poimirlem16

Description: Lemma for poimir establishing the vertices of the simplex of poimirlem17 . (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 )
	poimirlem18.3	⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ∃ 𝑝 ∈ ran 𝐹 ( 𝑝 ‘ 𝑛 ) ≠ 𝐾 )
	poimirlem18.4	⊢ ( 𝜑 → ( 2^nd ‘ 𝑇 ) = 0 )
Assertion	poimirlem16	⊢ ( 𝜑 → 𝐹 = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) + if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) , 1 , 0 ) ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 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		poimirlem18.3	⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ∃ 𝑝 ∈ ran 𝐹 ( 𝑝 ‘ 𝑛 ) ≠ 𝐾 )
6		poimirlem18.4	⊢ ( 𝜑 → ( 2^nd ‘ 𝑇 ) = 0 )
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 ) ) ) × { 1 } ) Fn ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) )
36	34 35	ax-mp	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) Fn ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) )
37		c0ex	⊢ 0 ∈ V
38		fnconstg	⊢ ( 0 ∈ V → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) )
39	37 38	ax-mp	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) )
40	36 39	pm3.2i	⊢ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) Fn ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∧ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 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 + 1 ) ) ∩ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∩ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) )
51	49 50	syl	⊢ ( 𝜑 → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 ... ( 𝑦 + 1 ) ) ∩ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∩ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) )
52		elfznn0	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 𝑦 ∈ ℕ₀ )
53		nn0p1nn	⊢ ( 𝑦 ∈ ℕ₀ → ( 𝑦 + 1 ) ∈ ℕ )
54	52 53	syl	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 𝑦 + 1 ) ∈ ℕ )
55	54	nnred	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 𝑦 + 1 ) ∈ ℝ )
56	55	ltp1d	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 𝑦 + 1 ) < ( ( 𝑦 + 1 ) + 1 ) )
57		fzdisj	⊢ ( ( 𝑦 + 1 ) < ( ( 𝑦 + 1 ) + 1 ) → ( ( 1 ... ( 𝑦 + 1 ) ) ∩ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) = ∅ )
58	56 57	syl	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 1 ... ( 𝑦 + 1 ) ) ∩ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) = ∅ )
59	58	imaeq2d	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 ... ( 𝑦 + 1 ) ) ∩ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ∅ ) )
60		ima0	⊢ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ∅ ) = ∅
61	59 60	eqtrdi	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 ... ( 𝑦 + 1 ) ) ∩ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ∅ )
62	51 61	sylan9req	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∩ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ∅ )
63		fnun	⊢ ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) Fn ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∧ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) ∧ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∩ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ∅ ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∪ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) )
64	40 62 63	sylancr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∪ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) )
65		imaundi	⊢ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 ... ( 𝑦 + 1 ) ) ∪ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∪ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) )
66		nnuz	⊢ ℕ = ( ℤ_≥ ‘ 1 )
67	54 66	eleqtrdi	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 𝑦 + 1 ) ∈ ( ℤ_≥ ‘ 1 ) )
68		peano2uz	⊢ ( ( 𝑦 + 1 ) ∈ ( ℤ_≥ ‘ 1 ) → ( ( 𝑦 + 1 ) + 1 ) ∈ ( ℤ_≥ ‘ 1 ) )
69	67 68	syl	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 𝑦 + 1 ) + 1 ) ∈ ( ℤ_≥ ‘ 1 ) )
70	1	nncnd	⊢ ( 𝜑 → 𝑁 ∈ ℂ )
71		npcan1	⊢ ( 𝑁 ∈ ℂ → ( ( 𝑁 − 1 ) + 1 ) = 𝑁 )
72	70 71	syl	⊢ ( 𝜑 → ( ( 𝑁 − 1 ) + 1 ) = 𝑁 )
73	72	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑁 − 1 ) + 1 ) = 𝑁 )
74		elfzuz3	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 𝑁 − 1 ) ∈ ( ℤ_≥ ‘ 𝑦 ) )
75		eluzp1p1	⊢ ( ( 𝑁 − 1 ) ∈ ( ℤ_≥ ‘ 𝑦 ) → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ_≥ ‘ ( 𝑦 + 1 ) ) )
76	74 75	syl	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ_≥ ‘ ( 𝑦 + 1 ) ) )
77	76	adantl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ_≥ ‘ ( 𝑦 + 1 ) ) )
78	73 77	eqeltrrd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 𝑁 ∈ ( ℤ_≥ ‘ ( 𝑦 + 1 ) ) )
79		fzsplit2	⊢ ( ( ( ( 𝑦 + 1 ) + 1 ) ∈ ( ℤ_≥ ‘ 1 ) ∧ 𝑁 ∈ ( ℤ_≥ ‘ ( 𝑦 + 1 ) ) ) → ( 1 ... 𝑁 ) = ( ( 1 ... ( 𝑦 + 1 ) ) ∪ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) )
80	69 78 79	syl2an2	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 1 ... 𝑁 ) = ( ( 1 ... ( 𝑦 + 1 ) ) ∪ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) )
81	80	imaeq2d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑁 ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 ... ( 𝑦 + 1 ) ) ∪ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) )
82		f1ofo	⊢ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) → ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –onto→ ( 1 ... 𝑁 ) )
83		foima	⊢ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –onto→ ( 1 ... 𝑁 ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑁 ) ) = ( 1 ... 𝑁 ) )
84	46 82 83	3syl	⊢ ( 𝜑 → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑁 ) ) = ( 1 ... 𝑁 ) )
85	84	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑁 ) ) = ( 1 ... 𝑁 ) )
86	81 85	eqtr3d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 ... ( 𝑦 + 1 ) ) ∪ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ( 1 ... 𝑁 ) )
87	65 86	eqtr3id	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∪ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ( 1 ... 𝑁 ) )
88	87	fneq2d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∪ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) ↔ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( 1 ... 𝑁 ) ) )
89	64 88	mpbid	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( 1 ... 𝑁 ) )
90		ovexd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 1 ... 𝑁 ) ∈ V )
91		inidm	⊢ ( ( 1 ... 𝑁 ) ∩ ( 1 ... 𝑁 ) ) = ( 1 ... 𝑁 )
92		eqidd	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) = ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) )
93		eqidd	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) )
94	33 89 90 90 91 92 93	offval	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) )
95		oveq1	⊢ ( 1 = if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) , 1 , 0 ) → ( 1 + ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) = ( if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) , 1 , 0 ) + ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) )
96	95	eqeq2d	⊢ ( 1 = if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) , 1 , 0 ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) = ( 1 + ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) ↔ ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) = ( if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) , 1 , 0 ) + ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) )
97		oveq1	⊢ ( 0 = if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) , 1 , 0 ) → ( 0 + ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) = ( if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) , 1 , 0 ) + ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) )
98	97	eqeq2d	⊢ ( 0 = if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) , 1 , 0 ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) = ( 0 + ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) ↔ ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) = ( if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) , 1 , 0 ) + ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) )
99		1p0e1	⊢ ( 1 + 0 ) = 1
100	99	eqcomi	⊢ 1 = ( 1 + 0 )
101		f1ofn	⊢ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) → ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) Fn ( 1 ... 𝑁 ) )
102	46 101	syl	⊢ ( 𝜑 → ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) Fn ( 1 ... 𝑁 ) )
103	102	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) Fn ( 1 ... 𝑁 ) )
104		fzss2	⊢ ( 𝑁 ∈ ( ℤ_≥ ‘ ( 𝑦 + 1 ) ) → ( 1 ... ( 𝑦 + 1 ) ) ⊆ ( 1 ... 𝑁 ) )
105	78 104	syl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 1 ... ( 𝑦 + 1 ) ) ⊆ ( 1 ... 𝑁 ) )
106		eluzfz1	⊢ ( ( 𝑦 + 1 ) ∈ ( ℤ_≥ ‘ 1 ) → 1 ∈ ( 1 ... ( 𝑦 + 1 ) ) )
107	67 106	syl	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 1 ∈ ( 1 ... ( 𝑦 + 1 ) ) )
108	107	adantl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 1 ∈ ( 1 ... ( 𝑦 + 1 ) ) )
109		fnfvima	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) Fn ( 1 ... 𝑁 ) ∧ ( 1 ... ( 𝑦 + 1 ) ) ⊆ ( 1 ... 𝑁 ) ∧ 1 ∈ ( 1 ... ( 𝑦 + 1 ) ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ∈ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) )
110	103 105 108 109	syl3anc	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ∈ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) )
111		fvun1	⊢ ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) Fn ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∧ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ∧ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∩ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ∅ ∧ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ∈ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) )
