poimirlem13 - Mathbox for Brendan Leahy

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Theorem poimirlem13 38552

Description: Lemma for poimir 38572- 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 739	. . . . . . . 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 739	. . . . . . . 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 789	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → 𝑧 ∈ 𝑆)
7		simprl 783	. . . . . . . 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 38549	. . . . . . 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 790	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → 𝑘 ∈ 𝑆)
10		simprr 785	. . . . . . . 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 38549	. . . . . . 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 2797	. . . . . 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 3640	. . . . . . . . . . . . . 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 2878	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
15		xp1st 8033	. . . . . . . . . . . . 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 18	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑆 → (1^st ‘𝑧) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
17		xp2nd 8034	. . . . . . . . . . . 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 18	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑆 → (2^nd ‘(1^st ‘𝑧)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
19		fvex 6898	. . . . . . . . . . . 12 ⊢ (2^nd ‘(1^st ‘𝑧)) ∈ V
20		f1oeq1 6812	. . . . . . . . . . . 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 3632	. . . . . . . . . . 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 221	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑆 → (2^nd ‘(1^st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
23		f1ofn 6825	. . . . . . . . . 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 18	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑆 → (2^nd ‘(1^st ‘𝑧)) Fn (1...𝑁))
25	24	adantr 486	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → (2^nd ‘(1^st ‘𝑧)) Fn (1...𝑁))
26	25	ad2antlr 740	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → (2^nd ‘(1^st ‘𝑧)) Fn (1...𝑁))
27		elrabi 3640	. . . . . . . . . . . . . 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 2878	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝑆 → 𝑘 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
29		xp1st 8033	. . . . . . . . . . . . 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 18	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝑆 → (1^st ‘𝑘) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
31		xp2nd 8034	. . . . . . . . . . . 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 18	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝑆 → (2^nd ‘(1^st ‘𝑘)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
33		fvex 6898	. . . . . . . . . . . 12 ⊢ (2^nd ‘(1^st ‘𝑘)) ∈ V
34		f1oeq1 6812	. . . . . . . . . . . 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 3632	. . . . . . . . . . 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 221	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝑆 → (2^nd ‘(1^st ‘𝑘)):(1...𝑁)–1-1-onto→(1...𝑁))
37		f1ofn 6825	. . . . . . . . . 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 18	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝑆 → (2^nd ‘(1^st ‘𝑘)) Fn (1...𝑁))
39	38	adantl 487	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → (2^nd ‘(1^st ‘𝑘)) Fn (1...𝑁))
40	39	ad2antlr 740	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → (2^nd ‘(1^st ‘𝑘)) Fn (1...𝑁))
41		eleq1 2848	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → (𝑚 ∈ (1...𝑁) ↔ 𝑛 ∈ (1...𝑁)))
42	41	anbi2d 642	. . . . . . . . . . . 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 7428	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
44	43	imaeq2d 6051	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → ((2^nd ‘(1^st ‘𝑧)) “ (1...𝑚)) = ((2^nd ‘(1^st ‘𝑧)) “ (1...𝑛)))
45	43	imaeq2d 6051	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → ((2^nd ‘(1^st ‘𝑘)) “ (1...𝑚)) = ((2^nd ‘(1^st ‘𝑘)) “ (1...𝑛)))
46	44, 45	eqeq12d 2776	. . . . . . . . . . . 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 347	. . . . . . . . . . 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 743	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑁 ∈ ℕ)
49	4	ad3antrrr 743	. . . . . . . . . . . . 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 488	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → 𝑧 ∈ 𝑆)
51	50	ad3antlr 744	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑧 ∈ 𝑆)
52		simplrl 789	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → (2^nd ‘𝑧) = 0)
