poimirlem13 - Mathbox for Brendan Leahy

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Theorem poimirlem13 37662

Description: Lemma for poimir 37682- for at most one simplex associated with a shared face is the opposite vertex first on the walk. (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)))

**Assertion**
Ref	Expression
poimirlem13	⊢ (𝜑 → ∃*𝑧 ∈ 𝑆 (2^nd ‘𝑧) = 0)

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

Proof of Theorem poimirlem13 Dummy variables 𝑘 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		poimir.0	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ)
2	1	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → 𝑁 ∈ ℕ)
3		poimirlem22.s	. . . . . . . 8 ⊢ 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}))))}
4		poimirlem22.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑_m (1...𝑁)))
5	4	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑_m (1...𝑁)))
6		simplrl 776	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → 𝑧 ∈ 𝑆)
7		simprl 770	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → (2^nd ‘𝑧) = 0)
8	2, 3, 5, 6, 7	poimirlem10 37659	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → ((𝐹‘(𝑁 − 1)) ∘_f − ((1...𝑁) × {1})) = (1^st ‘(1^st ‘𝑧)))
9		simplrr 777	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → 𝑘 ∈ 𝑆)
10		simprr 772	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → (2^nd ‘𝑘) = 0)
11	2, 3, 5, 9, 10	poimirlem10 37659	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → ((𝐹‘(𝑁 − 1)) ∘_f − ((1...𝑁) × {1})) = (1^st ‘(1^st ‘𝑘)))
12	8, 11	eqtr3d 2773	. . . . . 6 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → (1^st ‘(1^st ‘𝑧)) = (1^st ‘(1^st ‘𝑘)))
13		elrabi 3671	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ {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}))))} → 𝑧 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
14	13, 3	eleq2s 2853	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
15		xp1st 8025	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1^st ‘𝑧) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
16	14, 15	syl 17	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑆 → (1^st ‘𝑧) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
17		xp2nd 8026	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝑧) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2^nd ‘(1^st ‘𝑧)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
18	16, 17	syl 17	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑆 → (2^nd ‘(1^st ‘𝑧)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
19		fvex 6894	. . . . . . . . . . . 12 ⊢ (2^nd ‘(1^st ‘𝑧)) ∈ V
20		f1oeq1 6811	. . . . . . . . . . . 12 ⊢ (𝑓 = (2^nd ‘(1^st ‘𝑧)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2^nd ‘(1^st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁)))
21	19, 20	elab 3663	. . . . . . . . . . 11 ⊢ ((2^nd ‘(1^st ‘𝑧)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2^nd ‘(1^st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
22	18, 21	sylib 218	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑆 → (2^nd ‘(1^st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
23		f1ofn 6824	. . . . . . . . . 10 ⊢ ((2^nd ‘(1^st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^nd ‘(1^st ‘𝑧)) Fn (1...𝑁))
24	22, 23	syl 17	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑆 → (2^nd ‘(1^st ‘𝑧)) Fn (1...𝑁))