112	36 39 111	mp3an12	⊢ ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∩ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ∅ ∧ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ∈ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) )
113	62 110 112	syl2anc	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) )
114	34	fvconst2	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ∈ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) = 1 )
115	110 114	syl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) = 1 )
116	113 115	eqtrd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) = 1 )
117		simpr	⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) ) → 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) )
118	1	nnzd	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
119		peano2zm	⊢ ( 𝑁 ∈ ℤ → ( 𝑁 − 1 ) ∈ ℤ )
120	118 119	syl	⊢ ( 𝜑 → ( 𝑁 − 1 ) ∈ ℤ )
121		1z	⊢ 1 ∈ ℤ
122	120 121	jctil	⊢ ( 𝜑 → ( 1 ∈ ℤ ∧ ( 𝑁 − 1 ) ∈ ℤ ) )
123		elfzelz	⊢ ( 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) → 𝑛 ∈ ℤ )
124	123 121	jctir	⊢ ( 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) → ( 𝑛 ∈ ℤ ∧ 1 ∈ ℤ ) )
125		fzaddel	⊢ ( ( ( 1 ∈ ℤ ∧ ( 𝑁 − 1 ) ∈ ℤ ) ∧ ( 𝑛 ∈ ℤ ∧ 1 ∈ ℤ ) ) → ( 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) ↔ ( 𝑛 + 1 ) ∈ ( ( 1 + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) ) )
126	122 124 125	syl2an	⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) ) → ( 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) ↔ ( 𝑛 + 1 ) ∈ ( ( 1 + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) ) )
127	117 126	mpbid	⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) ) → ( 𝑛 + 1 ) ∈ ( ( 1 + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) )
128	72	oveq2d	⊢ ( 𝜑 → ( ( 1 + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) = ( ( 1 + 1 ) ... 𝑁 ) )
129	128	adantr	⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) ) → ( ( 1 + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) = ( ( 1 + 1 ) ... 𝑁 ) )
130	127 129	eleqtrd	⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) ) → ( 𝑛 + 1 ) ∈ ( ( 1 + 1 ) ... 𝑁 ) )
131	130	ralrimiva	⊢ ( 𝜑 → ∀ 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) ( 𝑛 + 1 ) ∈ ( ( 1 + 1 ) ... 𝑁 ) )
132		simpr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( 1 + 1 ) ... 𝑁 ) ) → 𝑦 ∈ ( ( 1 + 1 ) ... 𝑁 ) )
133		peano2z	⊢ ( 1 ∈ ℤ → ( 1 + 1 ) ∈ ℤ )
134	121 133	ax-mp	⊢ ( 1 + 1 ) ∈ ℤ
135	118 134	jctil	⊢ ( 𝜑 → ( ( 1 + 1 ) ∈ ℤ ∧ 𝑁 ∈ ℤ ) )
136		elfzelz	⊢ ( 𝑦 ∈ ( ( 1 + 1 ) ... 𝑁 ) → 𝑦 ∈ ℤ )
137	136 121	jctir	⊢ ( 𝑦 ∈ ( ( 1 + 1 ) ... 𝑁 ) → ( 𝑦 ∈ ℤ ∧ 1 ∈ ℤ ) )
138		fzsubel	⊢ ( ( ( ( 1 + 1 ) ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑦 ∈ ℤ ∧ 1 ∈ ℤ ) ) → ( 𝑦 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↔ ( 𝑦 − 1 ) ∈ ( ( ( 1 + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) )
139	135 137 138	syl2an	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( 1 + 1 ) ... 𝑁 ) ) → ( 𝑦 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↔ ( 𝑦 − 1 ) ∈ ( ( ( 1 + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) )
140	132 139	mpbid	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( 1 + 1 ) ... 𝑁 ) ) → ( 𝑦 − 1 ) ∈ ( ( ( 1 + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) )
141		ax-1cn	⊢ 1 ∈ ℂ
142	141 141	pncan3oi	⊢ ( ( 1 + 1 ) − 1 ) = 1
143	142	oveq1i	⊢ ( ( ( 1 + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) = ( 1 ... ( 𝑁 − 1 ) )
144	140 143	eleqtrdi	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( 1 + 1 ) ... 𝑁 ) ) → ( 𝑦 − 1 ) ∈ ( 1 ... ( 𝑁 − 1 ) ) )
145	136	zcnd	⊢ ( 𝑦 ∈ ( ( 1 + 1 ) ... 𝑁 ) → 𝑦 ∈ ℂ )
146		elfznn	⊢ ( 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) → 𝑛 ∈ ℕ )
147	146	nncnd	⊢ ( 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) → 𝑛 ∈ ℂ )
148		subadd2	⊢ ( ( 𝑦 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑛 ∈ ℂ ) → ( ( 𝑦 − 1 ) = 𝑛 ↔ ( 𝑛 + 1 ) = 𝑦 ) )
149	141 148	mp3an2	⊢ ( ( 𝑦 ∈ ℂ ∧ 𝑛 ∈ ℂ ) → ( ( 𝑦 − 1 ) = 𝑛 ↔ ( 𝑛 + 1 ) = 𝑦 ) )
150	149	bicomd	⊢ ( ( 𝑦 ∈ ℂ ∧ 𝑛 ∈ ℂ ) → ( ( 𝑛 + 1 ) = 𝑦 ↔ ( 𝑦 − 1 ) = 𝑛 ) )
151		eqcom	⊢ ( ( 𝑛 + 1 ) = 𝑦 ↔ 𝑦 = ( 𝑛 + 1 ) )
152		eqcom	⊢ ( ( 𝑦 − 1 ) = 𝑛 ↔ 𝑛 = ( 𝑦 − 1 ) )
153	150 151 152	3bitr3g	⊢ ( ( 𝑦 ∈ ℂ ∧ 𝑛 ∈ ℂ ) → ( 𝑦 = ( 𝑛 + 1 ) ↔ 𝑛 = ( 𝑦 − 1 ) ) )
154	145 147 153	syl2an	⊢ ( ( 𝑦 ∈ ( ( 1 + 1 ) ... 𝑁 ) ∧ 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) ) → ( 𝑦 = ( 𝑛 + 1 ) ↔ 𝑛 = ( 𝑦 − 1 ) ) )
155	154	ralrimiva	⊢ ( 𝑦 ∈ ( ( 1 + 1 ) ... 𝑁 ) → ∀ 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) ( 𝑦 = ( 𝑛 + 1 ) ↔ 𝑛 = ( 𝑦 − 1 ) ) )
156	155	adantl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( 1 + 1 ) ... 𝑁 ) ) → ∀ 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) ( 𝑦 = ( 𝑛 + 1 ) ↔ 𝑛 = ( 𝑦 − 1 ) ) )
157		reu6i	⊢ ( ( ( 𝑦 − 1 ) ∈ ( 1 ... ( 𝑁 − 1 ) ) ∧ ∀ 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) ( 𝑦 = ( 𝑛 + 1 ) ↔ 𝑛 = ( 𝑦 − 1 ) ) ) → ∃! 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) 𝑦 = ( 𝑛 + 1 ) )
158	144 156 157	syl2anc	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( 1 + 1 ) ... 𝑁 ) ) → ∃! 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) 𝑦 = ( 𝑛 + 1 ) )
159	158	ralrimiva	⊢ ( 𝜑 → ∀ 𝑦 ∈ ( ( 1 + 1 ) ... 𝑁 ) ∃! 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) 𝑦 = ( 𝑛 + 1 ) )
160		eqid	⊢ ( 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) ↦ ( 𝑛 + 1 ) ) = ( 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) ↦ ( 𝑛 + 1 ) )
161	160	f1ompt	⊢ ( ( 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) ↦ ( 𝑛 + 1 ) ) : ( 1 ... ( 𝑁 − 1 ) ) –1-1-onto→ ( ( 1 + 1 ) ... 𝑁 ) ↔ ( ∀ 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) ( 𝑛 + 1 ) ∈ ( ( 1 + 1 ) ... 𝑁 ) ∧ ∀ 𝑦 ∈ ( ( 1 + 1 ) ... 𝑁 ) ∃! 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) 𝑦 = ( 𝑛 + 1 ) ) )
162	131 159 161	sylanbrc	⊢ ( 𝜑 → ( 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) ↦ ( 𝑛 + 1 ) ) : ( 1 ... ( 𝑁 − 1 ) ) –1-1-onto→ ( ( 1 + 1 ) ... 𝑁 ) )
163		f1osng	⊢ ( ( 𝑁 ∈ ℕ ∧ 1 ∈ V ) → { ⟨ 𝑁 , 1 ⟩ } : { 𝑁 } –1-1-onto→ { 1 } )
164	1 34 163	sylancl	⊢ ( 𝜑 → { ⟨ 𝑁 , 1 ⟩ } : { 𝑁 } –1-1-onto→ { 1 } )
165	1	nnred	⊢ ( 𝜑 → 𝑁 ∈ ℝ )
166	165	ltm1d	⊢ ( 𝜑 → ( 𝑁 − 1 ) < 𝑁 )
167	120	zred	⊢ ( 𝜑 → ( 𝑁 − 1 ) ∈ ℝ )
168	167 165	ltnled	⊢ ( 𝜑 → ( ( 𝑁 − 1 ) < 𝑁 ↔ ¬ 𝑁 ≤ ( 𝑁 − 1 ) ) )
169	166 168	mpbid	⊢ ( 𝜑 → ¬ 𝑁 ≤ ( 𝑁 − 1 ) )
170		elfzle2	⊢ ( 𝑁 ∈ ( 1 ... ( 𝑁 − 1 ) ) → 𝑁 ≤ ( 𝑁 − 1 ) )
171	169 170	nsyl	⊢ ( 𝜑 → ¬ 𝑁 ∈ ( 1 ... ( 𝑁 − 1 ) ) )
172		disjsn	⊢ ( ( ( 1 ... ( 𝑁 − 1 ) ) ∩ { 𝑁 } ) = ∅ ↔ ¬ 𝑁 ∈ ( 1 ... ( 𝑁 − 1 ) ) )
173	171 172	sylibr	⊢ ( 𝜑 → ( ( 1 ... ( 𝑁 − 1 ) ) ∩ { 𝑁 } ) = ∅ )
174		1re	⊢ 1 ∈ ℝ
175	174	ltp1i	⊢ 1 < ( 1 + 1 )
176	174 174	readdcli	⊢ ( 1 + 1 ) ∈ ℝ
177	174 176	ltnlei	⊢ ( 1 < ( 1 + 1 ) ↔ ¬ ( 1 + 1 ) ≤ 1 )
178	175 177	mpbi	⊢ ¬ ( 1 + 1 ) ≤ 1
179		elfzle1	⊢ ( 1 ∈ ( ( 1 + 1 ) ... 𝑁 ) → ( 1 + 1 ) ≤ 1 )
180	178 179	mto	⊢ ¬ 1 ∈ ( ( 1 + 1 ) ... 𝑁 )
181		disjsn	⊢ ( ( ( ( 1 + 1 ) ... 𝑁 ) ∩ { 1 } ) = ∅ ↔ ¬ 1 ∈ ( ( 1 + 1 ) ... 𝑁 ) )
182	180 181	mpbir	⊢ ( ( ( 1 + 1 ) ... 𝑁 ) ∩ { 1 } ) = ∅
183		f1oun	⊢ ( ( ( ( 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) ↦ ( 𝑛 + 1 ) ) : ( 1 ... ( 𝑁 − 1 ) ) –1-1-onto→ ( ( 1 + 1 ) ... 𝑁 ) ∧ { ⟨ 𝑁 , 1 ⟩ } : { 𝑁 } –1-1-onto→ { 1 } ) ∧ ( ( ( 1 ... ( 𝑁 − 1 ) ) ∩ { 𝑁 } ) = ∅ ∧ ( ( ( 1 + 1 ) ... 𝑁 ) ∩ { 1 } ) = ∅ ) ) → ( ( 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) ↦ ( 𝑛 + 1 ) ) ∪ { ⟨ 𝑁 , 1 ⟩ } ) : ( ( 1 ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) –1-1-onto→ ( ( ( 1 + 1 ) ... 𝑁 ) ∪ { 1 } ) )
184	182 183	mpanr2	⊢ ( ( ( ( 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) ↦ ( 𝑛 + 1 ) ) : ( 1 ... ( 𝑁 − 1 ) ) –1-1-onto→ ( ( 1 + 1 ) ... 𝑁 ) ∧ { ⟨ 𝑁 , 1 ⟩ } : { 𝑁 } –1-1-onto→ { 1 } ) ∧ ( ( 1 ... ( 𝑁 − 1 ) ) ∩ { 𝑁 } ) = ∅ ) → ( ( 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) ↦ ( 𝑛 + 1 ) ) ∪ { ⟨ 𝑁 , 1 ⟩ } ) : ( ( 1 ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) –1-1-onto→ ( ( ( 1 + 1 ) ... 𝑁 ) ∪ { 1 } ) )
185	162 164 173 184	syl21anc	⊢ ( 𝜑 → ( ( 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) ↦ ( 𝑛 + 1 ) ) ∪ { ⟨ 𝑁 , 1 ⟩ } ) : ( ( 1 ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) –1-1-onto→ ( ( ( 1 + 1 ) ... 𝑁 ) ∪ { 1 } ) )
186	34	a1i	⊢ ( 𝜑 → 1 ∈ V )
187	1 66	eleqtrdi	⊢ ( 𝜑 → 𝑁 ∈ ( ℤ_≥ ‘ 1 ) )
188	72 187	eqeltrd	⊢ ( 𝜑 → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ_≥ ‘ 1 ) )
189		uzid	⊢ ( ( 𝑁 − 1 ) ∈ ℤ → ( 𝑁 − 1 ) ∈ ( ℤ_≥ ‘ ( 𝑁 − 1 ) ) )
190		peano2uz	⊢ ( ( 𝑁 − 1 ) ∈ ( ℤ_≥ ‘ ( 𝑁 − 1 ) ) → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ_≥ ‘ ( 𝑁 − 1 ) ) )
191	120 189 190	3syl	⊢ ( 𝜑 → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ_≥ ‘ ( 𝑁 − 1 ) ) )
192	72 191	eqeltrrd	⊢ ( 𝜑 → 𝑁 ∈ ( ℤ_≥ ‘ ( 𝑁 − 1 ) ) )
193		fzsplit2	⊢ ( ( ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ_≥ ‘ 1 ) ∧ 𝑁 ∈ ( ℤ_≥ ‘ ( 𝑁 − 1 ) ) ) → ( 1 ... 𝑁 ) = ( ( 1 ... ( 𝑁 − 1 ) ) ∪ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) )
194	188 192 193	syl2anc	⊢ ( 𝜑 → ( 1 ... 𝑁 ) = ( ( 1 ... ( 𝑁 − 1 ) ) ∪ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) )
195	72	oveq1d	⊢ ( 𝜑 → ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) = ( 𝑁 ... 𝑁 ) )
196		fzsn	⊢ ( 𝑁 ∈ ℤ → ( 𝑁 ... 𝑁 ) = { 𝑁 } )
197	118 196	syl	⊢ ( 𝜑 → ( 𝑁 ... 𝑁 ) = { 𝑁 } )
198	195 197	eqtrd	⊢ ( 𝜑 → ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) = { 𝑁 } )
199	198	uneq2d	⊢ ( 𝜑 → ( ( 1 ... ( 𝑁 − 1 ) ) ∪ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) = ( ( 1 ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) )
