
402	401 115	jctir	⊢ ( 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) → ( ( 𝑗 + 1 ) ∈ ℤ ∧ 1 ∈ ℤ ) )
403		fzsubel	⊢ ( ( ( ( 1 + 1 ) ∈ ℤ ∧ ( 𝑦 + 1 ) ∈ ℤ ) ∧ ( ( 𝑗 + 1 ) ∈ ℤ ∧ 1 ∈ ℤ ) ) → ( ( 𝑗 + 1 ) ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↔ ( ( 𝑗 + 1 ) − 1 ) ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) ) )
404	400 402 403	syl2an	⊢ ( ( 𝑦 ∈ ℤ ∧ 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) ) → ( ( 𝑗 + 1 ) ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↔ ( ( 𝑗 + 1 ) − 1 ) ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) ) )
405	398 404	mpbird	⊢ ( ( 𝑦 ∈ ℤ ∧ 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) ) → ( 𝑗 + 1 ) ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) )
406	395	eqcomd	⊢ ( 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) → 𝑗 = ( ( 𝑗 + 1 ) − 1 ) )
407	406	adantl	⊢ ( ( 𝑦 ∈ ℤ ∧ 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) ) → 𝑗 = ( ( 𝑗 + 1 ) − 1 ) )
408		oveq1	⊢ ( 𝑛 = ( 𝑗 + 1 ) → ( 𝑛 − 1 ) = ( ( 𝑗 + 1 ) − 1 ) )
409	408	rspceeqv	⊢ ( ( ( 𝑗 + 1 ) ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ∧ 𝑗 = ( ( 𝑗 + 1 ) − 1 ) ) → ∃ 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) 𝑗 = ( 𝑛 − 1 ) )
410	405 407 409	syl2anc	⊢ ( ( 𝑦 ∈ ℤ ∧ 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) ) → ∃ 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) 𝑗 = ( 𝑛 − 1 ) )
411	410	ex	⊢ ( 𝑦 ∈ ℤ → ( 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) → ∃ 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) 𝑗 = ( 𝑛 − 1 ) ) )
412		simpr	⊢ ( ( 𝑦 ∈ ℤ ∧ 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) → 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) )
413		elfzelz	⊢ ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) → 𝑛 ∈ ℤ )
414	413 115	jctir	⊢ ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) → ( 𝑛 ∈ ℤ ∧ 1 ∈ ℤ ) )
415		fzsubel	⊢ ( ( ( ( 1 + 1 ) ∈ ℤ ∧ ( 𝑦 + 1 ) ∈ ℤ ) ∧ ( 𝑛 ∈ ℤ ∧ 1 ∈ ℤ ) ) → ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↔ ( 𝑛 − 1 ) ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) ) )
416	400 414 415	syl2an	⊢ ( ( 𝑦 ∈ ℤ ∧ 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) → ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↔ ( 𝑛 − 1 ) ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) ) )
417	412 416	mpbid	⊢ ( ( 𝑦 ∈ ℤ ∧ 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) → ( 𝑛 − 1 ) ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) )
418		eleq1	⊢ ( 𝑗 = ( 𝑛 − 1 ) → ( 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) ↔ ( 𝑛 − 1 ) ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) ) )
419	417 418	syl5ibrcom	⊢ ( ( 𝑦 ∈ ℤ ∧ 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) → ( 𝑗 = ( 𝑛 − 1 ) → 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) ) )
420	419	rexlimdva	⊢ ( 𝑦 ∈ ℤ → ( ∃ 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) 𝑗 = ( 𝑛 − 1 ) → 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) ) )
421	411 420	impbid	⊢ ( 𝑦 ∈ ℤ → ( 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) ↔ ∃ 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) 𝑗 = ( 𝑛 − 1 ) ) )
422		eqid	⊢ ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↦ ( 𝑛 − 1 ) ) = ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↦ ( 𝑛 − 1 ) )
423	422	elrnmpt	⊢ ( 𝑗 ∈ V → ( 𝑗 ∈ ran ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↦ ( 𝑛 − 1 ) ) ↔ ∃ 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) 𝑗 = ( 𝑛 − 1 ) ) )
424	423	elv	⊢ ( 𝑗 ∈ ran ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↦ ( 𝑛 − 1 ) ) ↔ ∃ 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) 𝑗 = ( 𝑛 − 1 ) )
425	421 424	bitr4di	⊢ ( 𝑦 ∈ ℤ → ( 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) ↔ 𝑗 ∈ ran ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↦ ( 𝑛 − 1 ) ) ) )
426	425	eqrdv	⊢ ( 𝑦 ∈ ℤ → ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) = ran ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↦ ( 𝑛 − 1 ) ) )
427	391 426	eqtr3d	⊢ ( 𝑦 ∈ ℤ → ( 1 ... 𝑦 ) = ran ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↦ ( 𝑛 − 1 ) ) )
428	386 427	syl	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 1 ... 𝑦 ) = ran ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↦ ( 𝑛 − 1 ) ) )
429	428	adantl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 1 ... 𝑦 ) = ran ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↦ ( 𝑛 − 1 ) ) )
430		df-ima	⊢ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) = ran ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ↾ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) )
431		uzid	⊢ ( 1 ∈ ℤ → 1 ∈ ( ℤ_≥ ‘ 1 ) )
432		peano2uz	⊢ ( 1 ∈ ( ℤ_≥ ‘ 1 ) → ( 1 + 1 ) ∈ ( ℤ_≥ ‘ 1 ) )
433	115 431 432	mp2b	⊢ ( 1 + 1 ) ∈ ( ℤ_≥ ‘ 1 )
434		fzss1	⊢ ( ( 1 + 1 ) ∈ ( ℤ_≥ ‘ 1 ) → ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ⊆ ( 1 ... ( 𝑦 + 1 ) ) )
435	433 434	ax-mp	⊢ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ⊆ ( 1 ... ( 𝑦 + 1 ) )
436	435 272	sstrid	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ⊆ ( 1 ... 𝑁 ) )