25	24	adantr 480	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → (2^nd ‘(1^st ‘𝑧)) Fn (1...𝑁))
26	25	ad2antlr 727	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → (2^nd ‘(1^st ‘𝑧)) Fn (1...𝑁))
27		elrabi 3671	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ {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}))))} → 𝑘 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
28	27, 3	eleq2s 2853	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝑆 → 𝑘 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
29		xp1st 8025	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1^st ‘𝑘) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
30	28, 29	syl 17	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝑆 → (1^st ‘𝑘) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
31		xp2nd 8026	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝑘) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2^nd ‘(1^st ‘𝑘)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
32	30, 31	syl 17	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝑆 → (2^nd ‘(1^st ‘𝑘)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
33		fvex 6894	. . . . . . . . . . . 12 ⊢ (2^nd ‘(1^st ‘𝑘)) ∈ V
34		f1oeq1 6811	. . . . . . . . . . . 12 ⊢ (𝑓 = (2^nd ‘(1^st ‘𝑘)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2^nd ‘(1^st ‘𝑘)):(1...𝑁)–1-1-onto→(1...𝑁)))
35	33, 34	elab 3663	. . . . . . . . . . 11 ⊢ ((2^nd ‘(1^st ‘𝑘)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2^nd ‘(1^st ‘𝑘)):(1...𝑁)–1-1-onto→(1...𝑁))
36	32, 35	sylib 218	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝑆 → (2^nd ‘(1^st ‘𝑘)):(1...𝑁)–1-1-onto→(1...𝑁))
37		f1ofn 6824	. . . . . . . . . 10 ⊢ ((2^nd ‘(1^st ‘𝑘)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^nd ‘(1^st ‘𝑘)) Fn (1...𝑁))
38	36, 37	syl 17	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝑆 → (2^nd ‘(1^st ‘𝑘)) Fn (1...𝑁))
39	38	adantl 481	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → (2^nd ‘(1^st ‘𝑘)) Fn (1...𝑁))
40	39	ad2antlr 727	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → (2^nd ‘(1^st ‘𝑘)) Fn (1...𝑁))
41		eleq1 2823	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → (𝑚 ∈ (1...𝑁) ↔ 𝑛 ∈ (1...𝑁)))
42	41	anbi2d 630	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) ↔ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁))))
43		oveq2 7418	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
44	43	imaeq2d 6052	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → ((2^nd ‘(1^st ‘𝑧)) “ (1...𝑚)) = ((2^nd ‘(1^st ‘𝑧)) “ (1...𝑛)))
45	43	imaeq2d 6052	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → ((2^nd ‘(1^st ‘𝑘)) “ (1...𝑚)) = ((2^nd ‘(1^st ‘𝑘)) “ (1...𝑛)))
46	44, 45	eqeq12d 2752	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (((2^nd ‘(1^st ‘𝑧)) “ (1...𝑚)) = ((2^nd ‘(1^st ‘𝑘)) “ (1...𝑚)) ↔ ((2^nd ‘(1^st ‘𝑧)) “ (1...𝑛)) = ((2^nd ‘(1^st ‘𝑘)) “ (1...𝑛))))
47	42, 46	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → ((2^nd ‘(1^st ‘𝑧)) “ (1...𝑚)) = ((2^nd ‘(1^st ‘𝑘)) “ (1...𝑚))) ↔ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → ((2^nd ‘(1^st ‘𝑧)) “ (1...𝑛)) = ((2^nd ‘(1^st ‘𝑘)) “ (1...𝑛)))))
48	1	ad3antrrr 730	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑁 ∈ ℕ)
49	4	ad3antrrr 730	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑_m (1...𝑁)))