200	194 199	eqtr2d	⊢ ( 𝜑 → ( ( 1 ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) = ( 1 ... 𝑁 ) )
201		iftrue	⊢ ( 𝑛 = 𝑁 → if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) = 1 )
202	201	adantl	⊢ ( ( 𝜑 ∧ 𝑛 = 𝑁 ) → if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) = 1 )
203	1 186 200 202	fmptapd	⊢ ( 𝜑 → ( ( 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ∪ { ⟨ 𝑁 , 1 ⟩ } ) = ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) )
204		eleq1	⊢ ( 𝑛 = 𝑁 → ( 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) ↔ 𝑁 ∈ ( 1 ... ( 𝑁 − 1 ) ) ) )
205	204	notbid	⊢ ( 𝑛 = 𝑁 → ( ¬ 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) ↔ ¬ 𝑁 ∈ ( 1 ... ( 𝑁 − 1 ) ) ) )
206	171 205	syl5ibrcom	⊢ ( 𝜑 → ( 𝑛 = 𝑁 → ¬ 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) ) )
207	206	necon2ad	⊢ ( 𝜑 → ( 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) → 𝑛 ≠ 𝑁 ) )
208	207	imp	⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) ) → 𝑛 ≠ 𝑁 )
209		ifnefalse	⊢ ( 𝑛 ≠ 𝑁 → if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) = ( 𝑛 + 1 ) )
210	208 209	syl	⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) ) → if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) = ( 𝑛 + 1 ) )
211	210	mpteq2dva	⊢ ( 𝜑 → ( 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) = ( 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) ↦ ( 𝑛 + 1 ) ) )
212	211	uneq1d	⊢ ( 𝜑 → ( ( 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ∪ { ⟨ 𝑁 , 1 ⟩ } ) = ( ( 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) ↦ ( 𝑛 + 1 ) ) ∪ { ⟨ 𝑁 , 1 ⟩ } ) )
213	203 212	eqtr3d	⊢ ( 𝜑 → ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) = ( ( 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) ↦ ( 𝑛 + 1 ) ) ∪ { ⟨ 𝑁 , 1 ⟩ } ) )
214	194 199	eqtrd	⊢ ( 𝜑 → ( 1 ... 𝑁 ) = ( ( 1 ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) )
215		uzid	⊢ ( 1 ∈ ℤ → 1 ∈ ( ℤ_≥ ‘ 1 ) )
216		peano2uz	⊢ ( 1 ∈ ( ℤ_≥ ‘ 1 ) → ( 1 + 1 ) ∈ ( ℤ_≥ ‘ 1 ) )
217	121 215 216	mp2b	⊢ ( 1 + 1 ) ∈ ( ℤ_≥ ‘ 1 )
218		fzsplit2	⊢ ( ( ( 1 + 1 ) ∈ ( ℤ_≥ ‘ 1 ) ∧ 𝑁 ∈ ( ℤ_≥ ‘ 1 ) ) → ( 1 ... 𝑁 ) = ( ( 1 ... 1 ) ∪ ( ( 1 + 1 ) ... 𝑁 ) ) )
219	217 187 218	sylancr	⊢ ( 𝜑 → ( 1 ... 𝑁 ) = ( ( 1 ... 1 ) ∪ ( ( 1 + 1 ) ... 𝑁 ) ) )
220		fzsn	⊢ ( 1 ∈ ℤ → ( 1 ... 1 ) = { 1 } )
221	121 220	ax-mp	⊢ ( 1 ... 1 ) = { 1 }
222	221	uneq1i	⊢ ( ( 1 ... 1 ) ∪ ( ( 1 + 1 ) ... 𝑁 ) ) = ( { 1 } ∪ ( ( 1 + 1 ) ... 𝑁 ) )
223	222	equncomi	⊢ ( ( 1 ... 1 ) ∪ ( ( 1 + 1 ) ... 𝑁 ) ) = ( ( ( 1 + 1 ) ... 𝑁 ) ∪ { 1 } )
224	219 223	eqtrdi	⊢ ( 𝜑 → ( 1 ... 𝑁 ) = ( ( ( 1 + 1 ) ... 𝑁 ) ∪ { 1 } ) )
225	213 214 224	f1oeq123d	⊢ ( 𝜑 → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ↔ ( ( 𝑛 ∈ ( 1 ... ( 𝑁 − 1 ) ) ↦ ( 𝑛 + 1 ) ) ∪ { ⟨ 𝑁 , 1 ⟩ } ) : ( ( 1 ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) –1-1-onto→ ( ( ( 1 + 1 ) ... 𝑁 ) ∪ { 1 } ) ) )
226	185 225	mpbird	⊢ ( 𝜑 → ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) )
227		f1oco	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) )
228	46 226 227	syl2anc	⊢ ( 𝜑 → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) )
229		dff1o3	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ↔ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) : ( 1 ... 𝑁 ) –onto→ ( 1 ... 𝑁 ) ∧ Fun ^◡ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) ) )
230	229	simprbi	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) → Fun ^◡ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) )
231	228 230	syl	⊢ ( 𝜑 → Fun ^◡ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) )
232		imain	⊢ ( Fun ^◡ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 1 ... 𝑦 ) ∩ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) ∩ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) )
233	231 232	syl	⊢ ( 𝜑 → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 1 ... 𝑦 ) ∩ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) ∩ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) )
234	52	nn0red	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 𝑦 ∈ ℝ )
235	234	ltp1d	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 𝑦 < ( 𝑦 + 1 ) )
236		fzdisj	⊢ ( 𝑦 < ( 𝑦 + 1 ) → ( ( 1 ... 𝑦 ) ∩ ( ( 𝑦 + 1 ) ... 𝑁 ) ) = ∅ )
237	235 236	syl	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 1 ... 𝑦 ) ∩ ( ( 𝑦 + 1 ) ... 𝑁 ) ) = ∅ )
238	237	imaeq2d	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 1 ... 𝑦 ) ∩ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ∅ ) )
239		ima0	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ∅ ) = ∅
240	238 239	eqtrdi	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 1 ... 𝑦 ) ∩ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) = ∅ )
241	233 240	sylan9req	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) ∩ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) = ∅ )
242		imassrn	⊢ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ⊆ ran ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) )
243		f1of	⊢ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) → ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) : ( 1 ... 𝑁 ) ⟶ ( 1 ... 𝑁 ) )
244		frn	⊢ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) : ( 1 ... 𝑁 ) ⟶ ( 1 ... 𝑁 ) → ran ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ⊆ ( 1 ... 𝑁 ) )
245	226 243 244	3syl	⊢ ( 𝜑 → ran ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ⊆ ( 1 ... 𝑁 ) )
246	242 245	sstrid	⊢ ( 𝜑 → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ⊆ ( 1 ... 𝑁 ) )
247	246	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ⊆ ( 1 ... 𝑁 ) )
248		elfz1end	⊢ ( 𝑁 ∈ ℕ ↔ 𝑁 ∈ ( 1 ... 𝑁 ) )
249	1 248	sylib	⊢ ( 𝜑 → 𝑁 ∈ ( 1 ... 𝑁 ) )
250		eqid	⊢ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) = ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) )
251	201 250 34	fvmpt	⊢ ( 𝑁 ∈ ( 1 ... 𝑁 ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ‘ 𝑁 ) = 1 )
252	249 251	syl	⊢ ( 𝜑 → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ‘ 𝑁 ) = 1 )
253	252	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ‘ 𝑁 ) = 1 )
254		f1ofn	⊢ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) → ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) Fn ( 1 ... 𝑁 ) )
255	226 254	syl	⊢ ( 𝜑 → ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) Fn ( 1 ... 𝑁 ) )
256	255	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) Fn ( 1 ... 𝑁 ) )
257		fzss1	⊢ ( ( 𝑦 + 1 ) ∈ ( ℤ_≥ ‘ 1 ) → ( ( 𝑦 + 1 ) ... 𝑁 ) ⊆ ( 1 ... 𝑁 ) )
258	67 257	syl	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 𝑦 + 1 ) ... 𝑁 ) ⊆ ( 1 ... 𝑁 ) )
259	258	adantl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑦 + 1 ) ... 𝑁 ) ⊆ ( 1 ... 𝑁 ) )
260		eluzfz2	⊢ ( 𝑁 ∈ ( ℤ_≥ ‘ ( 𝑦 + 1 ) ) → 𝑁 ∈ ( ( 𝑦 + 1 ) ... 𝑁 ) )
261	78 260	syl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 𝑁 ∈ ( ( 𝑦 + 1 ) ... 𝑁 ) )
262		fnfvima	⊢ ( ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) Fn ( 1 ... 𝑁 ) ∧ ( ( 𝑦 + 1 ) ... 𝑁 ) ⊆ ( 1 ... 𝑁 ) ∧ 𝑁 ∈ ( ( 𝑦 + 1 ) ... 𝑁 ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ‘ 𝑁 ) ∈ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) )
263	256 259 261 262	syl3anc	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ‘ 𝑁 ) ∈ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) )
264	253 263	eqeltrrd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 1 ∈ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) )
265		fnfvima	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) Fn ( 1 ... 𝑁 ) ∧ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ⊆ ( 1 ... 𝑁 ) ∧ 1 ∈ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ∈ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) )
266	103 247 264 265	syl3anc	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ∈ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) )
267		imaco	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) )
268	266 267	eleqtrrdi	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ∈ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) )
269		fnconstg	⊢ ( 1 ∈ V → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) Fn ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) )
270	34 269	ax-mp	⊢ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) Fn ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) )
271		fnconstg	⊢ ( 0 ∈ V → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) )
272	37 271	ax-mp	⊢ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) )
273		fvun2	⊢ ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) Fn ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) ∧ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ∧ ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) ∩ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) = ∅ ∧ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ∈ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) )
274	270 272 273	mp3an12	⊢ ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) ∩ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) = ∅ ∧ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ∈ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) )
275	241 268 274	syl2anc	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) )
276	37	fvconst2	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ∈ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) = 0 )
277	268 276	syl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) = 0 )
278	275 277	eqtrd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) = 0 )
279	278	oveq2d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 1 + ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) ) = ( 1 + 0 ) )