437	436	resmptd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ↾ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) = ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) )
438		elfzle1	⊢ ( 1 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) → ( 1 + 1 ) ≤ 1 )
439	171 438	mto	⊢ ¬ 1 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) )
440		eleq1	⊢ ( 𝑛 = 1 → ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↔ 1 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) )
441	439 440	mtbiri	⊢ ( 𝑛 = 1 → ¬ 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) )
442	441	necon2ai	⊢ ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) → 𝑛 ≠ 1 )
443	442 182	syl	⊢ ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) → if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) = ( 𝑛 − 1 ) )
444	443	mpteq2ia	⊢ ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) = ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↦ ( 𝑛 − 1 ) )
445	437 444	eqtrdi	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ↾ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) = ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↦ ( 𝑛 − 1 ) ) )
446	445	rneqd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ran ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ↾ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) = ran ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↦ ( 𝑛 − 1 ) ) )
447	430 446	syl5eq	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) = ran ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↦ ( 𝑛 − 1 ) ) )
448	429 447	eqtr4d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 1 ... 𝑦 ) = ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) )
449	448	imaeq2d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ) )
450	266	sneqd	⊢ ( 𝜑 → { ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ‘ 1 ) } = { 𝑁 } )
451		fnsnfv	⊢ ( ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) Fn ( 1 ... 𝑁 ) ∧ 1 ∈ ( 1 ... 𝑁 ) ) → { ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ‘ 1 ) } = ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ { 1 } ) )
452	269 265 451	syl2anc	⊢ ( 𝜑 → { ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ‘ 1 ) } = ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ { 1 } ) )
453	450 452	eqtr3d	⊢ ( 𝜑 → { 𝑁 } = ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ { 1 } ) )
454	453	imaeq2d	⊢ ( 𝜑 → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ { 𝑁 } ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ { 1 } ) ) )
455	352 454	eqtrd	⊢ ( 𝜑 → { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ { 1 } ) ) )
456	455	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ { 1 } ) ) )
457	449 456	uneq12d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∪ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ) ∪ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ { 1 } ) ) ) )
458	379 385 457	3eqtr4a	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∪ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) )
459	458	xpeq1d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∪ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) × { 1 } ) )
460		xpundir	⊢ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∪ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) × { 1 } ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 1 } ) )
461	459 460	eqtrdi	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 1 } ) ) )
462		imaco	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) )
463		df-ima	⊢ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) = ran ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ↾ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) )
464		fzss1	⊢ ( ( ( 𝑦 + 1 ) + 1 ) ∈ ( ℤ_≥ ‘ 1 ) → ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ⊆ ( 1 ... 𝑁 ) )
465	230 464	syl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ⊆ ( 1 ... 𝑁 ) )
466	465	resmptd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ↾ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) = ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) )
467		1red	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 1 ∈ ℝ )
468	65	nnzd	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 𝑦 + 1 ) ∈ ℤ )
469	468	peano2zd	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 𝑦 + 1 ) + 1 ) ∈ ℤ )
470	469	zred	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 𝑦 + 1 ) + 1 ) ∈ ℝ )
471	65	nnge1d	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 1 ≤ ( 𝑦 + 1 ) )
472	467 217 470 471 218	lelttrd	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 1 < ( ( 𝑦 + 1 ) + 1 ) )
473	467 470	ltnled	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 1 < ( ( 𝑦 + 1 ) + 1 ) ↔ ¬ ( ( 𝑦 + 1 ) + 1 ) ≤ 1 ) )
474	472 473	mpbid	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ¬ ( ( 𝑦 + 1 ) + 1 ) ≤ 1 )
475		elfzle1	⊢ ( 1 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) → ( ( 𝑦 + 1 ) + 1 ) ≤ 1 )
476	474 475	nsyl	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ¬ 1 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) )
477		eleq1	⊢ ( 𝑛 = 1 → ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↔ 1 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) )