50		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → 𝑧 ∈ 𝑆)
51	50	ad3antlr 731	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑧 ∈ 𝑆)
52		simplrl 776	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → (2^nd ‘𝑧) = 0)
53		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → 𝑘 ∈ 𝑆)
54	53	ad3antlr 731	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑘 ∈ 𝑆)
55		simplrr 777	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → (2^nd ‘𝑘) = 0)
56		simpr 484	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑚 ∈ (1...𝑁))
57	48, 3, 49, 51, 52, 54, 55, 56	poimirlem11 37660	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → ((2^nd ‘(1^st ‘𝑧)) “ (1...𝑚)) ⊆ ((2^nd ‘(1^st ‘𝑘)) “ (1...𝑚)))
58	48, 3, 49, 54, 55, 51, 52, 56	poimirlem11 37660	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → ((2^nd ‘(1^st ‘𝑘)) “ (1...𝑚)) ⊆ ((2^nd ‘(1^st ‘𝑧)) “ (1...𝑚)))
59	57, 58	eqssd 3981	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → ((2^nd ‘(1^st ‘𝑧)) “ (1...𝑚)) = ((2^nd ‘(1^st ‘𝑘)) “ (1...𝑚)))
60	47, 59	chvarvv 1989	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → ((2^nd ‘(1^st ‘𝑧)) “ (1...𝑛)) = ((2^nd ‘(1^st ‘𝑘)) “ (1...𝑛)))
61		simpll 766	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → 𝜑)
62		elfznn 13575	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℕ)
63		nnm1nn0 12547	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ₀)
64	62, 63	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ∈ ℕ₀)
65	64	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1) → (𝑛 − 1) ∈ ℕ₀)
66	62	nncnd 12261	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℂ)
67		ax-1cn 11192	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℂ
68		subeq0 11514	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) = 0 ↔ 𝑛 = 1))
69	66, 67, 68	sylancl 586	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) = 0 ↔ 𝑛 = 1))
70	69	necon3abid 2969	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) ≠ 0 ↔ ¬ 𝑛 = 1))
71	70	biimpar 477	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1) → (𝑛 − 1) ≠ 0)
72		elnnne0 12520	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 − 1) ∈ ℕ ↔ ((𝑛 − 1) ∈ ℕ₀ ∧ (𝑛 − 1) ≠ 0))
73	65, 71, 72	sylanbrc 583	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1) → (𝑛 − 1) ∈ ℕ)
74	73	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → (𝑛 − 1) ∈ ℕ)
75	64	nn0red 12568	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ∈ ℝ)
76	75	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ∈ ℝ)
77	62	nnred 12260	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℝ)
78	77	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℝ)
79	1	nnred 12260	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℝ)
80	79	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → 𝑁 ∈ ℝ)
81	77	lem1d 12180	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ≤ 𝑛)
82	81	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ≤ 𝑛)
83		elfzle2 13550	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ≤ 𝑁)
84	83	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ≤ 𝑁)
85	76, 78, 80, 82, 84	letrd 11397	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ≤ 𝑁)
86	85	adantrr 717	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → (𝑛 − 1) ≤ 𝑁)
87	1	nnzd 12620	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ ℤ)
