poimirlem20 - Mathbox for Brendan Leahy

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Theorem poimirlem20 35797

Description: Lemma for poimir 35810 establishing existence for poimirlem21 35798. (Contributed by Brendan Leahy, 21-Aug-2020.)

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

**Assertion**
Ref	Expression
poimirlem20	⊢ (𝜑 → ∃𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)

Distinct variable groups: 𝑓,𝑗,𝑛,𝑝,𝑡,𝑦,𝑧 𝜑,𝑗,𝑛,𝑦 𝑗,𝐹,𝑛,𝑦 𝑗,𝑁,𝑛,𝑦 𝑇,𝑗,𝑛,𝑦 𝜑,𝑝,𝑡 𝑓,𝐾,𝑗,𝑛,𝑝,𝑡 𝑓,𝑁,𝑝,𝑡 𝑇,𝑓,𝑝 𝜑,𝑧 𝑓,𝐹,𝑝,𝑡,𝑧 𝑧,𝐾 𝑧,𝑁 𝑡,𝑇,𝑧 𝑆,𝑗,𝑛,𝑝,𝑡,𝑦,𝑧 Allowed substitution hints: 𝜑(𝑓) 𝑆(𝑓) 𝐾(𝑦)

**Proof of Theorem poimirlem20**
Step	Hyp	Ref	Expression
1		oveq2 7283	. . . . . . . . 9 ⊢ (1 = if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0) → (((1^st ‘(1^st ‘𝑇))‘𝑛) − 1) = (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0)))
2	1	eleq1d 2823	. . . . . . . 8 ⊢ (1 = if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0) → ((((1^st ‘(1^st ‘𝑇))‘𝑛) − 1) ∈ (0..^𝐾) ↔ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0)) ∈ (0..^𝐾)))
3		oveq2 7283	. . . . . . . . 9 ⊢ (0 = if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0) → (((1^st ‘(1^st ‘𝑇))‘𝑛) − 0) = (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0)))
4	3	eleq1d 2823	. . . . . . . 8 ⊢ (0 = if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0) → ((((1^st ‘(1^st ‘𝑇))‘𝑛) − 0) ∈ (0..^𝐾) ↔ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0)) ∈ (0..^𝐾)))
5		fveq2 6774	. . . . . . . . . . . 12 ⊢ (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁) → ((1^st ‘(1^st ‘𝑇))‘𝑛) = ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)))
6	5	oveq1d 7290	. . . . . . . . . . 11 ⊢ (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁) → (((1^st ‘(1^st ‘𝑇))‘𝑛) − 1) = (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) − 1))
7	6	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁)) → (((1^st ‘(1^st ‘𝑇))‘𝑛) − 1) = (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) − 1))
8		poimirlem22.2	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑇 ∈ 𝑆)
9		elrabi 3618	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (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)))
10		poimirlem22.s	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 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}))))}
11	9, 10	eleq2s 2857	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑇 ∈ 𝑆 → 𝑇 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
12	8, 11	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑇 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
13		xp1st 7863	. . . . . . . . . . . . . . . . . . 19 ⊢ (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)}))
14	12, 13	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1^st ‘𝑇) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
15		xp1st 7863	. . . . . . . . . . . . . . . . . 18 ⊢ ((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)))
16	14, 15	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑇)) ∈ ((0..^𝐾) ↑_m (1...𝑁)))
17		elmapi 8637	. . . . . . . . . . . . . . . . 17 ⊢ ((1^st ‘(1^st ‘𝑇)) ∈ ((0..^𝐾) ↑_m (1...𝑁)) → (1^st ‘(1^st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
18	16, 17	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
19		xp2nd 7864	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((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)})
20	14, 19	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
21		fvex 6787	. . . . . . . . . . . . . . . . . . . 20 ⊢ (2^nd ‘(1^st ‘𝑇)) ∈ V
22		f1oeq1 6704	. . . . . . . . . . . . . . . . . . . 20 ⊢ (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)))
23	21, 22	elab 3609	. . . . . . . . . . . . . . . . . . 19 ⊢ ((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))
24	20, 23	sylib 217	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
25		f1of 6716	. . . . . . . . . . . . . . . . . 18 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)⟶(1...𝑁))
26	24, 25	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)⟶(1...𝑁))
27		poimir.0	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℕ)
28		elfz1end 13286	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (1...𝑁))
29	27, 28	sylib 217	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ (1...𝑁))
30	26, 29	ffvelrnd 6962	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇))‘𝑁) ∈ (1...𝑁))
31	18, 30	ffvelrnd 6962	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) ∈ (0..^𝐾))
32		elfzonn0 13432	. . . . . . . . . . . . . . 15 ⊢ (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) ∈ (0..^𝐾) → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) ∈ ℕ₀)
33	31, 32	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) ∈ ℕ₀)
34		fvex 6787	. . . . . . . . . . . . . . . . 17 ⊢ ((2^nd ‘(1^st ‘𝑇))‘𝑁) ∈ V
35		eleq1 2826	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁) → (𝑛 ∈ (1...𝑁) ↔ ((2^nd ‘(1^st ‘𝑇))‘𝑁) ∈ (1...𝑁)))
36	35	anbi2d 629	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁) → ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ↔ (𝜑 ∧ ((2^nd ‘(1^st ‘𝑇))‘𝑁) ∈ (1...𝑁))))
37		fveq2 6774	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁) → (𝑝‘𝑛) = (𝑝‘((2^nd ‘(1^st ‘𝑇))‘𝑁)))
38	37	neeq1d 3003	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁) → ((𝑝‘𝑛) ≠ 0 ↔ (𝑝‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) ≠ 0))
39	38	rexbidv 3226	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁) → (∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝐹(𝑝‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) ≠ 0))
40	36, 39	imbi12d 345	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁) → (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 0) ↔ ((𝜑 ∧ ((2^nd ‘(1^st ‘𝑇))‘𝑁) ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) ≠ 0)))
41		poimirlem22.3	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 0)
42	34, 40, 41	vtocl 3498	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((2^nd ‘(1^st ‘𝑇))‘𝑁) ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) ≠ 0)
43	30, 42	mpdan 684	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∃𝑝 ∈ ran 𝐹(𝑝‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) ≠ 0)