478	477	notbid	⊢ ( 𝑛 = 1 → ( ¬ 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↔ ¬ 1 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) )
479	476 478	syl5ibrcom	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 𝑛 = 1 → ¬ 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) )
480	479	necon2ad	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) → 𝑛 ≠ 1 ) )
481	480	imp	⊢ ( ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ∧ 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) → 𝑛 ≠ 1 )
482	481 182	syl	⊢ ( ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ∧ 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) → if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) = ( 𝑛 − 1 ) )
483	482	mpteq2dva	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) = ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) )
484	483	adantl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) = ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) )
485	466 484	eqtrd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ↾ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) = ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) )
486	485	rneqd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ran ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ↾ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) = ran ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) )
487	463 486	syl5eq	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) = ran ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) )
488		eqid	⊢ ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) = ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) )
489	488	elrnmpt	⊢ ( 𝑗 ∈ V → ( 𝑗 ∈ ran ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) ↔ ∃ 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) 𝑗 = ( 𝑛 − 1 ) ) )
490	489	elv	⊢ ( 𝑗 ∈ ran ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) ↔ ∃ 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) 𝑗 = ( 𝑛 − 1 ) )
491		simpr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) → 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) )
492	114 469	anim12ci	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 𝑦 + 1 ) + 1 ) ∈ ℤ ∧ 𝑁 ∈ ℤ ) )
493		elfzelz	⊢ ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) → 𝑛 ∈ ℤ )
494	493 115	jctir	⊢ ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) → ( 𝑛 ∈ ℤ ∧ 1 ∈ ℤ ) )
495		fzsubel	⊢ ( ( ( ( ( 𝑦 + 1 ) + 1 ) ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑛 ∈ ℤ ∧ 1 ∈ ℤ ) ) → ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↔ ( 𝑛 − 1 ) ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) )
496	492 494 495	syl2an	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) → ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↔ ( 𝑛 − 1 ) ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) )
497	491 496	mpbid	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) → ( 𝑛 − 1 ) ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) )
498		eleq1	⊢ ( 𝑗 = ( 𝑛 − 1 ) → ( 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ↔ ( 𝑛 − 1 ) ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) )
499	497 498	syl5ibrcom	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) → ( 𝑗 = ( 𝑛 − 1 ) → 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) )
500	499	rexlimdva	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ∃ 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) 𝑗 = ( 𝑛 − 1 ) → 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) )
501		elfzelz	⊢ ( 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) → 𝑗 ∈ ℤ )
502	501	zcnd	⊢ ( 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) → 𝑗 ∈ ℂ )
503	502 394	syl	⊢ ( 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) → ( ( 𝑗 + 1 ) − 1 ) = 𝑗 )
504	503	eleq1d	⊢ ( 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) → ( ( ( 𝑗 + 1 ) − 1 ) ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ↔ 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) )
505	504	ibir	⊢ ( 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) → ( ( 𝑗 + 1 ) − 1 ) ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) )
506	505	adantl	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) → ( ( 𝑗 + 1 ) − 1 ) ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) )
507	501	peano2zd	⊢ ( 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) → ( 𝑗 + 1 ) ∈ ℤ )
508	507 115	jctir	⊢ ( 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) → ( ( 𝑗 + 1 ) ∈ ℤ ∧ 1 ∈ ℤ ) )
509		fzsubel	⊢ ( ( ( ( ( 𝑦 + 1 ) + 1 ) ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( ( 𝑗 + 1 ) ∈ ℤ ∧ 1 ∈ ℤ ) ) → ( ( 𝑗 + 1 ) ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↔ ( ( 𝑗 + 1 ) − 1 ) ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) )
510	492 508 509	syl2an	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) → ( ( 𝑗 + 1 ) ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↔ ( ( 𝑗 + 1 ) − 1 ) ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) )