88		fznn 13614	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℤ → ((𝑛 − 1) ∈ (1...𝑁) ↔ ((𝑛 − 1) ∈ ℕ ∧ (𝑛 − 1) ≤ 𝑁)))
89	87, 88	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑛 − 1) ∈ (1...𝑁) ↔ ((𝑛 − 1) ∈ ℕ ∧ (𝑛 − 1) ≤ 𝑁)))
90	89	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → ((𝑛 − 1) ∈ (1...𝑁) ↔ ((𝑛 − 1) ∈ ℕ ∧ (𝑛 − 1) ≤ 𝑁)))
91	74, 86, 90	mpbir2and 713	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → (𝑛 − 1) ∈ (1...𝑁))
92	61, 91	sylan 580	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → (𝑛 − 1) ∈ (1...𝑁))
93		ovex 7443	. . . . . . . . . . . . . 14 ⊢ (𝑛 − 1) ∈ V
94		eleq1 2823	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (𝑛 − 1) → (𝑚 ∈ (1...𝑁) ↔ (𝑛 − 1) ∈ (1...𝑁)))
95	94	anbi2d 630	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (𝑛 − 1) → ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) ↔ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ (𝑛 − 1) ∈ (1...𝑁))))
96		oveq2 7418	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = (𝑛 − 1) → (1...𝑚) = (1...(𝑛 − 1)))
97	96	imaeq2d 6052	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (𝑛 − 1) → ((2^nd ‘(1^st ‘𝑧)) “ (1...𝑚)) = ((2^nd ‘(1^st ‘𝑧)) “ (1...(𝑛 − 1))))
98	96	imaeq2d 6052	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (𝑛 − 1) → ((2^nd ‘(1^st ‘𝑘)) “ (1...𝑚)) = ((2^nd ‘(1^st ‘𝑘)) “ (1...(𝑛 − 1))))
99	97, 98	eqeq12d 2752	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (𝑛 − 1) → (((2^nd ‘(1^st ‘𝑧)) “ (1...𝑚)) = ((2^nd ‘(1^st ‘𝑘)) “ (1...𝑚)) ↔ ((2^nd ‘(1^st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2^nd ‘(1^st ‘𝑘)) “ (1...(𝑛 − 1)))))
100	95, 99	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑛 − 1) → (((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → ((2^nd ‘(1^st ‘𝑧)) “ (1...𝑚)) = ((2^nd ‘(1^st ‘𝑘)) “ (1...𝑚))) ↔ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ (𝑛 − 1) ∈ (1...𝑁)) → ((2^nd ‘(1^st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2^nd ‘(1^st ‘𝑘)) “ (1...(𝑛 − 1))))))
101	93, 100, 59	vtocl 3542	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ (𝑛 − 1) ∈ (1...𝑁)) → ((2^nd ‘(1^st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2^nd ‘(1^st ‘𝑘)) “ (1...(𝑛 − 1))))
102	92, 101	syldan 591	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → ((2^nd ‘(1^st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2^nd ‘(1^st ‘𝑘)) “ (1...(𝑛 − 1))))
103	102	expr 456	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → (¬ 𝑛 = 1 → ((2^nd ‘(1^st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2^nd ‘(1^st ‘𝑘)) “ (1...(𝑛 − 1)))))
104		ima0 6069	. . . . . . . . . . . . 13 ⊢ ((2^nd ‘(1^st ‘𝑧)) “ ∅) = ∅
105		ima0 6069	. . . . . . . . . . . . 13 ⊢ ((2^nd ‘(1^st ‘𝑘)) “ ∅) = ∅
106	104, 105	eqtr4i 2762	. . . . . . . . . . . 12 ⊢ ((2^nd ‘(1^st ‘𝑧)) “ ∅) = ((2^nd ‘(1^st ‘𝑘)) “ ∅)
107		oveq1 7417	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 1 → (𝑛 − 1) = (1 − 1))
108		1m1e0 12317	. . . . . . . . . . . . . . . 16 ⊢ (1 − 1) = 0
109	107, 108	eqtrdi 2787	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 1 → (𝑛 − 1) = 0)
110	109	oveq2d 7426	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 1 → (1...(𝑛 − 1)) = (1...0))
111		fz10 13567	. . . . . . . . . . . . . 14 ⊢ (1...0) = ∅
112	110, 111	eqtrdi 2787	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1 → (1...(𝑛 − 1)) = ∅)