44		fveq1 6773	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))) → (𝑝‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) = (((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)))
45	18	ffnd 6601	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑇)) Fn (1...𝑁))
46	45	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1^st ‘(1^st ‘𝑇)) Fn (1...𝑁))
47		1ex 10971	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 1 ∈ V
48		fnconstg 6662	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (1 ∈ V → (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)))
49	47, 48	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦))
50		c0ex 10969	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 0 ∈ V
51		fnconstg 6662	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (0 ∈ V → (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)))
52	50, 51	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁))
53	49, 52	pm3.2i 471	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) ∧ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)))
54		dff1o3 6722	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((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))))
55	54	simprbi 497	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ^◡(2^nd ‘(1^st ‘𝑇)))
56	24, 55	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → Fun ^◡(2^nd ‘(1^st ‘𝑇)))
57		imain 6519	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (Fun ^◡(2^nd ‘(1^st ‘𝑇)) → ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) ∩ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁))))
58	56, 57	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) ∩ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁))))
59		elfznn0 13349	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℕ₀)
60	59	nn0red 12294	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℝ)
61	60	ltp1d 11905	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 < (𝑦 + 1))
62		fzdisj 13283	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 < (𝑦 + 1) → ((1...𝑦) ∩ ((𝑦 + 1)...𝑁)) = ∅)
63	61, 62	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((1...𝑦) ∩ ((𝑦 + 1)...𝑁)) = ∅)
64	63	imaeq2d 5969	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = ((2^nd ‘(1^st ‘𝑇)) “ ∅))
65		ima0 5985	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((2^nd ‘(1^st ‘𝑇)) “ ∅) = ∅
66	64, 65	eqtrdi 2794	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = ∅)
67	58, 66	sylan9req 2799	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) ∩ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁))) = ∅)
68		fnun 6545	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) ∧ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁))) ∧ (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) ∩ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁))) = ∅) → ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁))))
69	53, 67, 68	sylancr 587	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁))))
70		imaundi 6053	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)))
71		nn0p1nn 12272	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑦 ∈ ℕ₀ → (𝑦 + 1) ∈ ℕ)
72		nnuz 12621	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ℕ = (ℤ_≥‘1)
73	71, 72	eleqtrdi 2849	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 ∈ ℕ₀ → (𝑦 + 1) ∈ (ℤ_≥‘1))
74	59, 73	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ (ℤ_≥‘1))
75	74	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 + 1) ∈ (ℤ_≥‘1))
76	27	nncnd 11989	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → 𝑁 ∈ ℂ)
77		npcan1 11400	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
78	76, 77	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
79	78	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) = 𝑁)
80		elfzuz3 13253	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈ (ℤ_≥‘𝑦))
81		peano2uz 12641	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑁 − 1) ∈ (ℤ_≥‘𝑦) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘𝑦))
82	80, 81	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘𝑦))
83	82	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘𝑦))
84	79, 83	eqeltrrd 2840	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ_≥‘𝑦))
85		fzsplit2 13281	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑦 + 1) ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ (ℤ_≥‘𝑦)) → (1...𝑁) = ((1...𝑦) ∪ ((𝑦 + 1)...𝑁)))
86	75, 84, 85	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) = ((1...𝑦) ∪ ((𝑦 + 1)...𝑁)))
87	86	imaeq2d 5969	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))))
88		f1ofo 6723	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–onto→(1...𝑁))
89		foima 6693	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
90	24, 88, 89	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
91	90	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
92	87, 91	eqtr3d 2780	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))) = (1...𝑁))
93	70, 92	eqtr3id 2792	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁))) = (1...𝑁))
94	93	fneq2d 6527	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁))) ↔ ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (1...𝑁)))
95	69, 94	mpbid 231	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (1...𝑁))
96		ovex 7308	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (1...𝑁) ∈ V
97	96	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) ∈ V)
98		inidm 4152	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
99		eqidd 2739	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ((2^nd ‘(1^st ‘𝑇))‘𝑁) ∈ (1...𝑁)) → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) = ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)))
100		f1ofn 6717	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁))
101	24, 100	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁))
102	101	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁))
103		fzss1 13295	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑦 + 1) ∈ (ℤ_≥‘1) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁))
104	74, 103	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁))
105	104	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁))