511	506 510	mpbird	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) → ( 𝑗 + 1 ) ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) )
512	503	eqcomd	⊢ ( 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) → 𝑗 = ( ( 𝑗 + 1 ) − 1 ) )
513	512	adantl	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) → 𝑗 = ( ( 𝑗 + 1 ) − 1 ) )
514	408	rspceeqv	⊢ ( ( ( 𝑗 + 1 ) ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ∧ 𝑗 = ( ( 𝑗 + 1 ) − 1 ) ) → ∃ 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) 𝑗 = ( 𝑛 − 1 ) )
515	511 513 514	syl2anc	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) → ∃ 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) 𝑗 = ( 𝑛 − 1 ) )
516	515	ex	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) → ∃ 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) 𝑗 = ( 𝑛 − 1 ) ) )
517	500 516	impbid	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ∃ 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) 𝑗 = ( 𝑛 − 1 ) ↔ 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) )
518	490 517	syl5bb	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑗 ∈ ran ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) ↔ 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) )
519	518	eqrdv	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ran ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) = ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) )
520	65	nncnd	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 𝑦 + 1 ) ∈ ℂ )
521		pncan1	⊢ ( ( 𝑦 + 1 ) ∈ ℂ → ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) = ( 𝑦 + 1 ) )
522	520 521	syl	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) = ( 𝑦 + 1 ) )
523	522	oveq1d	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) = ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) )
524	523	adantl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) = ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) )
525	487 519 524	3eqtrd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) = ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) )
526	525	imaeq2d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) )
527	462 526	syl5eq	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) )
528	527	xpeq1d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) )
529	461 528	uneq12d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 1 } ) ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) )
530		un23	⊢ ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 1 } ) ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 1 } ) )
531	529 530	eqtrdi	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 1 } ) ) )
532	531	fveq1d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 1 } ) ) ‘ 𝑛 ) )
533	532	ad2antrr	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) ∧ ¬ 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 1 } ) ) ‘ 𝑛 ) )
534		imaundi	⊢ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ∪ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ { 𝑁 } ) )
535		fzsplit2	⊢ ( ( ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ_≥ ‘ ( 𝑦 + 1 ) ) ∧ 𝑁 ∈ ( ℤ_≥ ‘ ( 𝑁 − 1 ) ) ) → ( ( 𝑦 + 1 ) ... 𝑁 ) = ( ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ∪ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) )
536	232 200 535	syl2anr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑦 + 1 ) ... 𝑁 ) = ( ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ∪ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) )
537	206	uneq2d	⊢ ( 𝜑 → ( ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ∪ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) = ( ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) )
538	537	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ∪ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) = ( ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) )
539	536 538	eqtrd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑦 + 1 ) ... 𝑁 ) = ( ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) )
540	539	imaeq2d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) ) )
541	352	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ { 𝑁 } ) )
542	541	uneq2d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ∪ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ∪ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ { 𝑁 } ) ) )
543	534 540 542	3eqtr4a	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ∪ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) )
544	543	xpeq1d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ∪ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) × { 0 } ) )
545		xpundir	⊢ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ∪ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) × { 0 } ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 0 } ) )
546	544 545	eqtrdi	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 0 } ) ) )
547	546	uneq2d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 0 } ) ) ) )
548		unass	⊢ ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 0 } ) ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 0 } ) ) )