280	100 116 279	3eqtr4a	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) = ( 1 + ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) ) )
281		fveq2	⊢ ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) )
282		fveq2	⊢ ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) )
283	282	oveq2d	⊢ ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) → ( 1 + ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) = ( 1 + ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) ) )
284	281 283	eqeq12d	⊢ ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) = ( 1 + ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) ↔ ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) = ( 1 + ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) ) ) )
285	280 284	syl5ibrcom	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) = ( 1 + ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) )
286	285	imp	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) = ( 1 + ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) )
287	286	adantlr	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) = ( 1 + ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) )
288		eldifsn	⊢ ( 𝑛 ∈ ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) ↔ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑛 ≠ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) )
289		df-ne	⊢ ( 𝑛 ≠ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ↔ ¬ 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) )
290	289	anbi2i	⊢ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑛 ≠ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) ↔ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ ¬ 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) )
291	288 290	bitri	⊢ ( 𝑛 ∈ ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) ↔ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ ¬ 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) )
292		fnconstg	⊢ ( 1 ∈ V → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) Fn ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) )
293	34 292	ax-mp	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) Fn ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) )
294	293 39	pm3.2i	⊢ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) Fn ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ∧ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) )
295		imain	⊢ ( Fun ^◡ ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ∩ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ∩ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) )
296	49 295	syl	⊢ ( 𝜑 → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ∩ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ∩ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) )
297		fzdisj	⊢ ( ( 𝑦 + 1 ) < ( ( 𝑦 + 1 ) + 1 ) → ( ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ∩ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) = ∅ )
298	56 297	syl	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ∩ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) = ∅ )
299	298	imaeq2d	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ∩ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ∅ ) )
300	299 60	eqtrdi	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ∩ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ∅ )
301	296 300	sylan9req	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ∩ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ∅ )
302		fnun	⊢ ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) Fn ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ∧ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) ∧ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ∩ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ∅ ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ∪ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) )
303	294 301 302	sylancr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ∪ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) )
304		imaundi	⊢ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ∪ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ∪ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) )
305		fzpred	⊢ ( 𝑁 ∈ ( ℤ_≥ ‘ 1 ) → ( 1 ... 𝑁 ) = ( { 1 } ∪ ( ( 1 + 1 ) ... 𝑁 ) ) )
306	187 305	syl	⊢ ( 𝜑 → ( 1 ... 𝑁 ) = ( { 1 } ∪ ( ( 1 + 1 ) ... 𝑁 ) ) )
307		uncom	⊢ ( { 1 } ∪ ( ( 1 + 1 ) ... 𝑁 ) ) = ( ( ( 1 + 1 ) ... 𝑁 ) ∪ { 1 } )
308	306 307	eqtrdi	⊢ ( 𝜑 → ( 1 ... 𝑁 ) = ( ( ( 1 + 1 ) ... 𝑁 ) ∪ { 1 } ) )
309	308	difeq1d	⊢ ( 𝜑 → ( ( 1 ... 𝑁 ) ∖ { 1 } ) = ( ( ( ( 1 + 1 ) ... 𝑁 ) ∪ { 1 } ) ∖ { 1 } ) )
310		difun2	⊢ ( ( ( ( 1 + 1 ) ... 𝑁 ) ∪ { 1 } ) ∖ { 1 } ) = ( ( ( 1 + 1 ) ... 𝑁 ) ∖ { 1 } )
311		disj3	⊢ ( ( ( ( 1 + 1 ) ... 𝑁 ) ∩ { 1 } ) = ∅ ↔ ( ( 1 + 1 ) ... 𝑁 ) = ( ( ( 1 + 1 ) ... 𝑁 ) ∖ { 1 } ) )
312	182 311	mpbi	⊢ ( ( 1 + 1 ) ... 𝑁 ) = ( ( ( 1 + 1 ) ... 𝑁 ) ∖ { 1 } )
313	310 312	eqtr4i	⊢ ( ( ( ( 1 + 1 ) ... 𝑁 ) ∪ { 1 } ) ∖ { 1 } ) = ( ( 1 + 1 ) ... 𝑁 )
314	309 313	eqtrdi	⊢ ( 𝜑 → ( ( 1 ... 𝑁 ) ∖ { 1 } ) = ( ( 1 + 1 ) ... 𝑁 ) )
315	314	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 1 ... 𝑁 ) ∖ { 1 } ) = ( ( 1 + 1 ) ... 𝑁 ) )
316		eluzp1p1	⊢ ( ( 𝑦 + 1 ) ∈ ( ℤ_≥ ‘ 1 ) → ( ( 𝑦 + 1 ) + 1 ) ∈ ( ℤ_≥ ‘ ( 1 + 1 ) ) )
317	67 316	syl	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 𝑦 + 1 ) + 1 ) ∈ ( ℤ_≥ ‘ ( 1 + 1 ) ) )
318		fzsplit2	⊢ ( ( ( ( 𝑦 + 1 ) + 1 ) ∈ ( ℤ_≥ ‘ ( 1 + 1 ) ) ∧ 𝑁 ∈ ( ℤ_≥ ‘ ( 𝑦 + 1 ) ) ) → ( ( 1 + 1 ) ... 𝑁 ) = ( ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ∪ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) )
319	317 78 318	syl2an2	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 1 + 1 ) ... 𝑁 ) = ( ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ∪ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) )
320	315 319	eqtrd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 1 ... 𝑁 ) ∖ { 1 } ) = ( ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ∪ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) )
321	320	imaeq2d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑁 ) ∖ { 1 } ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ∪ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) )
322		imadif	⊢ ( Fun ^◡ ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑁 ) ∖ { 1 } ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑁 ) ) ∖ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ { 1 } ) ) )
323	49 322	syl	⊢ ( 𝜑 → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑁 ) ∖ { 1 } ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑁 ) ) ∖ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ { 1 } ) ) )
324		eluzfz1	⊢ ( 𝑁 ∈ ( ℤ_≥ ‘ 1 ) → 1 ∈ ( 1 ... 𝑁 ) )
325	187 324	syl	⊢ ( 𝜑 → 1 ∈ ( 1 ... 𝑁 ) )
326		fnsnfv	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) Fn ( 1 ... 𝑁 ) ∧ 1 ∈ ( 1 ... 𝑁 ) ) → { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ { 1 } ) )
327	102 325 326	syl2anc	⊢ ( 𝜑 → { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ { 1 } ) )
328	327	eqcomd	⊢ ( 𝜑 → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ { 1 } ) = { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } )
329	84 328	difeq12d	⊢ ( 𝜑 → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑁 ) ) ∖ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ { 1 } ) ) = ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) )
330	323 329	eqtrd	⊢ ( 𝜑 → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑁 ) ∖ { 1 } ) ) = ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) )
331	330	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑁 ) ∖ { 1 } ) ) = ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) )
332	321 331	eqtr3d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ∪ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) )
333	304 332	eqtr3id	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ∪ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) )
334	333	fneq2d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ∪ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) ↔ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) ) )
335	303 334	mpbid	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) )
336		disjdifr	⊢ ( ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) ∩ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) = ∅
337		fnconstg	⊢ ( 1 ∈ V → ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 1 } ) Fn { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } )
338	34 337	ax-mp	⊢ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 1 } ) Fn { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) }
339		fvun1	⊢ ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) ∧ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 1 } ) Fn { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ∧ ( ( ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) ∩ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) = ∅ ∧ 𝑛 ∈ ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 1 } ) ) ‘ 𝑛 ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) )
340	338 339	mp3an2	⊢ ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) ∧ ( ( ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) ∩ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) = ∅ ∧ 𝑛 ∈ ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 1 } ) ) ‘ 𝑛 ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) )
341		fnconstg	⊢ ( 0 ∈ V → ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 0 } ) Fn { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } )
342	37 341	ax-mp	⊢ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 0 } ) Fn { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) }
343		fvun1	⊢ ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) ∧ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 0 } ) Fn { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ∧ ( ( ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) ∩ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) = ∅ ∧ 𝑛 ∈ ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) )
344	342 343	mp3an2	⊢ ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) ∧ ( ( ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) ∩ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) = ∅ ∧ 𝑛 ∈ ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) )
345	340 344	eqtr4d	⊢ ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) ∧ ( ( ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) ∩ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) = ∅ ∧ 𝑛 ∈ ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 1 } ) ) ‘ 𝑛 ) = ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 0 } ) ) ‘ 𝑛 ) )
346	336 345	mpanr1	⊢ ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) ∧ 𝑛 ∈ ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 1 } ) ) ‘ 𝑛 ) = ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 0 } ) ) ‘ 𝑛 ) )
347	335 346	sylan	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( ( 1 ... 𝑁 ) ∖ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 1 } ) ) ‘ 𝑛 ) = ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 0 } ) ) ‘ 𝑛 ) )
348	291 347	sylan2br	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ ¬ 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 1 } ) ) ‘ 𝑛 ) = ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 0 } ) ) ‘ 𝑛 ) )
349	348	anassrs	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) ∧ ¬ 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 1 } ) ) ‘ 𝑛 ) = ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 0 } ) ) ‘ 𝑛 ) )
350		fzpred	⊢ ( ( 𝑦 + 1 ) ∈ ( ℤ_≥ ‘ 1 ) → ( 1 ... ( 𝑦 + 1 ) ) = ( { 1 } ∪ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) )
351	67 350	syl	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 1 ... ( 𝑦 + 1 ) ) = ( { 1 } ∪ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) )
352	351	imaeq2d	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( { 1 } ∪ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ) )
353	352	adantl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( { 1 } ∪ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ) )
354	327	uneq1d	⊢ ( 𝜑 → ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ∪ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ { 1 } ) ∪ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ) )
355		uncom	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ∪ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) = ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ∪ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) )
356		imaundi	⊢ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( { 1 } ∪ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ { 1 } ) ∪ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) )
357	354 355 356	3eqtr4g	⊢ ( 𝜑 → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ∪ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( { 1 } ∪ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ) )
358	357	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ∪ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( { 1 } ∪ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ) )
359	353 358	eqtr4d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ∪ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) )
360	359	xpeq1d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ∪ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) × { 1 } ) )
361		xpundir	⊢ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ∪ { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ) × { 1 } ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 1 } ) )
362	360 361	eqtrdi	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 1 } ) ) )
363	362	uneq1d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 1 } ) ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) )
364		un23	⊢ ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 1 } ) ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 1 } ) )
365	363 364	eqtrdi	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 1 } ) ) )
366	365	fveq1d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 1 } ) ) ‘ 𝑛 ) )
367	366	ad2antrr	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) ∧ ¬ 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 1 } ) ) ‘ 𝑛 ) )
368		imaco	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) “ ( 1 ... 𝑦 ) ) )
369		df-ima	⊢ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) “ ( 1 ... 𝑦 ) ) = ran ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ↾ ( 1 ... 𝑦 ) )
370		peano2uz	⊢ ( ( 𝑁 − 1 ) ∈ ( ℤ_≥ ‘ 𝑦 ) → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ_≥ ‘ 𝑦 ) )
371	74 370	syl	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ_≥ ‘ 𝑦 ) )
372	371	adantl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ_≥ ‘ 𝑦 ) )
373	73 372	eqeltrrd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 𝑁 ∈ ( ℤ_≥ ‘ 𝑦 ) )
374		fzss2	⊢ ( 𝑁 ∈ ( ℤ_≥ ‘ 𝑦 ) → ( 1 ... 𝑦 ) ⊆ ( 1 ... 𝑁 ) )
375	373 374	syl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 1 ... 𝑦 ) ⊆ ( 1 ... 𝑁 ) )
376	375	resmptd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ↾ ( 1 ... 𝑦 ) ) = ( 𝑛 ∈ ( 1 ... 𝑦 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) )
377		fzss2	⊢ ( ( 𝑁 − 1 ) ∈ ( ℤ_≥ ‘ 𝑦 ) → ( 1 ... 𝑦 ) ⊆ ( 1 ... ( 𝑁 − 1 ) ) )
378	74 377	syl	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 1 ... 𝑦 ) ⊆ ( 1 ... ( 𝑁 − 1 ) ) )
379	378	adantl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 1 ... 𝑦 ) ⊆ ( 1 ... ( 𝑁 − 1 ) ) )
380	171	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ¬ 𝑁 ∈ ( 1 ... ( 𝑁 − 1 ) ) )
381	379 380	ssneldd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ¬ 𝑁 ∈ ( 1 ... 𝑦 ) )
382		eleq1	⊢ ( 𝑛 = 𝑁 → ( 𝑛 ∈ ( 1 ... 𝑦 ) ↔ 𝑁 ∈ ( 1 ... 𝑦 ) ) )
383	382	notbid	⊢ ( 𝑛 = 𝑁 → ( ¬ 𝑛 ∈ ( 1 ... 𝑦 ) ↔ ¬ 𝑁 ∈ ( 1 ... 𝑦 ) ) )
384	381 383	syl5ibrcom	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑛 = 𝑁 → ¬ 𝑛 ∈ ( 1 ... 𝑦 ) ) )
385	384	necon2ad	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑛 ∈ ( 1 ... 𝑦 ) → 𝑛 ≠ 𝑁 ) )
386	385	imp	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑦 ) ) → 𝑛 ≠ 𝑁 )
387	386 209	syl	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑦 ) ) → if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) = ( 𝑛 + 1 ) )
388	387	mpteq2dva	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑛 ∈ ( 1 ... 𝑦 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) = ( 𝑛 ∈ ( 1 ... 𝑦 ) ↦ ( 𝑛 + 1 ) ) )
389	376 388	eqtrd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ↾ ( 1 ... 𝑦 ) ) = ( 𝑛 ∈ ( 1 ... 𝑦 ) ↦ ( 𝑛 + 1 ) ) )
390	389	rneqd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ran ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ↾ ( 1 ... 𝑦 ) ) = ran ( 𝑛 ∈ ( 1 ... 𝑦 ) ↦ ( 𝑛 + 1 ) ) )
391	369 390	syl5eq	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) “ ( 1 ... 𝑦 ) ) = ran ( 𝑛 ∈ ( 1 ... 𝑦 ) ↦ ( 𝑛 + 1 ) ) )
392		eqid	⊢ ( 𝑛 ∈ ( 1 ... 𝑦 ) ↦ ( 𝑛 + 1 ) ) = ( 𝑛 ∈ ( 1 ... 𝑦 ) ↦ ( 𝑛 + 1 ) )
393	392	elrnmpt	⊢ ( 𝑗 ∈ V → ( 𝑗 ∈ ran ( 𝑛 ∈ ( 1 ... 𝑦 ) ↦ ( 𝑛 + 1 ) ) ↔ ∃ 𝑛 ∈ ( 1 ... 𝑦 ) 𝑗 = ( 𝑛 + 1 ) ) )
394	393	elv	⊢ ( 𝑗 ∈ ran ( 𝑛 ∈ ( 1 ... 𝑦 ) ↦ ( 𝑛 + 1 ) ) ↔ ∃ 𝑛 ∈ ( 1 ... 𝑦 ) 𝑗 = ( 𝑛 + 1 ) )
395		elfzelz	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 𝑦 ∈ ℤ )
396	395	adantl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 𝑦 ∈ ℤ )
397		simpr	⊢ ( ( 𝑦 ∈ ℤ ∧ 𝑛 ∈ ( 1 ... 𝑦 ) ) → 𝑛 ∈ ( 1 ... 𝑦 ) )
398	121	jctl	⊢ ( 𝑦 ∈ ℤ → ( 1 ∈ ℤ ∧ 𝑦 ∈ ℤ ) )
399		elfzelz	⊢ ( 𝑛 ∈ ( 1 ... 𝑦 ) → 𝑛 ∈ ℤ )
400	399 121	jctir	⊢ ( 𝑛 ∈ ( 1 ... 𝑦 ) → ( 𝑛 ∈ ℤ ∧ 1 ∈ ℤ ) )
401		fzaddel	⊢ ( ( ( 1 ∈ ℤ ∧ 𝑦 ∈ ℤ ) ∧ ( 𝑛 ∈ ℤ ∧ 1 ∈ ℤ ) ) → ( 𝑛 ∈ ( 1 ... 𝑦 ) ↔ ( 𝑛 + 1 ) ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) )
402	398 400 401	syl2an	⊢ ( ( 𝑦 ∈ ℤ ∧ 𝑛 ∈ ( 1 ... 𝑦 ) ) → ( 𝑛 ∈ ( 1 ... 𝑦 ) ↔ ( 𝑛 + 1 ) ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) )
403	397 402	mpbid	⊢ ( ( 𝑦 ∈ ℤ ∧ 𝑛 ∈ ( 1 ... 𝑦 ) ) → ( 𝑛 + 1 ) ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) )
404		eleq1	⊢ ( 𝑗 = ( 𝑛 + 1 ) → ( 𝑗 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↔ ( 𝑛 + 1 ) ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) )
405	403 404	syl5ibrcom	⊢ ( ( 𝑦 ∈ ℤ ∧ 𝑛 ∈ ( 1 ... 𝑦 ) ) → ( 𝑗 = ( 𝑛 + 1 ) → 𝑗 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) )
406	405	rexlimdva	⊢ ( 𝑦 ∈ ℤ → ( ∃ 𝑛 ∈ ( 1 ... 𝑦 ) 𝑗 = ( 𝑛 + 1 ) → 𝑗 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) )
407		elfzelz	⊢ ( 𝑗 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) → 𝑗 ∈ ℤ )
408	407	zcnd	⊢ ( 𝑗 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) → 𝑗 ∈ ℂ )
409		npcan1	⊢ ( 𝑗 ∈ ℂ → ( ( 𝑗 − 1 ) + 1 ) = 𝑗 )
410	408 409	syl	⊢ ( 𝑗 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) → ( ( 𝑗 − 1 ) + 1 ) = 𝑗 )
411	410	eleq1d	⊢ ( 𝑗 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) → ( ( ( 𝑗 − 1 ) + 1 ) ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↔ 𝑗 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) )
412	411	ibir	⊢ ( 𝑗 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) → ( ( 𝑗 − 1 ) + 1 ) ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) )