106		eluzp1p1 12610	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑁 − 1) ∈ (ℤ_≥‘𝑦) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑦 + 1)))
107		uzss 12605	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑦 + 1)) → (ℤ_≥‘((𝑁 − 1) + 1)) ⊆ (ℤ_≥‘(𝑦 + 1)))
108	80, 106, 107	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (ℤ_≥‘((𝑁 − 1) + 1)) ⊆ (ℤ_≥‘(𝑦 + 1)))
109	108	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (ℤ_≥‘((𝑁 − 1) + 1)) ⊆ (ℤ_≥‘(𝑦 + 1)))
110	27	nnzd 12425	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → 𝑁 ∈ ℤ)
111	110	uzidd 12598	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑁))
112	78	fveq2d 6778	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → (ℤ_≥‘((𝑁 − 1) + 1)) = (ℤ_≥‘𝑁))
113	111, 112	eleqtrrd 2842	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘((𝑁 − 1) + 1)))
114	113	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ_≥‘((𝑁 − 1) + 1)))
115	109, 114	sseldd 3922	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ_≥‘(𝑦 + 1)))
116		eluzfz2 13264	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑁 ∈ (ℤ_≥‘(𝑦 + 1)) → 𝑁 ∈ ((𝑦 + 1)...𝑁))
117	115, 116	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ((𝑦 + 1)...𝑁))
118		fnfvima 7109	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁) ∧ ((𝑦 + 1)...𝑁) ⊆ (1...𝑁) ∧ 𝑁 ∈ ((𝑦 + 1)...𝑁)) → ((2^nd ‘(1^st ‘𝑇))‘𝑁) ∈ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)))
119	102, 105, 117, 118	syl3anc 1370	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^nd ‘(1^st ‘𝑇))‘𝑁) ∈ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)))
120		fvun2 6860	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) ∧ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) ∧ ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) ∩ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁))) = ∅ ∧ ((2^nd ‘(1^st ‘𝑇))‘𝑁) ∈ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)))) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) = ((((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})‘((2^nd ‘(1^st ‘𝑇))‘𝑁)))
121	49, 52, 120	mp3an12 1450	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) ∩ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁))) = ∅ ∧ ((2^nd ‘(1^st ‘𝑇))‘𝑁) ∈ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁))) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) = ((((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})‘((2^nd ‘(1^st ‘𝑇))‘𝑁)))
122	67, 119, 121	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) = ((((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})‘((2^nd ‘(1^st ‘𝑇))‘𝑁)))
123	50	fvconst2 7079	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((2^nd ‘(1^st ‘𝑇))‘𝑁) ∈ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) → ((((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) = 0)
124	119, 123	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) = 0)
125	122, 124	eqtrd 2778	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) = 0)
126	125	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ((2^nd ‘(1^st ‘𝑇))‘𝑁) ∈ (1...𝑁)) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) = 0)
127	46, 95, 97, 97, 98, 99, 126	ofval 7544	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ((2^nd ‘(1^st ‘𝑇))‘𝑁) ∈ (1...𝑁)) → (((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) = (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) + 0))
128	30, 127	mpidan 686	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) = (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) + 0))
129	33	nn0cnd 12295	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) ∈ ℂ)
130	129	addid1d 11175	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) + 0) = ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)))
131	130	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) + 0) = ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)))
132	128, 131	eqtrd 2778	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) = ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)))
133	44, 132	sylan9eqr 2800	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑝 = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))) → (𝑝‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) = ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)))
134	133	adantllr 716	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑝 ∈ ran 𝐹) ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑝 = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))) → (𝑝‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) = ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)))
135		fveq2 6774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑡 = 𝑇 → (2^nd ‘𝑡) = (2^nd ‘𝑇))
136	135	breq2d 5086	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑡 = 𝑇 → (𝑦 < (2^nd ‘𝑡) ↔ 𝑦 < (2^nd ‘𝑇)))
137	136	ifbid 4482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑡 = 𝑇 → if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)))
138		2fveq3 6779	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑡 = 𝑇 → (1^st ‘(1^st ‘𝑡)) = (1^st ‘(1^st ‘𝑇)))
139		2fveq3 6779	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑡 = 𝑇 → (2^nd ‘(1^st ‘𝑡)) = (2^nd ‘(1^st ‘𝑇)))
140	139	imaeq1d 5968	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑡 = 𝑇 → ((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) = ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)))
141	140	xpeq1d 5618	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑡 = 𝑇 → (((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}))
142	139	imaeq1d 5968	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑡 = 𝑇 → ((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)))
143	142	xpeq1d 5618	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑡 = 𝑇 → (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
144	141, 143	uneq12d 4098	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (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})))
145	138, 144	oveq12d 7293	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (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}))))