113	112	imaeq2d 6052	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → ((2^nd ‘(1^st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2^nd ‘(1^st ‘𝑧)) “ ∅))
114	112	imaeq2d 6052	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → ((2^nd ‘(1^st ‘𝑘)) “ (1...(𝑛 − 1))) = ((2^nd ‘(1^st ‘𝑘)) “ ∅))
115	106, 113, 114	3eqtr4a 2797	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → ((2^nd ‘(1^st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2^nd ‘(1^st ‘𝑘)) “ (1...(𝑛 − 1))))
116	103, 115	pm2.61d2 181	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → ((2^nd ‘(1^st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2^nd ‘(1^st ‘𝑘)) “ (1...(𝑛 − 1))))
117	60, 116	difeq12d 4107	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → (((2^nd ‘(1^st ‘𝑧)) “ (1...𝑛)) ∖ ((2^nd ‘(1^st ‘𝑧)) “ (1...(𝑛 − 1)))) = (((2^nd ‘(1^st ‘𝑘)) “ (1...𝑛)) ∖ ((2^nd ‘(1^st ‘𝑘)) “ (1...(𝑛 − 1)))))
118		fnsnfv 6963	. . . . . . . . . . . 12 ⊢ (((2^nd ‘(1^st ‘𝑧)) Fn (1...𝑁) ∧ 𝑛 ∈ (1...𝑁)) → {((2^nd ‘(1^st ‘𝑧))‘𝑛)} = ((2^nd ‘(1^st ‘𝑧)) “ {𝑛}))
119	24, 118	sylan 580	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → {((2^nd ‘(1^st ‘𝑧))‘𝑛)} = ((2^nd ‘(1^st ‘𝑧)) “ {𝑛}))
120	62	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℕ)
121		uncom 4138	. . . . . . . . . . . . . . . 16 ⊢ ((1...(𝑛 − 1)) ∪ {𝑛}) = ({𝑛} ∪ (1...(𝑛 − 1)))
122	121	difeq1i 4102	. . . . . . . . . . . . . . 15 ⊢ (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))) = (({𝑛} ∪ (1...(𝑛 − 1))) ∖ (1...(𝑛 − 1)))
123		difun2 4461	. . . . . . . . . . . . . . 15 ⊢ (({𝑛} ∪ (1...(𝑛 − 1))) ∖ (1...(𝑛 − 1))) = ({𝑛} ∖ (1...(𝑛 − 1)))
124	122, 123	eqtri 2759	. . . . . . . . . . . . . 14 ⊢ (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))) = ({𝑛} ∖ (1...(𝑛 − 1)))
125		nncn 12253	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
126		npcan1 11667	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℂ → ((𝑛 − 1) + 1) = 𝑛)
127	125, 126	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) = 𝑛)
128		elnnuz 12901	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ_≥‘1))
129	128	biimpi 216	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ_≥‘1))
130	127, 129	eqeltrd 2835	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) ∈ (ℤ_≥‘1))
131	63	nn0zd 12619	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℤ)
132		uzid 12872	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 − 1) ∈ ℤ → (𝑛 − 1) ∈ (ℤ_≥‘(𝑛 − 1)))
133	131, 132	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ (ℤ_≥‘(𝑛 − 1)))
134		peano2uz 12922	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 − 1) ∈ (ℤ_≥‘(𝑛 − 1)) → ((𝑛 − 1) + 1) ∈ (ℤ_≥‘(𝑛 − 1)))
135	133, 134	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) ∈ (ℤ_≥‘(𝑛 − 1)))
136	127, 135	eqeltrrd 2836	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ_≥‘(𝑛 − 1)))
137		fzsplit2 13571	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 − 1) + 1) ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘(𝑛 − 1))) → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)))
138	130, 136, 137	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)))
139	127	oveq1d 7425	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → (((𝑛 − 1) + 1)...𝑛) = (𝑛...𝑛))
140		nnz 12614	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
141		fzsn 13588	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℤ → (𝑛...𝑛) = {𝑛})