413	412	adantl	⊢ ( ( 𝑦 ∈ ℤ ∧ 𝑗 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) → ( ( 𝑗 − 1 ) + 1 ) ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) )
414		peano2zm	⊢ ( 𝑗 ∈ ℤ → ( 𝑗 − 1 ) ∈ ℤ )
415	407 414	syl	⊢ ( 𝑗 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) → ( 𝑗 − 1 ) ∈ ℤ )
416	415 121	jctir	⊢ ( 𝑗 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) → ( ( 𝑗 − 1 ) ∈ ℤ ∧ 1 ∈ ℤ ) )
417		fzaddel	⊢ ( ( ( 1 ∈ ℤ ∧ 𝑦 ∈ ℤ ) ∧ ( ( 𝑗 − 1 ) ∈ ℤ ∧ 1 ∈ ℤ ) ) → ( ( 𝑗 − 1 ) ∈ ( 1 ... 𝑦 ) ↔ ( ( 𝑗 − 1 ) + 1 ) ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) )
418	398 416 417	syl2an	⊢ ( ( 𝑦 ∈ ℤ ∧ 𝑗 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) → ( ( 𝑗 − 1 ) ∈ ( 1 ... 𝑦 ) ↔ ( ( 𝑗 − 1 ) + 1 ) ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) )
419	413 418	mpbird	⊢ ( ( 𝑦 ∈ ℤ ∧ 𝑗 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) → ( 𝑗 − 1 ) ∈ ( 1 ... 𝑦 ) )
420	408	adantl	⊢ ( ( 𝑦 ∈ ℤ ∧ 𝑗 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) → 𝑗 ∈ ℂ )
421	409	eqcomd	⊢ ( 𝑗 ∈ ℂ → 𝑗 = ( ( 𝑗 − 1 ) + 1 ) )
422	420 421	syl	⊢ ( ( 𝑦 ∈ ℤ ∧ 𝑗 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) → 𝑗 = ( ( 𝑗 − 1 ) + 1 ) )
423		oveq1	⊢ ( 𝑛 = ( 𝑗 − 1 ) → ( 𝑛 + 1 ) = ( ( 𝑗 − 1 ) + 1 ) )
424	423	rspceeqv	⊢ ( ( ( 𝑗 − 1 ) ∈ ( 1 ... 𝑦 ) ∧ 𝑗 = ( ( 𝑗 − 1 ) + 1 ) ) → ∃ 𝑛 ∈ ( 1 ... 𝑦 ) 𝑗 = ( 𝑛 + 1 ) )
425	419 422 424	syl2anc	⊢ ( ( 𝑦 ∈ ℤ ∧ 𝑗 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) → ∃ 𝑛 ∈ ( 1 ... 𝑦 ) 𝑗 = ( 𝑛 + 1 ) )
426	425	ex	⊢ ( 𝑦 ∈ ℤ → ( 𝑗 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) → ∃ 𝑛 ∈ ( 1 ... 𝑦 ) 𝑗 = ( 𝑛 + 1 ) ) )
427	406 426	impbid	⊢ ( 𝑦 ∈ ℤ → ( ∃ 𝑛 ∈ ( 1 ... 𝑦 ) 𝑗 = ( 𝑛 + 1 ) ↔ 𝑗 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) )
428	396 427	syl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ∃ 𝑛 ∈ ( 1 ... 𝑦 ) 𝑗 = ( 𝑛 + 1 ) ↔ 𝑗 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) )
429	394 428	syl5bb	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑗 ∈ ran ( 𝑛 ∈ ( 1 ... 𝑦 ) ↦ ( 𝑛 + 1 ) ) ↔ 𝑗 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) )
430	429	eqrdv	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ran ( 𝑛 ∈ ( 1 ... 𝑦 ) ↦ ( 𝑛 + 1 ) ) = ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) )
431	391 430	eqtrd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) “ ( 1 ... 𝑦 ) ) = ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) )
432	431	imaeq2d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) “ ( 1 ... 𝑦 ) ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) )
433	368 432	syl5eq	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) )
434	433	xpeq1d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) )
435		imaundi	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( { 𝑁 } ∪ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ { 𝑁 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) )
436		imaco	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ { 𝑁 } ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) “ { 𝑁 } ) )
437		imaco	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) )
438	436 437	uneq12i	⊢ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ { 𝑁 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) “ { 𝑁 } ) ) ∪ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) )
439	435 438	eqtri	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( { 𝑁 } ∪ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) “ { 𝑁 } ) ) ∪ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) )
440	192	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 𝑁 ∈ ( ℤ_≥ ‘ ( 𝑁 − 1 ) ) )
441		fzsplit2	⊢ ( ( ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ_≥ ‘ ( 𝑦 + 1 ) ) ∧ 𝑁 ∈ ( ℤ_≥ ‘ ( 𝑁 − 1 ) ) ) → ( ( 𝑦 + 1 ) ... 𝑁 ) = ( ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ∪ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) )
442	76 440 441	syl2an2	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑦 + 1 ) ... 𝑁 ) = ( ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ∪ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) )
443	198	uneq2d	⊢ ( 𝜑 → ( ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ∪ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) = ( ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) )
444	443	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ∪ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) = ( ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) )
445	442 444	eqtrd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑦 + 1 ) ... 𝑁 ) = ( ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) )
446		uncom	⊢ ( ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) = ( { 𝑁 } ∪ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) )
447	445 446	eqtrdi	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑦 + 1 ) ... 𝑁 ) = ( { 𝑁 } ∪ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) )
448	447	imaeq2d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( { 𝑁 } ∪ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) )
449	252	sneqd	⊢ ( 𝜑 → { ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ‘ 𝑁 ) } = { 1 } )
450		fnsnfv	⊢ ( ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) Fn ( 1 ... 𝑁 ) ∧ 𝑁 ∈ ( 1 ... 𝑁 ) ) → { ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ‘ 𝑁 ) } = ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) “ { 𝑁 } ) )
451	255 249 450	syl2anc	⊢ ( 𝜑 → { ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ‘ 𝑁 ) } = ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) “ { 𝑁 } ) )
452	449 451	eqtr3d	⊢ ( 𝜑 → { 1 } = ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) “ { 𝑁 } ) )
453	452	imaeq2d	⊢ ( 𝜑 → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ { 1 } ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) “ { 𝑁 } ) ) )
454	327 453	eqtrd	⊢ ( 𝜑 → { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) “ { 𝑁 } ) ) )
455	454	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) “ { 𝑁 } ) ) )
456		df-ima	⊢ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) = ran ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ↾ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) )
457		fzss1	⊢ ( ( 𝑦 + 1 ) ∈ ( ℤ_≥ ‘ 1 ) → ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ⊆ ( 1 ... ( 𝑁 − 1 ) ) )
458	67 457	syl	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ⊆ ( 1 ... ( 𝑁 − 1 ) ) )
459		fzss2	⊢ ( 𝑁 ∈ ( ℤ_≥ ‘ ( 𝑁 − 1 ) ) → ( 1 ... ( 𝑁 − 1 ) ) ⊆ ( 1 ... 𝑁 ) )
460	192 459	syl	⊢ ( 𝜑 → ( 1 ... ( 𝑁 − 1 ) ) ⊆ ( 1 ... 𝑁 ) )
461	458 460	sylan9ssr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ⊆ ( 1 ... 𝑁 ) )
462	461	resmptd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ↾ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) = ( 𝑛 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) )
463		elfzle2	⊢ ( 𝑁 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) → 𝑁 ≤ ( 𝑁 − 1 ) )
464	169 463	nsyl	⊢ ( 𝜑 → ¬ 𝑁 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) )
465		eleq1	⊢ ( 𝑛 = 𝑁 → ( 𝑛 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ↔ 𝑁 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) )
466	465	notbid	⊢ ( 𝑛 = 𝑁 → ( ¬ 𝑛 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ↔ ¬ 𝑁 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) )
467	464 466	syl5ibrcom	⊢ ( 𝜑 → ( 𝑛 = 𝑁 → ¬ 𝑛 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) )
468	467	con2d	⊢ ( 𝜑 → ( 𝑛 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) → ¬ 𝑛 = 𝑁 ) )
469	468	imp	⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) → ¬ 𝑛 = 𝑁 )
470	469	iffalsed	⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) → if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) = ( 𝑛 + 1 ) )
471	470	mpteq2dva	⊢ ( 𝜑 → ( 𝑛 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) = ( 𝑛 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ↦ ( 𝑛 + 1 ) ) )
472	471	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑛 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) = ( 𝑛 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ↦ ( 𝑛 + 1 ) ) )
473	462 472	eqtrd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ↾ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) = ( 𝑛 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ↦ ( 𝑛 + 1 ) ) )
474	473	rneqd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ran ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ↾ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) = ran ( 𝑛 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ↦ ( 𝑛 + 1 ) ) )
475	456 474	syl5eq	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) = ran ( 𝑛 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ↦ ( 𝑛 + 1 ) ) )
476		elfzelz	⊢ ( 𝑗 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) → 𝑗 ∈ ℤ )
477	476	zcnd	⊢ ( 𝑗 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) → 𝑗 ∈ ℂ )
478	477 409	syl	⊢ ( 𝑗 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) → ( ( 𝑗 − 1 ) + 1 ) = 𝑗 )
479	478	eleq1d	⊢ ( 𝑗 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) → ( ( ( 𝑗 − 1 ) + 1 ) ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) ↔ 𝑗 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) ) )
480	479	ibir	⊢ ( 𝑗 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) → ( ( 𝑗 − 1 ) + 1 ) ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) )
481	480	adantl	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑗 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) ) → ( ( 𝑗 − 1 ) + 1 ) ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) )