146	137, 145	csbeq12dv 3841	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (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}))))
147	146	mpteq2dv 5176	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (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})))))
148	147	eqeq2d 2749	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (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}))))))
149	148, 10	elrab2 3627	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (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}))))))
150	149	simprbi 497	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (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})))))
151	8, 150	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (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})))))
152	60	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ∈ ℝ)
153		peano2zm 12363	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
154	110, 153	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → (𝑁 − 1) ∈ ℤ)
155	154	zred 12426	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → (𝑁 − 1) ∈ ℝ)
156	155	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) ∈ ℝ)
157	27	nnred 11988	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → 𝑁 ∈ ℝ)
158	157	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℝ)
159		elfzle2 13260	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ≤ (𝑁 − 1))
160	159	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ≤ (𝑁 − 1))
161	157	ltm1d 11907	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → (𝑁 − 1) < 𝑁)
162	161	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) < 𝑁)
163	152, 156, 158, 160, 162	lelttrd 11133	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 < 𝑁)
164		poimirlem21.4	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → (2^nd ‘𝑇) = 𝑁)
165	164	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (2^nd ‘𝑇) = 𝑁)
166	163, 165	breqtrrd 5102	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 < (2^nd ‘𝑇))
167	166	iftrued 4467	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) = 𝑦)
168	167	csbeq1d 3836	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋𝑦 / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
169		vex 3436	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑦 ∈ V
170		oveq2 7283	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑗 = 𝑦 → (1...𝑗) = (1...𝑦))
171	170	imaeq2d 5969	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 = 𝑦 → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) = ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)))
172	171	xpeq1d 5618	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 = 𝑦 → (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}))
173		oveq1 7282	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑗 = 𝑦 → (𝑗 + 1) = (𝑦 + 1))
174	173	oveq1d 7290	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑗 = 𝑦 → ((𝑗 + 1)...𝑁) = ((𝑦 + 1)...𝑁))
175	174	imaeq2d 5969	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 = 𝑦 → ((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)))
176	175	xpeq1d 5618	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 = 𝑦 → (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))
177	172, 176	uneq12d 4098	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 = 𝑦 → ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))
178	177	oveq2d 7291	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 = 𝑦 → ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))))
179	169, 178	csbie 3868	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ⦋𝑦 / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))
180	168, 179	eqtrdi 2794	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))))
181	180	mpteq2dva 5174	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (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 ∈ (0...(𝑁 − 1)) ↦ ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))))
182	151, 181	eqtrd 2778	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))))
183	182	rneqd 5847	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ran 𝐹 = ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))))
184	183	eleq2d 2824	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑝 ∈ ran 𝐹 ↔ 𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))))))
185		eqid 2738	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))))
186		ovex 7308	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))) ∈ V
187	185, 186	elrnmpti 5869	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))) ↔ ∃𝑦 ∈ (0...(𝑁 − 1))𝑝 = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))))
188	184, 187	bitrdi 287	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑝 ∈ ran 𝐹 ↔ ∃𝑦 ∈ (0...(𝑁 − 1))𝑝 = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))))
189	188	biimpa 477	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑝 ∈ ran 𝐹) → ∃𝑦 ∈ (0...(𝑁 − 1))𝑝 = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))))
190	134, 189	r19.29a 3218	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑝 ∈ ran 𝐹) → (𝑝‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) = ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)))
191	190	neeq1d 3003	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ ran 𝐹) → ((𝑝‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) ≠ 0 ↔ ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) ≠ 0))
192	191	biimpd 228	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑝 ∈ ran 𝐹) → ((𝑝‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) ≠ 0 → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) ≠ 0))
193	192	rexlimdva 3213	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (∃𝑝 ∈ ran 𝐹(𝑝‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) ≠ 0 → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) ≠ 0))
194	43, 193	mpd 15	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) ≠ 0)
195		elnnne0 12247	. . . . . . . . . . . . . 14 ⊢ (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) ∈ ℕ ↔ (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) ∈ ℕ₀ ∧ ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) ≠ 0))
196	33, 194, 195	sylanbrc 583	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) ∈ ℕ)