142	140, 141	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → (𝑛...𝑛) = {𝑛})
143	139, 142	eqtrd 2771	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → (((𝑛 − 1) + 1)...𝑛) = {𝑛})
144	143	uneq2d 4148	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)) = ((1...(𝑛 − 1)) ∪ {𝑛}))
145	138, 144	eqtrd 2771	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → (1...𝑛) = ((1...(𝑛 − 1)) ∪ {𝑛}))
146	145	difeq1d 4105	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → ((1...𝑛) ∖ (1...(𝑛 − 1))) = (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))))
147		nnre 12252	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
148		ltm1 12088	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℝ → (𝑛 − 1) < 𝑛)
149		peano2rem 11555	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℝ → (𝑛 − 1) ∈ ℝ)
150		ltnle 11319	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 − 1) ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((𝑛 − 1) < 𝑛 ↔ ¬ 𝑛 ≤ (𝑛 − 1)))
151	149, 150	mpancom 688	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℝ → ((𝑛 − 1) < 𝑛 ↔ ¬ 𝑛 ≤ (𝑛 − 1)))
152	148, 151	mpbid 232	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℝ → ¬ 𝑛 ≤ (𝑛 − 1))
153		elfzle2 13550	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (1...(𝑛 − 1)) → 𝑛 ≤ (𝑛 − 1))
154	152, 153	nsyl 140	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℝ → ¬ 𝑛 ∈ (1...(𝑛 − 1)))
155	147, 154	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → ¬ 𝑛 ∈ (1...(𝑛 − 1)))
156		incom 4189	. . . . . . . . . . . . . . . . 17 ⊢ ((1...(𝑛 − 1)) ∩ {𝑛}) = ({𝑛} ∩ (1...(𝑛 − 1)))
157	156	eqeq1i 2741	. . . . . . . . . . . . . . . 16 ⊢ (((1...(𝑛 − 1)) ∩ {𝑛}) = ∅ ↔ ({𝑛} ∩ (1...(𝑛 − 1))) = ∅)
158		disjsn 4692	. . . . . . . . . . . . . . . 16 ⊢ (((1...(𝑛 − 1)) ∩ {𝑛}) = ∅ ↔ ¬ 𝑛 ∈ (1...(𝑛 − 1)))
159		disj3 4434	. . . . . . . . . . . . . . . 16 ⊢ (({𝑛} ∩ (1...(𝑛 − 1))) = ∅ ↔ {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1))))
160	157, 158, 159	3bitr3i 301	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑛 ∈ (1...(𝑛 − 1)) ↔ {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1))))
161	155, 160	sylib 218	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1))))
162	124, 146, 161	3eqtr4a 2797	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → ((1...𝑛) ∖ (1...(𝑛 − 1))) = {𝑛})
163	120, 162	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → ((1...𝑛) ∖ (1...(𝑛 − 1))) = {𝑛})
164	163	imaeq2d 6052	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → ((2^nd ‘(1^st ‘𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = ((2^nd ‘(1^st ‘𝑧)) “ {𝑛}))
165		dff1o3 6829	. . . . . . . . . . . . . . 15 ⊢ ((2^nd ‘(1^st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2^nd ‘(1^st ‘𝑧)):(1...𝑁)–onto→(1...𝑁) ∧ Fun ^◡(2^nd ‘(1^st ‘𝑧))))
166	165	simprbi 496	. . . . . . . . . . . . . 14 ⊢ ((2^nd ‘(1^st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ^◡(2^nd ‘(1^st ‘𝑧)))
167	22, 166	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑆 → Fun ^◡(2^nd ‘(1^st ‘𝑧)))
168		imadif 6625	. . . . . . . . . . . . 13 ⊢ (Fun ^◡(2^nd ‘(1^st ‘𝑧)) → ((2^nd ‘(1^st ‘𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = (((2^nd ‘(1^st ‘𝑧)) “ (1...𝑛)) ∖ ((2^nd ‘(1^st ‘𝑧)) “ (1...(𝑛 − 1)))))