53		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → 𝑘 ∈ 𝑆)
54	53	ad3antlr 744	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑘 ∈ 𝑆)
55		simplrr 790	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → (2^nd ‘𝑘) = 0)
56		simpr 490	. . . . . . . . . . . . 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 38550	. . . . . . . . . . . 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 38550	. . . . . . . . . . . 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 3925	. . . . . . . . . . 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 2021	. . . . . . . . . 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 779	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → 𝜑)
62		elfznn 13687	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℕ)
63		nnm1nn0 12647	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ₀)
64	62, 63	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ∈ ℕ₀)
65	64	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1) → (𝑛 − 1) ∈ ℕ₀)
66	62	nncnd 12351	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℂ)
67		ax-1cn 11258	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℂ
68		subeq0 11584	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) = 0 ↔ 𝑛 = 1))
69	66, 67, 68	sylancl 598	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) = 0 ↔ 𝑛 = 1))
70	69	necon3abid 2991	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) ≠ 0 ↔ ¬ 𝑛 = 1))
71	70	biimpar 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1) → (𝑛 − 1) ≠ 0)
72		elnnne0 12620	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 − 1) ∈ ℕ ↔ ((𝑛 − 1) ∈ ℕ₀ ∧ (𝑛 − 1) ≠ 0))
73	65, 71, 72	sylanbrc 595	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1) → (𝑛 − 1) ∈ ℕ)
74	73	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → (𝑛 − 1) ∈ ℕ)
75	64	nn0red 12668	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ∈ ℝ)
76	75	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ∈ ℝ)
77	62	nnred 12350	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℝ)
78	77	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℝ)
79	1	nnred 12350	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℝ)
80	79	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → 𝑁 ∈ ℝ)
81	77	lem1d 12250	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ≤ 𝑛)
82	81	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ≤ 𝑛)
83		elfzle2 13661	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ≤ 𝑁)
84	83	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ≤ 𝑁)
85	76, 78, 80, 82, 84	letrd 11467	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ≤ 𝑁)
86	85	adantrr 730	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → (𝑛 − 1) ≤ 𝑁)
87	1	nnzd 12719	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ ℤ)
88		fznn 13726	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℤ → ((𝑛 − 1) ∈ (1...𝑁) ↔ ((𝑛 − 1) ∈ ℕ ∧ (𝑛 − 1) ≤ 𝑁)))
89	87, 88	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑛 − 1) ∈ (1...𝑁) ↔ ((𝑛 − 1) ∈ ℕ ∧ (𝑛 − 1) ≤ 𝑁)))
90	89	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → ((𝑛 − 1) ∈ (1...𝑁) ↔ ((𝑛 − 1) ∈ ℕ ∧ (𝑛 − 1) ≤ 𝑁)))
91	74, 86, 90	mpbir2and 726	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → (𝑛 − 1) ∈ (1...𝑁))
92	61, 91	sylan 592	. . . . . . . . . . . . 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 7453	. . . . . . . . . . . . . 14 ⊢ (𝑛 − 1) ∈ V
94		eleq1 2848	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (𝑛 − 1) → (𝑚 ∈ (1...𝑁) ↔ (𝑛 − 1) ∈ (1...𝑁)))
95	94	anbi2d 642	. . . . . . . . . . . . . . 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 7428	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = (𝑛 − 1) → (1...𝑚) = (1...(𝑛 − 1)))
97	96	imaeq2d 6051	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (𝑛 − 1) → ((2^nd ‘(1^st ‘𝑧)) “ (1...𝑚)) = ((2^nd ‘(1^st ‘𝑧)) “ (1...(𝑛 − 1))))
98	96	imaeq2d 6051	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (𝑛 − 1) → ((2^nd ‘(1^st ‘𝑘)) “ (1...𝑚)) = ((2^nd ‘(1^st ‘𝑘)) “ (1...(𝑛 − 1))))
99	97, 98	eqeq12d 2776	. . . . . . . . . . . . . . 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 347	. . . . . . . . . . . . . 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 3520	. . . . . . . . . . . . 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 603	. . . . . . . . . . . 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 462	. . . . . . . . . . 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 6074	. . . . . . . . . . . . 13 ⊢ ((2^nd ‘(1^st ‘𝑧)) “ ∅) = ∅
105		ima0 6074	. . . . . . . . . . . . 13 ⊢ ((2^nd ‘(1^st ‘𝑘)) “ ∅) = ∅
106	104, 105	eqtr4i 2786	. . . . . . . . . . . 12 ⊢ ((2^nd ‘(1^st ‘𝑧)) “ ∅) = ((2^nd ‘(1^st ‘𝑘)) “ ∅)
107		oveq1 7427	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 1 → (𝑛 − 1) = (1 − 1))