197		nnm1nn0 12274	. . . . . . . . . . . . 13 ⊢ (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) ∈ ℕ → (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) − 1) ∈ ℕ₀)
198	196, 197	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) − 1) ∈ ℕ₀)
199		elfzo0 13428	. . . . . . . . . . . . . 14 ⊢ (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) ∈ (0..^𝐾) ↔ (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) ∈ ℕ₀ ∧ 𝐾 ∈ ℕ ∧ ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) < 𝐾))
200	31, 199	sylib 217	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) ∈ ℕ₀ ∧ 𝐾 ∈ ℕ ∧ ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) < 𝐾))
201	200	simp2d 1142	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ ℕ)
202	198	nn0red 12294	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) − 1) ∈ ℝ)
203	33	nn0red 12294	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) ∈ ℝ)
204	201	nnred 11988	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ ℝ)
205	203	ltm1d 11907	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) − 1) < ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)))
206		elfzolt2 13396	. . . . . . . . . . . . . 14 ⊢ (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) ∈ (0..^𝐾) → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) < 𝐾)
207	31, 206	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) < 𝐾)
208	202, 203, 204, 205, 207	lttrd 11136	. . . . . . . . . . . 12 ⊢ (𝜑 → (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) − 1) < 𝐾)
209		elfzo0 13428	. . . . . . . . . . . 12 ⊢ ((((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) − 1) ∈ (0..^𝐾) ↔ ((((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) − 1) ∈ ℕ₀ ∧ 𝐾 ∈ ℕ ∧ (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) − 1) < 𝐾))
210	198, 201, 208, 209	syl3anbrc 1342	. . . . . . . . . . 11 ⊢ (𝜑 → (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) − 1) ∈ (0..^𝐾))
211	210	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁)) → (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘𝑁)) − 1) ∈ (0..^𝐾))
212	7, 211	eqeltrd 2839	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁)) → (((1^st ‘(1^st ‘𝑇))‘𝑛) − 1) ∈ (0..^𝐾))
213	212	adantlr 712	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁)) → (((1^st ‘(1^st ‘𝑇))‘𝑛) − 1) ∈ (0..^𝐾))
214	18	ffvelrnda 6961	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1^st ‘(1^st ‘𝑇))‘𝑛) ∈ (0..^𝐾))
215		elfzonn0 13432	. . . . . . . . . . . . 13 ⊢ (((1^st ‘(1^st ‘𝑇))‘𝑛) ∈ (0..^𝐾) → ((1^st ‘(1^st ‘𝑇))‘𝑛) ∈ ℕ₀)
216	214, 215	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1^st ‘(1^st ‘𝑇))‘𝑛) ∈ ℕ₀)
217	216	nn0cnd 12295	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1^st ‘(1^st ‘𝑇))‘𝑛) ∈ ℂ)
218	217	subid1d 11321	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((1^st ‘(1^st ‘𝑇))‘𝑛) − 0) = ((1^st ‘(1^st ‘𝑇))‘𝑛))
219	218, 214	eqeltrd 2839	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((1^st ‘(1^st ‘𝑇))‘𝑛) − 0) ∈ (0..^𝐾))
220	219	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁)) → (((1^st ‘(1^st ‘𝑇))‘𝑛) − 0) ∈ (0..^𝐾))
221	2, 4, 213, 220	ifbothda 4497	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0)) ∈ (0..^𝐾))
222	221	fmpttd 6989	. . . . . 6 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))):(1...𝑁)⟶(0..^𝐾))
223		ovex 7308	. . . . . . 7 ⊢ (0..^𝐾) ∈ V
224	223, 96	elmap 8659	. . . . . 6 ⊢ ((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))) ∈ ((0..^𝐾) ↑_m (1...𝑁)) ↔ (𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))):(1...𝑁)⟶(0..^𝐾))
225	222, 224	sylibr 233	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))) ∈ ((0..^𝐾) ↑_m (1...𝑁)))
226		simpr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ((1 + 1)...𝑁)) → 𝑛 ∈ ((1 + 1)...𝑁))
227		1z 12350	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℤ
228		peano2z 12361	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ ℤ → (1 + 1) ∈ ℤ)
229	227, 228	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (1 + 1) ∈ ℤ
230	110, 229	jctil 520	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((1 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ))
231		elfzelz 13256	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ((1 + 1)...𝑁) → 𝑛 ∈ ℤ)
232	231, 227	jctir 521	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ((1 + 1)...𝑁) → (𝑛 ∈ ℤ ∧ 1 ∈ ℤ))
233		fzsubel 13292	. . . . . . . . . . . . . 14 ⊢ ((((1 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑛 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑛 ∈ ((1 + 1)...𝑁) ↔ (𝑛 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1))))
234	230, 232, 233	syl2an 596	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ((1 + 1)...𝑁)) → (𝑛 ∈ ((1 + 1)...𝑁) ↔ (𝑛 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1))))
235	226, 234	mpbid 231	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ((1 + 1)...𝑁)) → (𝑛 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1)))
236		ax-1cn 10929	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
237	236, 236	pncan3oi 11237	. . . . . . . . . . . . 13 ⊢ ((1 + 1) − 1) = 1
238	237	oveq1i 7285	. . . . . . . . . . . 12 ⊢ (((1 + 1) − 1)...(𝑁 − 1)) = (1...(𝑁 − 1))
239	235, 238	eleqtrdi 2849	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ((1 + 1)...𝑁)) → (𝑛 − 1) ∈ (1...(𝑁 − 1)))
240	239	ralrimiva 3103	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑛 ∈ ((1 + 1)...𝑁)(𝑛 − 1) ∈ (1...(𝑁 − 1)))
241		simpr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) → 𝑦 ∈ (1...(𝑁 − 1)))
242	154, 227	jctil 520	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ))
243		elfzelz 13256	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (1...(𝑁 − 1)) → 𝑦 ∈ ℤ)
244	243, 227	jctir 521	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (1...(𝑁 − 1)) → (𝑦 ∈ ℤ ∧ 1 ∈ ℤ))
245		fzaddel 13290	. . . . . . . . . . . . . . 15 ⊢ (((1 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) ∧ (𝑦 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑦 ∈ (1...(𝑁 − 1)) ↔ (𝑦 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1))))
246	242, 244, 245	syl2an 596	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) → (𝑦 ∈ (1...(𝑁 − 1)) ↔ (𝑦 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1))))