169	167, 168	syl 17	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑆 → ((2^nd ‘(1^st ‘𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = (((2^nd ‘(1^st ‘𝑧)) “ (1...𝑛)) ∖ ((2^nd ‘(1^st ‘𝑧)) “ (1...(𝑛 − 1)))))
170	169	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → ((2^nd ‘(1^st ‘𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = (((2^nd ‘(1^st ‘𝑧)) “ (1...𝑛)) ∖ ((2^nd ‘(1^st ‘𝑧)) “ (1...(𝑛 − 1)))))
171	119, 164, 170	3eqtr2d 2777	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → {((2^nd ‘(1^st ‘𝑧))‘𝑛)} = (((2^nd ‘(1^st ‘𝑧)) “ (1...𝑛)) ∖ ((2^nd ‘(1^st ‘𝑧)) “ (1...(𝑛 − 1)))))
172	6, 171	sylan 580	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → {((2^nd ‘(1^st ‘𝑧))‘𝑛)} = (((2^nd ‘(1^st ‘𝑧)) “ (1...𝑛)) ∖ ((2^nd ‘(1^st ‘𝑧)) “ (1...(𝑛 − 1)))))
173		eleq1 2823	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑘 → (𝑧 ∈ 𝑆 ↔ 𝑘 ∈ 𝑆))
174	173	anbi1d 631	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑘 → ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) ↔ (𝑘 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁))))
175		2fveq3 6886	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑘 → (2^nd ‘(1^st ‘𝑧)) = (2^nd ‘(1^st ‘𝑘)))
176	175	fveq1d 6883	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑘 → ((2^nd ‘(1^st ‘𝑧))‘𝑛) = ((2^nd ‘(1^st ‘𝑘))‘𝑛))
177	176	sneqd 4618	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑘 → {((2^nd ‘(1^st ‘𝑧))‘𝑛)} = {((2^nd ‘(1^st ‘𝑘))‘𝑛)})
178	175	imaeq1d 6051	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑘 → ((2^nd ‘(1^st ‘𝑧)) “ (1...𝑛)) = ((2^nd ‘(1^st ‘𝑘)) “ (1...𝑛)))
179	175	imaeq1d 6051	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑘 → ((2^nd ‘(1^st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2^nd ‘(1^st ‘𝑘)) “ (1...(𝑛 − 1))))
180	178, 179	difeq12d 4107	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑘 → (((2^nd ‘(1^st ‘𝑧)) “ (1...𝑛)) ∖ ((2^nd ‘(1^st ‘𝑧)) “ (1...(𝑛 − 1)))) = (((2^nd ‘(1^st ‘𝑘)) “ (1...𝑛)) ∖ ((2^nd ‘(1^st ‘𝑘)) “ (1...(𝑛 − 1)))))
181	177, 180	eqeq12d 2752	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑘 → ({((2^nd ‘(1^st ‘𝑧))‘𝑛)} = (((2^nd ‘(1^st ‘𝑧)) “ (1...𝑛)) ∖ ((2^nd ‘(1^st ‘𝑧)) “ (1...(𝑛 − 1)))) ↔ {((2^nd ‘(1^st ‘𝑘))‘𝑛)} = (((2^nd ‘(1^st ‘𝑘)) “ (1...𝑛)) ∖ ((2^nd ‘(1^st ‘𝑘)) “ (1...(𝑛 − 1))))))
182	174, 181	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑘 → (((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → {((2^nd ‘(1^st ‘𝑧))‘𝑛)} = (((2^nd ‘(1^st ‘𝑧)) “ (1...𝑛)) ∖ ((2^nd ‘(1^st ‘𝑧)) “ (1...(𝑛 − 1))))) ↔ ((𝑘 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → {((2^nd ‘(1^st ‘𝑘))‘𝑛)} = (((2^nd ‘(1^st ‘𝑘)) “ (1...𝑛)) ∖ ((2^nd ‘(1^st ‘𝑘)) “ (1...(𝑛 − 1)))))))
183	182, 171	chvarvv 1989	. . . . . . . . . 10 ⊢ ((𝑘 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → {((2^nd ‘(1^st ‘𝑘))‘𝑛)} = (((2^nd ‘(1^st ‘𝑘)) “ (1...𝑛)) ∖ ((2^nd ‘(1^st ‘𝑘)) “ (1...(𝑛 − 1)))))
184	9, 183	sylan 580	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → {((2^nd ‘(1^st ‘𝑘))‘𝑛)} = (((2^nd ‘(1^st ‘𝑘)) “ (1...𝑛)) ∖ ((2^nd ‘(1^st ‘𝑘)) “ (1...(𝑛 − 1)))))
185	117, 172, 184	3eqtr4d 2781	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → {((2^nd ‘(1^st ‘𝑧))‘𝑛)} = {((2^nd ‘(1^st ‘𝑘))‘𝑛)})
186		fvex 6894	. . . . . . . . 9 ⊢ ((2^nd ‘(1^st ‘𝑧))‘𝑛) ∈ V
187	186	sneqr 4821	. . . . . . . 8 ⊢ ({((2^nd ‘(1^st ‘𝑧))‘𝑛)} = {((2^nd ‘(1^st ‘𝑘))‘𝑛)} → ((2^nd ‘(1^st ‘𝑧))‘𝑛) = ((2^nd ‘(1^st ‘𝑘))‘𝑛))