482	54	nnzd	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 𝑦 + 1 ) ∈ ℤ )
483	120 482	anim12ci	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑦 + 1 ) ∈ ℤ ∧ ( 𝑁 − 1 ) ∈ ℤ ) )
484	476 414	syl	⊢ ( 𝑗 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) → ( 𝑗 − 1 ) ∈ ℤ )
485	484 121	jctir	⊢ ( 𝑗 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) → ( ( 𝑗 − 1 ) ∈ ℤ ∧ 1 ∈ ℤ ) )
486		fzaddel	⊢ ( ( ( ( 𝑦 + 1 ) ∈ ℤ ∧ ( 𝑁 − 1 ) ∈ ℤ ) ∧ ( ( 𝑗 − 1 ) ∈ ℤ ∧ 1 ∈ ℤ ) ) → ( ( 𝑗 − 1 ) ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ↔ ( ( 𝑗 − 1 ) + 1 ) ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) ) )
487	483 485 486	syl2an	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑗 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) ) → ( ( 𝑗 − 1 ) ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ↔ ( ( 𝑗 − 1 ) + 1 ) ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) ) )
488	481 487	mpbird	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑗 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) ) → ( 𝑗 − 1 ) ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) )
489	477 421	syl	⊢ ( 𝑗 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) → 𝑗 = ( ( 𝑗 − 1 ) + 1 ) )
490	489	adantl	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑗 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) ) → 𝑗 = ( ( 𝑗 − 1 ) + 1 ) )
491	423	rspceeqv	⊢ ( ( ( 𝑗 − 1 ) ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ∧ 𝑗 = ( ( 𝑗 − 1 ) + 1 ) ) → ∃ 𝑛 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) 𝑗 = ( 𝑛 + 1 ) )
492	488 490 491	syl2anc	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑗 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) ) → ∃ 𝑛 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) 𝑗 = ( 𝑛 + 1 ) )
493	492	ex	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑗 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) → ∃ 𝑛 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) 𝑗 = ( 𝑛 + 1 ) ) )
494		simpr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) → 𝑛 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) )
495		elfzelz	⊢ ( 𝑛 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) → 𝑛 ∈ ℤ )
496	495 121	jctir	⊢ ( 𝑛 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) → ( 𝑛 ∈ ℤ ∧ 1 ∈ ℤ ) )
497		fzaddel	⊢ ( ( ( ( 𝑦 + 1 ) ∈ ℤ ∧ ( 𝑁 − 1 ) ∈ ℤ ) ∧ ( 𝑛 ∈ ℤ ∧ 1 ∈ ℤ ) ) → ( 𝑛 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ↔ ( 𝑛 + 1 ) ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) ) )
498	483 496 497	syl2an	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) → ( 𝑛 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ↔ ( 𝑛 + 1 ) ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) ) )
499	494 498	mpbid	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) → ( 𝑛 + 1 ) ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) )
500		eleq1	⊢ ( 𝑗 = ( 𝑛 + 1 ) → ( 𝑗 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) ↔ ( 𝑛 + 1 ) ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) ) )
501	499 500	syl5ibrcom	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) → ( 𝑗 = ( 𝑛 + 1 ) → 𝑗 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) ) )
502	501	rexlimdva	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ∃ 𝑛 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) 𝑗 = ( 𝑛 + 1 ) → 𝑗 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) ) )
503	493 502	impbid	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑗 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) ↔ ∃ 𝑛 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) 𝑗 = ( 𝑛 + 1 ) ) )
504		eqid	⊢ ( 𝑛 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ↦ ( 𝑛 + 1 ) ) = ( 𝑛 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ↦ ( 𝑛 + 1 ) )
505	504	elrnmpt	⊢ ( 𝑗 ∈ V → ( 𝑗 ∈ ran ( 𝑛 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ↦ ( 𝑛 + 1 ) ) ↔ ∃ 𝑛 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) 𝑗 = ( 𝑛 + 1 ) ) )
506	505	elv	⊢ ( 𝑗 ∈ ran ( 𝑛 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ↦ ( 𝑛 + 1 ) ) ↔ ∃ 𝑛 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) 𝑗 = ( 𝑛 + 1 ) )
507	503 506	bitr4di	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑗 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) ↔ 𝑗 ∈ ran ( 𝑛 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ↦ ( 𝑛 + 1 ) ) ) )
508	507	eqrdv	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 𝑦 + 1 ) + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) = ran ( 𝑛 ∈ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ↦ ( 𝑛 + 1 ) ) )
509	72	oveq2d	⊢ ( 𝜑 → ( ( ( 𝑦 + 1 ) + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) = ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) )
510	509	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 𝑦 + 1 ) + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) = ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) )
511	475 508 510	3eqtr2rd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) = ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) )
512	511	imaeq2d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) )
513	455 512	uneq12d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ∪ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) “ { 𝑁 } ) ) ∪ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) ) )
514	439 448 513	3eqtr4a	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) = ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ∪ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) )
515	514	xpeq1d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) = ( ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ∪ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) × { 0 } ) )
516		xpundir	⊢ ( ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } ∪ ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) × { 0 } ) = ( ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 0 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) )
517	515 516	eqtrdi	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) = ( ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 0 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) )
518	434 517	uneq12d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 0 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
519		unass	⊢ ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 0 } ) ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 0 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) )
520		un23	⊢ ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 0 } ) ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 0 } ) )
521	519 520	eqtr3i	⊢ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 0 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 0 } ) )
522	518 521	eqtrdi	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 0 } ) ) )
523	522	fveq1d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 0 } ) ) ‘ 𝑛 ) )
524	523	ad2antrr	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) ∧ ¬ 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ∪ ( { ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) } × { 0 } ) ) ‘ 𝑛 ) )
525	349 367 524	3eqtr4d	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) ∧ ¬ 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) )
526		snssi	⊢ ( 1 ∈ ℂ → { 1 } ⊆ ℂ )
527	141 526	ax-mp	⊢ { 1 } ⊆ ℂ
528		0cn	⊢ 0 ∈ ℂ
529		snssi	⊢ ( 0 ∈ ℂ → { 0 } ⊆ ℂ )
530	528 529	ax-mp	⊢ { 0 } ⊆ ℂ
531	527 530	unssi	⊢ ( { 1 } ∪ { 0 } ) ⊆ ℂ
532	34	fconst	⊢ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) : ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) ⟶ { 1 }
533	37	fconst	⊢ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) : ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ⟶ { 0 }
534	532 533	pm3.2i	⊢ ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) : ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) ⟶ { 1 } ∧ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) : ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ⟶ { 0 } )
535		fun	⊢ ( ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) : ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) ⟶ { 1 } ∧ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) : ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ⟶ { 0 } ) ∧ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) ∩ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) = ∅ ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) : ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) ⟶ ( { 1 } ∪ { 0 } ) )
536	534 241 535	sylancr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) : ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) ⟶ ( { 1 } ∪ { 0 } ) )
537		imaundi	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 1 ... 𝑦 ) ∪ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) = ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) )
538		fzsplit2	⊢ ( ( ( 𝑦 + 1 ) ∈ ( ℤ_≥ ‘ 1 ) ∧ 𝑁 ∈ ( ℤ_≥ ‘ 𝑦 ) ) → ( 1 ... 𝑁 ) = ( ( 1 ... 𝑦 ) ∪ ( ( 𝑦 + 1 ) ... 𝑁 ) ) )
539	67 373 538	syl2an2	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 1 ... 𝑁 ) = ( ( 1 ... 𝑦 ) ∪ ( ( 𝑦 + 1 ) ... 𝑁 ) ) )
540	539	imaeq2d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑁 ) ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 1 ... 𝑦 ) ∪ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) )
541		f1ofo	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) : ( 1 ... 𝑁 ) –onto→ ( 1 ... 𝑁 ) )