247	241, 246	mpbid 231	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) → (𝑦 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1)))
248	78	oveq2d 7291	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 + 1)...𝑁))
249	248	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 + 1)...𝑁))
250	247, 249	eleqtrd 2841	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) → (𝑦 + 1) ∈ ((1 + 1)...𝑁))
251	231	zcnd 12427	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ((1 + 1)...𝑁) → 𝑛 ∈ ℂ)
252	243	zcnd 12427	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (1...(𝑁 − 1)) → 𝑦 ∈ ℂ)
253		subadd2 11225	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑛 − 1) = 𝑦 ↔ (𝑦 + 1) = 𝑛))
254	236, 253	mp3an2 1448	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑛 − 1) = 𝑦 ↔ (𝑦 + 1) = 𝑛))
255		eqcom 2745	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑛 − 1) ↔ (𝑛 − 1) = 𝑦)
256		eqcom 2745	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (𝑦 + 1) ↔ (𝑦 + 1) = 𝑛)
257	254, 255, 256	3bitr4g 314	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1)))
258	251, 252, 257	syl2anr 597	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ (1...(𝑁 − 1)) ∧ 𝑛 ∈ ((1 + 1)...𝑁)) → (𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1)))
259	258	ralrimiva 3103	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (1...(𝑁 − 1)) → ∀𝑛 ∈ ((1 + 1)...𝑁)(𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1)))
260	259	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) → ∀𝑛 ∈ ((1 + 1)...𝑁)(𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1)))
261		reu6i 3663	. . . . . . . . . . . 12 ⊢ (((𝑦 + 1) ∈ ((1 + 1)...𝑁) ∧ ∀𝑛 ∈ ((1 + 1)...𝑁)(𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1))) → ∃!𝑛 ∈ ((1 + 1)...𝑁)𝑦 = (𝑛 − 1))
262	250, 260, 261	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑁 − 1))) → ∃!𝑛 ∈ ((1 + 1)...𝑁)𝑦 = (𝑛 − 1))
263	262	ralrimiva 3103	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑦 ∈ (1...(𝑁 − 1))∃!𝑛 ∈ ((1 + 1)...𝑁)𝑦 = (𝑛 − 1))
264		eqid 2738	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) = (𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1))
265	264	f1ompt 6985	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)):((1 + 1)...𝑁)–1-1-onto→(1...(𝑁 − 1)) ↔ (∀𝑛 ∈ ((1 + 1)...𝑁)(𝑛 − 1) ∈ (1...(𝑁 − 1)) ∧ ∀𝑦 ∈ (1...(𝑁 − 1))∃!𝑛 ∈ ((1 + 1)...𝑁)𝑦 = (𝑛 − 1)))
266	240, 263, 265	sylanbrc 583	. . . . . . . . 9 ⊢ (𝜑 → (𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)):((1 + 1)...𝑁)–1-1-onto→(1...(𝑁 − 1)))
267		f1osng 6757	. . . . . . . . . 10 ⊢ ((1 ∈ V ∧ 𝑁 ∈ ℕ) → {⟨1, 𝑁⟩}:{1}–1-1-onto→{𝑁})
268	47, 27, 267	sylancr 587	. . . . . . . . 9 ⊢ (𝜑 → {⟨1, 𝑁⟩}:{1}–1-1-onto→{𝑁})
269	155, 157	ltnled 11122	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁 − 1) < 𝑁 ↔ ¬ 𝑁 ≤ (𝑁 − 1)))
270	161, 269	mpbid 231	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝑁 ≤ (𝑁 − 1))
271		elfzle2 13260	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (1...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1))
272	270, 271	nsyl 140	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑁 ∈ (1...(𝑁 − 1)))
273		disjsn 4647	. . . . . . . . . 10 ⊢ (((1...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (1...(𝑁 − 1)))
274	272, 273	sylibr 233	. . . . . . . . 9 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∩ {𝑁}) = ∅)
275		1re 10975	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ
276	275	ltp1i 11879	. . . . . . . . . . . . 13 ⊢ 1 < (1 + 1)
277	229	zrei 12325	. . . . . . . . . . . . . 14 ⊢ (1 + 1) ∈ ℝ
278	275, 277	ltnlei 11096	. . . . . . . . . . . . 13 ⊢ (1 < (1 + 1) ↔ ¬ (1 + 1) ≤ 1)
279	276, 278	mpbi 229	. . . . . . . . . . . 12 ⊢ ¬ (1 + 1) ≤ 1
280		elfzle1 13259	. . . . . . . . . . . 12 ⊢ (1 ∈ ((1 + 1)...𝑁) → (1 + 1) ≤ 1)
281	279, 280	mto 196	. . . . . . . . . . 11 ⊢ ¬ 1 ∈ ((1 + 1)...𝑁)
282		disjsn 4647	. . . . . . . . . . 11 ⊢ ((((1 + 1)...𝑁) ∩ {1}) = ∅ ↔ ¬ 1 ∈ ((1 + 1)...𝑁))
283	281, 282	mpbir 230	. . . . . . . . . 10 ⊢ (((1 + 1)...𝑁) ∩ {1}) = ∅
284		f1oun 6735	. . . . . . . . . 10 ⊢ ((((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)):((1 + 1)...𝑁)–1-1-onto→(1...(𝑁 − 1)) ∧ {⟨1, 𝑁⟩}:{1}–1-1-onto→{𝑁}) ∧ ((((1 + 1)...𝑁) ∩ {1}) = ∅ ∧ ((1...(𝑁 − 1)) ∩ {𝑁}) = ∅)) → ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {⟨1, 𝑁⟩}):(((1 + 1)...𝑁) ∪ {1})–1-1-onto→((1...(𝑁 − 1)) ∪ {𝑁}))
285	283, 284	mpanr1 700	. . . . . . . . 9 ⊢ ((((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)):((1 + 1)...𝑁)–1-1-onto→(1...(𝑁 − 1)) ∧ {⟨1, 𝑁⟩}:{1}–1-1-onto→{𝑁}) ∧ ((1...(𝑁 − 1)) ∩ {𝑁}) = ∅) → ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {⟨1, 𝑁⟩}):(((1 + 1)...𝑁) ∪ {1})–1-1-onto→((1...(𝑁 − 1)) ∪ {𝑁}))
286	266, 268, 274, 285	syl21anc 835	. . . . . . . 8 ⊢ (𝜑 → ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {⟨1, 𝑁⟩}):(((1 + 1)...𝑁) ∪ {1})–1-1-onto→((1...(𝑁 − 1)) ∪ {𝑁}))
287		eleq1 2826	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 1 → (𝑛 ∈ ((1 + 1)...𝑁) ↔ 1 ∈ ((1 + 1)...𝑁)))
288	281, 287	mtbiri 327	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 1 → ¬ 𝑛 ∈ ((1 + 1)...𝑁))
289	288	necon2ai 2973	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ((1 + 1)...𝑁) → 𝑛 ≠ 1)
290		ifnefalse 4471	. . . . . . . . . . . . 13 ⊢ (𝑛 ≠ 1 → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = (𝑛 − 1))
291	289, 290	syl 17	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ((1 + 1)...𝑁) → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = (𝑛 − 1))
292	291	mpteq2ia 5177	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ((1 + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) = (𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1))
293	292	uneq1i 4093	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ((1 + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ∪ {⟨1, 𝑁⟩}) = ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {⟨1, 𝑁⟩})
294	47	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ V)
295	27, 72	eleqtrdi 2849	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘1))
296		fzpred 13304	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ_≥‘1) → (1...𝑁) = ({1} ∪ ((1 + 1)...𝑁)))
297	295, 296	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (1...𝑁) = ({1} ∪ ((1 + 1)...𝑁)))