188	185, 187	syl 17	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → ((2^nd ‘(1^st ‘𝑧))‘𝑛) = ((2^nd ‘(1^st ‘𝑘))‘𝑛))
189	26, 40, 188	eqfnfvd 7029	. . . . . 6 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → (2^nd ‘(1^st ‘𝑧)) = (2^nd ‘(1^st ‘𝑘)))
190		xpopth 8034	. . . . . . . 8 ⊢ (((1^st ‘𝑧) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (1^st ‘𝑘) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) → (((1^st ‘(1^st ‘𝑧)) = (1^st ‘(1^st ‘𝑘)) ∧ (2^nd ‘(1^st ‘𝑧)) = (2^nd ‘(1^st ‘𝑘))) ↔ (1^st ‘𝑧) = (1^st ‘𝑘)))
191	16, 30, 190	syl2an 596	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → (((1^st ‘(1^st ‘𝑧)) = (1^st ‘(1^st ‘𝑘)) ∧ (2^nd ‘(1^st ‘𝑧)) = (2^nd ‘(1^st ‘𝑘))) ↔ (1^st ‘𝑧) = (1^st ‘𝑘)))
192	191	ad2antlr 727	. . . . . 6 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → (((1^st ‘(1^st ‘𝑧)) = (1^st ‘(1^st ‘𝑘)) ∧ (2^nd ‘(1^st ‘𝑧)) = (2^nd ‘(1^st ‘𝑘))) ↔ (1^st ‘𝑧) = (1^st ‘𝑘)))
193	12, 189, 192	mpbi2and 712	. . . . 5 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → (1^st ‘𝑧) = (1^st ‘𝑘))
194		eqtr3 2758	. . . . . 6 ⊢ (((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0) → (2^nd ‘𝑧) = (2^nd ‘𝑘))
195	194	adantl 481	. . . . 5 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → (2^nd ‘𝑧) = (2^nd ‘𝑘))
196		xpopth 8034	. . . . . . 7 ⊢ ((𝑧 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑘 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (((1^st ‘𝑧) = (1^st ‘𝑘) ∧ (2^nd ‘𝑧) = (2^nd ‘𝑘)) ↔ 𝑧 = 𝑘))
197	14, 28, 196	syl2an 596	. . . . . 6 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → (((1^st ‘𝑧) = (1^st ‘𝑘) ∧ (2^nd ‘𝑧) = (2^nd ‘𝑘)) ↔ 𝑧 = 𝑘))
198	197	ad2antlr 727	. . . . 5 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → (((1^st ‘𝑧) = (1^st ‘𝑘) ∧ (2^nd ‘𝑧) = (2^nd ‘𝑘)) ↔ 𝑧 = 𝑘))
199	193, 195, 198	mpbi2and 712	. . . 4 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → 𝑧 = 𝑘)
200	199	ex 412	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) → (((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0) → 𝑧 = 𝑘))
201	200	ralrimivva 3188	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 ∀𝑘 ∈ 𝑆 (((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0) → 𝑧 = 𝑘))
202		fveqeq2 6890	. . 3 ⊢ (𝑧 = 𝑘 → ((2^nd ‘𝑧) = 0 ↔ (2^nd ‘𝑘) = 0))
203	202	rmo4 3718	. 2 ⊢ (∃*𝑧 ∈ 𝑆 (2^nd ‘𝑧) = 0 ↔ ∀𝑧 ∈ 𝑆 ∀𝑘 ∈ 𝑆 (((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0) → 𝑧 = 𝑘))
204	201, 203	sylibr 234	1 ⊢ (𝜑 → ∃*𝑧 ∈ 𝑆 (2^nd ‘𝑧) = 0)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 {cab 2714 ≠ wne 2933 ∀wral 3052 ∃*wrmo 3363 {crab 3420 ⦋csb 3879 ∖ cdif 3928 ∪ cun 3929 ∩ cin 3930 ∅c0 4313 ifcif 4505 {csn 4606 class class class wbr 5124 ↦ cmpt 5206 × cxp 5657 ^◡ccnv 5658 “ cima 5662 Fun wfun 6530 Fn wfn 6531 ⟶wf 6532 –onto→wfo 6534 –1-1-onto→wf1o 6535 ‘cfv 6536 (class class class)co 7410 ∘_f cof 7674 1^st c1st 7991 2^nd c2nd 7992 ↑_m cmap 8845 ℂcc 11132 ℝcr 11133 0cc0 11134 1c1 11135 + caddc 11137 < clt 11274 ≤ cle 11275 − cmin 11471 ℕcn 12245 ℕ₀cn0 12506 ℤcz 12593 ℤ_≥cuz 12857 ...cfz 13529 ..^cfzo 13676

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7676 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8724 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-nn 12246 df-n0 12507 df-z 12594 df-uz 12858 df-fz 13530 df-fzo 13677

This theorem is referenced by: poimirlem18 37667 poimirlem21 37670

W3C validator