542		foima	⊢ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) : ( 1 ... 𝑁 ) –onto→ ( 1 ... 𝑁 ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑁 ) ) = ( 1 ... 𝑁 ) )
543	228 541 542	3syl	⊢ ( 𝜑 → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑁 ) ) = ( 1 ... 𝑁 ) )
544	543	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑁 ) ) = ( 1 ... 𝑁 ) )
545	540 544	eqtr3d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 1 ... 𝑦 ) ∪ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) = ( 1 ... 𝑁 ) )
546	537 545	eqtr3id	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) = ( 1 ... 𝑁 ) )
547	546	feq2d	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) : ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) ⟶ ( { 1 } ∪ { 0 } ) ↔ ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( { 1 } ∪ { 0 } ) ) )
548	536 547	mpbid	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( { 1 } ∪ { 0 } ) )
549	548	ffvelrnda	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ∈ ( { 1 } ∪ { 0 } ) )
550	531 549	sselid	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ∈ ℂ )
551	550	addid2d	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( 0 + ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) = ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) )
552	551	adantr	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) ∧ ¬ 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) → ( 0 + ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) = ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) )
553	525 552	eqtr4d	⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) ∧ ¬ 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) = ( 0 + ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) )
554	96 98 287 553	ifbothda	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) = ( if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) , 1 , 0 ) + ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 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 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) = ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) + ( if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) , 1 , 0 ) + ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) )
556		elmapi	⊢ ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∈ ( ( 0 ..^ 𝐾 ) ↑_m ( 1 ... 𝑁 ) ) → ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ..^ 𝐾 ) )
557	30 556	syl	⊢ ( 𝜑 → ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ..^ 𝐾 ) )
558	557	ffvelrnda	⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) ∈ ( 0 ..^ 𝐾 ) )
559		elfzonn0	⊢ ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) ∈ ( 0 ..^ 𝐾 ) → ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) ∈ ℕ₀ )
560	558 559	syl	⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) ∈ ℕ₀ )
561	560	nn0cnd	⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) ∈ ℂ )
562	561	adantlr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) ∈ ℂ )
563	141 528	ifcli	⊢ if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) , 1 , 0 ) ∈ ℂ
564	563	a1i	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) , 1 , 0 ) ∈ ℂ )
565	562 564 550	addassd	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) + if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) , 1 , 0 ) ) + ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) = ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) + ( if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) , 1 , 0 ) + ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) )
566	555 565	eqtr4d	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) = ( ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) + if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) , 1 , 0 ) ) + ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) )
567	566	mpteq2dva	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) = ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) + if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) , 1 , 0 ) ) + ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) )
568	94 567	eqtrd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ∘_f + ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) + if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) , 1 , 0 ) ) + ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) )
569	6	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 2^nd ‘ 𝑇 ) = 0 )
570		elfzle1	⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 0 ≤ 𝑦 )
571	570	adantl	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 0 ≤ 𝑦 )
572	569 571	eqbrtrd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 2^nd ‘ 𝑇 ) ≤ 𝑦 )
573		0re	⊢ 0 ∈ ℝ
574	6 573	eqeltrdi	⊢ ( 𝜑 → ( 2^nd ‘ 𝑇 ) ∈ ℝ )
575		lenlt	⊢ ( ( ( 2^nd ‘ 𝑇 ) ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ↔ ¬ 𝑦 < ( 2^nd ‘ 𝑇 ) ) )
576	574 234 575	syl2an	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2^nd ‘ 𝑇 ) ≤ 𝑦 ↔ ¬ 𝑦 < ( 2^nd ‘ 𝑇 ) ) )
577	572 576	mpbid	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ¬ 𝑦 < ( 2^nd ‘ 𝑇 ) )
578	577	iffalsed	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → if ( 𝑦 < ( 2^nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) = ( 𝑦 + 1 ) )
579	578	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 + 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 } ) ) ) )
580		ovex	⊢ ( 𝑦 + 1 ) ∈ V
581		oveq2	⊢ ( 𝑗 = ( 𝑦 + 1 ) → ( 1 ... 𝑗 ) = ( 1 ... ( 𝑦 + 1 ) ) )
582	581	imaeq2d	⊢ ( 𝑗 = ( 𝑦 + 1 ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) )
583	582	xpeq1d	⊢ ( 𝑗 = ( 𝑦 + 1 ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) )
584		oveq1	⊢ ( 𝑗 = ( 𝑦 + 1 ) → ( 𝑗 + 1 ) = ( ( 𝑦 + 1 ) + 1 ) )
585	584	oveq1d	⊢ ( 𝑗 = ( 𝑦 + 1 ) → ( ( 𝑗 + 1 ) ... 𝑁 ) = ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) )
586	585	imaeq2d	⊢ ( 𝑗 = ( 𝑦 + 1 ) → ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) )
587	586	xpeq1d	⊢ ( 𝑗 = ( 𝑦 + 1 ) → ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) = ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) )
588	583 587	uneq12d	⊢ ( 𝑗 = ( 𝑦 + 1 ) → ( ( ( ( 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 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) )
589	588	oveq2d	⊢ ( 𝑗 = ( 𝑦 + 1 ) → ( ( 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 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
590	580 589	csbie	⊢ ⦋ ( 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 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) )
591	579 590	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 ) ) ) × { 1 } ) ∪ ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) )
592		ovexd	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) + if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) , 1 , 0 ) ) ∈ V )
593		fvexd	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ∈ V )
594		eqidd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) + if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) , 1 , 0 ) ) ) = ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) + if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) , 1 , 0 ) ) ) )
595	548	ffnd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( 1 ... 𝑁 ) )
596		nfcv	⊢ Ⅎ 𝑛 ( 2^nd ‘ ( 1^st ‘ 𝑇 ) )
597		nfmpt1	⊢ Ⅎ 𝑛 ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) )
598	596 597	nfco	⊢ Ⅎ 𝑛 ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) )
599		nfcv	⊢ Ⅎ 𝑛 ( 1 ... 𝑦 )
600	598 599	nfima	⊢ Ⅎ 𝑛 ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) )
601		nfcv	⊢ Ⅎ 𝑛 { 1 }
602	600 601	nfxp	⊢ Ⅎ 𝑛 ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } )
603		nfcv	⊢ Ⅎ 𝑛 ( ( 𝑦 + 1 ) ... 𝑁 )
604	598 603	nfima	⊢ Ⅎ 𝑛 ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) )
605		nfcv	⊢ Ⅎ 𝑛 { 0 }
606	604 605	nfxp	⊢ Ⅎ 𝑛 ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } )
607	602 606	nfun	⊢ Ⅎ 𝑛 ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) )
608	607	dffn5f	⊢ ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( 1 ... 𝑁 ) ↔ ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) )
609	595 608	sylib	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) )
610	90 592 593 594 609	offval2	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) + if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) , 1 , 0 ) ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) + if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) , 1 , 0 ) ) + ( ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) )
611	568 591 610	3eqtr4rd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) + if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) , 1 , 0 ) ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 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 } ) ) ) )
612	611	mpteq2dva	⊢ ( 𝜑 → ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) + if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) , 1 , 0 ) ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 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 } ) ) ) ) )
613	23 612	eqtr4d	⊢ ( 𝜑 → 𝐹 = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( 1^st ‘ ( 1^st ‘ 𝑇 ) ) ‘ 𝑛 ) + if ( 𝑛 = ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ‘ 1 ) , 1 , 0 ) ) ) ∘_f + ( ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2^nd ‘ ( 1^st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 𝑁 , 1 , ( 𝑛 + 1 ) ) ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) )

Metamath Proof Explorer

Theorem poimirlem16

Proof