298		uncom 4087	. . . . . . . . . . . 12 ⊢ ({1} ∪ ((1 + 1)...𝑁)) = (((1 + 1)...𝑁) ∪ {1})
299	297, 298	eqtr2di 2795	. . . . . . . . . . 11 ⊢ (𝜑 → (((1 + 1)...𝑁) ∪ {1}) = (1...𝑁))
300		iftrue 4465	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = 𝑁)
301	300	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 = 1) → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = 𝑁)
302	294, 27, 299, 301	fmptapd 7043	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑛 ∈ ((1 + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ∪ {⟨1, 𝑁⟩}) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))
303	293, 302	eqtr3id 2792	. . . . . . . . 9 ⊢ (𝜑 → ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {⟨1, 𝑁⟩}) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))
304	78, 295	eqeltrd 2839	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘1))
305		uzid 12597	. . . . . . . . . . . . 13 ⊢ ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ_≥‘(𝑁 − 1)))
306		peano2uz 12641	. . . . . . . . . . . . 13 ⊢ ((𝑁 − 1) ∈ (ℤ_≥‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑁 − 1)))
307	154, 305, 306	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑁 − 1)))
308	78, 307	eqeltrrd 2840	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘(𝑁 − 1)))
309		fzsplit2 13281	. . . . . . . . . . 11 ⊢ ((((𝑁 − 1) + 1) ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ (ℤ_≥‘(𝑁 − 1))) → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
310	304, 308, 309	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
311	78	oveq1d 7290	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = (𝑁...𝑁))
312		fzsn 13298	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
313	110, 312	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁...𝑁) = {𝑁})
314	311, 313	eqtrd 2778	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = {𝑁})
315	314	uneq2d 4097	. . . . . . . . . 10 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = ((1...(𝑁 − 1)) ∪ {𝑁}))
316	310, 315	eqtr2d 2779	. . . . . . . . 9 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∪ {𝑁}) = (1...𝑁))
317	303, 299, 316	f1oeq123d 6710	. . . . . . . 8 ⊢ (𝜑 → (((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {⟨1, 𝑁⟩}):(((1 + 1)...𝑁) ∪ {1})–1-1-onto→((1...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)–1-1-onto→(1...𝑁)))
318	286, 317	mpbid 231	. . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)–1-1-onto→(1...𝑁))
319		f1oco 6739	. . . . . . 7 ⊢ (((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ∧ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)–1-1-onto→(1...𝑁)) → ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–1-1-onto→(1...𝑁))
320	24, 318, 319	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–1-1-onto→(1...𝑁))
321	96	mptex 7099	. . . . . . . 8 ⊢ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ∈ V
322	21, 321	coex 7777	. . . . . . 7 ⊢ ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) ∈ V
323		f1oeq1 6704	. . . . . . 7 ⊢ (𝑓 = ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–1-1-onto→(1...𝑁)))
324	322, 323	elab 3609	. . . . . 6 ⊢ (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–1-1-onto→(1...𝑁))
325	320, 324	sylibr 233	. . . . 5 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
326	225, 325	opelxpd 5627	. . . 4 ⊢ (𝜑 → ⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩ ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
327	27	nnnn0d 12293	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
328		0elfz 13353	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → 0 ∈ (0...𝑁))
329	327, 328	syl 17	. . . 4 ⊢ (𝜑 → 0 ∈ (0...𝑁))
330	326, 329	opelxpd 5627	. . 3 ⊢ (𝜑 → ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
331		poimirlem22.1	. . . . 5 ⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑_m (1...𝑁)))
332	27, 10, 331, 8, 41, 164	poimirlem19 35796	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))))
333		elfzle1 13259	. . . . . . . . 9 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 0 ≤ 𝑦)
334		0re 10977	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
335		lenlt 11053	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (0 ≤ 𝑦 ↔ ¬ 𝑦 < 0))
336	334, 60, 335	sylancr 587	. . . . . . . . 9 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (0 ≤ 𝑦 ↔ ¬ 𝑦 < 0))
337	333, 336	mpbid 231	. . . . . . . 8 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ¬ 𝑦 < 0)
338	337	iffalsed 4470	. . . . . . 7 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → if(𝑦 < 0, 𝑦, (𝑦 + 1)) = (𝑦 + 1))
339	338	csbeq1d 3836	. . . . . 6 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ⦋if(𝑦 < 0, 𝑦, (𝑦 + 1)) / 𝑗⦌((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋(𝑦 + 1) / 𝑗⦌((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0}))))
340		ovex 7308	. . . . . . 7 ⊢ (𝑦 + 1) ∈ V
341		oveq2 7283	. . . . . . . . . . 11 ⊢ (𝑗 = (𝑦 + 1) → (1...𝑗) = (1...(𝑦 + 1)))
342	341	imaeq2d 5969	. . . . . . . . . 10 ⊢ (𝑗 = (𝑦 + 1) → (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) = (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))))
343	342	xpeq1d 5618	. . . . . . . . 9 ⊢ (𝑗 = (𝑦 + 1) → ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) × {1}) = ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}))
344		oveq1 7282	. . . . . . . . . . . 12 ⊢ (𝑗 = (𝑦 + 1) → (𝑗 + 1) = ((𝑦 + 1) + 1))
345	344	oveq1d 7290	. . . . . . . . . . 11 ⊢ (𝑗 = (𝑦 + 1) → ((𝑗 + 1)...𝑁) = (((𝑦 + 1) + 1)...𝑁))
346	345	imaeq2d 5969	. . . . . . . . . 10 ⊢ (𝑗 = (𝑦 + 1) → (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) = (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)))
347	346	xpeq1d 5618	. . . . . . . . 9 ⊢ (𝑗 = (𝑦 + 1) → ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0}) = ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))
348	343, 347	uneq12d 4098	. . . . . . . 8 ⊢ (𝑗 = (𝑦 + 1) → (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0})) = (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))
349	348	oveq2d 7291	. . . . . . 7 ⊢ (𝑗 = (𝑦 + 1) → ((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))
350	340, 349	csbie 3868	. . . . . 6 ⊢ ⦋(𝑦 + 1) / 𝑗⦌((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))