549	547 548	eqtr4di	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 0 } ) ) )
550	549	fveq1d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 0 } ) ) ‘ 𝑛 ) )
551	550	ad2antrr	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) ∧ ¬ 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 0 } ) ) ‘ 𝑛 ) )
552	374 533 551	3eqtr4d	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) ∧ ¬ 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) )
553	316 552	eqtrd	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) ∧ ¬ 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) ) → ( ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − 0 ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) )
554	250 252 314 553	ifbothda	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) )
555	554	oveq2d	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) + ( ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) ) = ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) )
556	248 555	eqtr2d	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) = ( ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) + ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) )
557	556	mpteq2dva	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) = ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) + ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) )
558	93 557	eqtrd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) + ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) )
559	53	adantl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 𝑦 ∈ ℝ )
560	160	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑁 − 1 ) ∈ ℝ )
561	158	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 𝑁 ∈ ℝ )
562		elfzle2	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 𝑦 ≤ ( 𝑁 − 1 ) )
563	562	adantl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 𝑦 ≤ ( 𝑁 − 1 ) )
564	159	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑁 − 1 ) < 𝑁 )
565	559 560 561 563 564	lelttrd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 𝑦 < 𝑁 )
566	6	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 2^nd ‘ 𝑇 ) = 𝑁 )
567	565 566	breqtrrd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 𝑦 < ( 2^nd ‘ 𝑇 ) )
568	567	iftrued	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) = 𝑦 )
569	568	csbeq1d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ⦋ 𝑦 / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
570		vex	⊢ 𝑦 ∈ V
571		oveq2	⊢ ( 𝑗 = 𝑦 → ( 1 ... 𝑗 ) = ( 1 ... 𝑦 ) )
572	571	imaeq2d	⊢ ( 𝑗 = 𝑦 → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) )
573	572	xpeq1d	⊢ ( 𝑗 = 𝑦 → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) )
574		oveq1	⊢ ( 𝑗 = 𝑦 → ( 𝑗 + 1 ) = ( 𝑦 + 1 ) )
575	574	oveq1d	⊢ ( 𝑗 = 𝑦 → ( ( 𝑗 + 1 ) ... 𝑁 ) = ( ( 𝑦 + 1 ) ... 𝑁 ) )
576	575	imaeq2d	⊢ ( 𝑗 = 𝑦 → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) )
577	576	xpeq1d	⊢ ( 𝑗 = 𝑦 → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) )
578	573 577	uneq12d	⊢ ( 𝑗 = 𝑦 → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) )
579	578	oveq2d	⊢ ( 𝑗 = 𝑦 → ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
580	570 579	csbie	⊢ ⦋ 𝑦 / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) )
581	569 580	eqtrdi	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
582		ovexd	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) ∈ V )
583		fvexd	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ∈ V )
584		eqidd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) ) = ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) ) )
585	245	ffnd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( 1 ... 𝑁 ) )
586		nfcv	⊢ Ⅎ 𝑛 ( 2^nd ‘ ( 1^st ‘ 𝑇 ) )
587		nfmpt1	⊢ Ⅎ 𝑛 ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) )
588	586 587	nfco	⊢ Ⅎ 𝑛 ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) )
589		nfcv	⊢ Ⅎ 𝑛 ( 1 ... ( 𝑦 + 1 ) )
590	588 589	nfima	⊢ Ⅎ 𝑛 ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) )
591		nfcv	⊢ Ⅎ 𝑛 { 1 }
592	590 591	nfxp	⊢ Ⅎ 𝑛 ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } )
593		nfcv	⊢ Ⅎ 𝑛 ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 )
594	588 593	nfima	⊢ Ⅎ 𝑛 ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) )
595		nfcv	⊢ Ⅎ 𝑛 { 0 }
596	594 595	nfxp	⊢ Ⅎ 𝑛 ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } )
597	592 596	nfun	⊢ Ⅎ 𝑛 ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) )
598	597	dffn5f	⊢ ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( 1 ... 𝑁 ) ↔ ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) )
599	585 598	sylib	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) )
600	89 582 583 584 599	offval2	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) + ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) )
601	558 581 600	3eqtr4rd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
602	601	mpteq2dva	⊢ ( 𝜑 → ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) )
603	23 602	eqtr4d	⊢ ( 𝜑 → 𝐹 = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) )

Metamath Proof Explorer

Theorem poimirlem19

Proof