351	339, 350	eqtrdi 2794	. . . . 5 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ⦋if(𝑦 < 0, 𝑦, (𝑦 + 1)) / 𝑗⦌((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))
352	351	mpteq2ia 5177	. . . 4 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 0, 𝑦, (𝑦 + 1)) / 𝑗⦌((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))
353	332, 352	eqtr4di 2796	. . 3 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 0, 𝑦, (𝑦 + 1)) / 𝑗⦌((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0})))))
354		opex 5379	. . . . . . . . . 10 ⊢ ⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩ ∈ V
355	354, 50	op2ndd 7842	. . . . . . . . 9 ⊢ (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → (2^nd ‘𝑡) = 0)
356	355	breq2d 5086	. . . . . . . 8 ⊢ (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → (𝑦 < (2^nd ‘𝑡) ↔ 𝑦 < 0))
357	356	ifbid 4482	. . . . . . 7 ⊢ (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < 0, 𝑦, (𝑦 + 1)))
358	354, 50	op1std 7841	. . . . . . . . 9 ⊢ (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → (1^st ‘𝑡) = ⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩)
359	96	mptex 7099	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))) ∈ V
360	359, 322	op1std 7841	. . . . . . . . 9 ⊢ ((1^st ‘𝑡) = ⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩ → (1^st ‘(1^st ‘𝑡)) = (𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))))
361	358, 360	syl 17	. . . . . . . 8 ⊢ (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → (1^st ‘(1^st ‘𝑡)) = (𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))))
362	359, 322	op2ndd 7842	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝑡) = ⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩ → (2^nd ‘(1^st ‘𝑡)) = ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))))
363	358, 362	syl 17	. . . . . . . . . . 11 ⊢ (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → (2^nd ‘(1^st ‘𝑡)) = ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))))
364	363	imaeq1d 5968	. . . . . . . . . 10 ⊢ (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → ((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) = (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)))
365	364	xpeq1d 5618	. . . . . . . . 9 ⊢ (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → (((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) = ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) × {1}))
366	363	imaeq1d 5968	. . . . . . . . . 10 ⊢ (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → ((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)))
367	366	xpeq1d 5618	. . . . . . . . 9 ⊢ (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0}))
368	365, 367	uneq12d 4098	. . . . . . . 8 ⊢ (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0})))
369	361, 368	oveq12d 7293	. . . . . . 7 ⊢ (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → ((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0}))))
370	357, 369	csbeq12dv 3841	. . . . . 6 ⊢ (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 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}))) = ⦋if(𝑦 < 0, 𝑦, (𝑦 + 1)) / 𝑗⦌((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0}))))
371	370	mpteq2dv 5176	. . . . 5 ⊢ (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 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})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 0, 𝑦, (𝑦 + 1)) / 𝑗⦌((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0})))))
372	371	eqeq2d 2749	. . . 4 ⊢ (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 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})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 0, 𝑦, (𝑦 + 1)) / 𝑗⦌((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0}))))))
373	372, 10	elrab2 3627	. . 3 ⊢ (⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ ∈ 𝑆 ↔ (⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 0, 𝑦, (𝑦 + 1)) / 𝑗⦌((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0}))))))
374	330, 353, 373	sylanbrc 583	. 2 ⊢ (𝜑 → ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ ∈ 𝑆)
375	354, 50	op2ndd 7842	. . . . . 6 ⊢ (𝑇 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → (2^nd ‘𝑇) = 0)
376	375	eqcoms 2746	. . . . 5 ⊢ (⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ = 𝑇 → (2^nd ‘𝑇) = 0)
377	27	nnne0d 12023	. . . . . . 7 ⊢ (𝜑 → 𝑁 ≠ 0)
378	377	necomd 2999	. . . . . 6 ⊢ (𝜑 → 0 ≠ 𝑁)
379		neeq1 3006	. . . . . 6 ⊢ ((2^nd ‘𝑇) = 0 → ((2^nd ‘𝑇) ≠ 𝑁 ↔ 0 ≠ 𝑁))
380	378, 379	syl5ibrcom 246	. . . . 5 ⊢ (𝜑 → ((2^nd ‘𝑇) = 0 → (2^nd ‘𝑇) ≠ 𝑁))
381	376, 380	syl5 34	. . . 4 ⊢ (𝜑 → (⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ = 𝑇 → (2^nd ‘𝑇) ≠ 𝑁))
382	381	necon2d 2966	. . 3 ⊢ (𝜑 → ((2^nd ‘𝑇) = 𝑁 → ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ ≠ 𝑇))
383	164, 382	mpd 15	. 2 ⊢ (𝜑 → ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ ≠ 𝑇)
384		neeq1 3006	. . 3 ⊢ (𝑧 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → (𝑧 ≠ 𝑇 ↔ ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ ≠ 𝑇))
385	384	rspcev 3561	. 2 ⊢ ((⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ ∈ 𝑆 ∧ ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) − if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘𝑁), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ ≠ 𝑇) → ∃𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)
386	374, 383, 385	syl2anc 584	1 ⊢ (𝜑 → ∃𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 {cab 2715 ≠ wne 2943 ∀wral 3064 ∃wrex 3065 ∃!wreu 3066 {crab 3068 Vcvv 3432 ⦋csb 3832 ∪ cun 3885 ∩ cin 3886 ⊆ wss 3887 ∅c0 4256 ifcif 4459 {csn 4561 ⟨cop 4567 class class class wbr 5074 ↦ cmpt 5157 × cxp 5587 ^◡ccnv 5588 ran crn 5590 “ cima 5592 ∘ ccom 5593 Fun wfun 6427 Fn wfn 6428 ⟶wf 6429 –onto→wfo 6431 –1-1-onto→wf1o 6432 ‘cfv 6433 (class class class)co 7275 ∘_f cof 7531 1^st c1st 7829 2^nd c2nd 7830 ↑_m cmap 8615 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 < clt 11009 ≤ cle 11010 − cmin 11205 ℕcn 11973 ℕ₀cn0 12233 ℤcz 12319 ℤ_≥cuz 12582 ...cfz 13239 ..^cfzo 13382

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-fzo 13383

This theorem is referenced by: poimirlem21 35798

W3C validator