108		1m1e0 12415	. . . . . . . . . . . . . . . 16 ⊢ (1 − 1) = 0
109	107, 108	eqtrdi 2811	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 1 → (𝑛 − 1) = 0)
110	109	oveq2d 7436	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 1 → (1...(𝑛 − 1)) = (1...0))
111		fz10 13678	. . . . . . . . . . . . . 14 ⊢ (1...0) = ∅
112	110, 111	eqtrdi 2811	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1 → (1...(𝑛 − 1)) = ∅)
113	112	imaeq2d 6051	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → ((2^nd ‘(1^st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2^nd ‘(1^st ‘𝑧)) “ ∅))
114	112	imaeq2d 6051	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → ((2^nd ‘(1^st ‘𝑘)) “ (1...(𝑛 − 1))) = ((2^nd ‘(1^st ‘𝑘)) “ ∅))
115	106, 113, 114	3eqtr4a 2821	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → ((2^nd ‘(1^st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2^nd ‘(1^st ‘𝑘)) “ (1...(𝑛 − 1))))
116	103, 115	pm2.61d2 183	. . . . . . . . . 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 4064	. . . . . . . . 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 6964	. . . . . . . . . . . 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 592	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → {((2^nd ‘(1^st ‘𝑧))‘𝑛)} = ((2^nd ‘(1^st ‘𝑧)) “ {𝑛}))
120	62	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℕ)
121		uncom 4157	. . . . . . . . . . . . . . . 16 ⊢ ((1...(𝑛 − 1)) ∪ {𝑛}) = ({𝑛} ∪ (1...(𝑛 − 1)))
122	121	difeq1i 4059	. . . . . . . . . . . . . . 15 ⊢ (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))) = (({𝑛} ∪ (1...(𝑛 − 1))) ∖ (1...(𝑛 − 1)))
123		difun2 4436	. . . . . . . . . . . . . . 15 ⊢ (({𝑛} ∪ (1...(𝑛 − 1))) ∖ (1...(𝑛 − 1))) = ({𝑛} ∖ (1...(𝑛 − 1)))
124	122, 123	eqtri 2783	. . . . . . . . . . . . . 14 ⊢ (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))) = ({𝑛} ∖ (1...(𝑛 − 1)))
125		nncn 12343	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
126		npcan1 11741	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℂ → ((𝑛 − 1) + 1) = 𝑛)
127	125, 126	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) = 𝑛)
128		elnnuz 13005	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ_≥‘1))
129	128	biimpi 219	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ_≥‘1))
130	127, 129	eqeltrd 2860	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) ∈ (ℤ_≥‘1))
131	63	nn0zd 12718	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℤ)
132		uzid 12980	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 − 1) ∈ ℤ → (𝑛 − 1) ∈ (ℤ_≥‘(𝑛 − 1)))
133	131, 132	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ (ℤ_≥‘(𝑛 − 1)))
134		peano2uz 13028	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 − 1) ∈ (ℤ_≥‘(𝑛 − 1)) → ((𝑛 − 1) + 1) ∈ (ℤ_≥‘(𝑛 − 1)))
135	133, 134	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) ∈ (ℤ_≥‘(𝑛 − 1)))
136	127, 135	eqeltrrd 2861	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ_≥‘(𝑛 − 1)))
137		fzsplit2 13683	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 − 1) + 1) ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘(𝑛 − 1))) → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)))
138	130, 136, 137	syl2anc 596	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)))
139	127	oveq1d 7435	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → (((𝑛 − 1) + 1)...𝑛) = (𝑛...𝑛))
140		nnz 12714	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
141		fzsn 13700	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℤ → (𝑛...𝑛) = {𝑛})
142	140, 141	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → (𝑛...𝑛) = {𝑛})
143	139, 142	eqtrd 2795	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → (((𝑛 − 1) + 1)...𝑛) = {𝑛})
144	143	uneq2d 4167	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)) = ((1...(𝑛 − 1)) ∪ {𝑛}))
145	138, 144	eqtrd 2795	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → (1...𝑛) = ((1...(𝑛 − 1)) ∪ {𝑛}))
146	145	difeq1d 4062	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → ((1...𝑛) ∖ (1...(𝑛 − 1))) = (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))))
147		nnre 12342	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
148		ltm1 12159	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℝ → (𝑛 − 1) < 𝑛)
149		peano2rem 11625	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℝ → (𝑛 − 1) ∈ ℝ)
150		ltnle 11389	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 − 1) ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((𝑛 − 1) < 𝑛 ↔ ¬ 𝑛 ≤ (𝑛 − 1)))
151	149, 150	mpancom 701	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℝ → ((𝑛 − 1) < 𝑛 ↔ ¬ 𝑛 ≤ (𝑛 − 1)))
152	148, 151	mpbid 235	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℝ → ¬ 𝑛 ≤ (𝑛 − 1))
153		elfzle2 13661	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (1...(𝑛 − 1)) → 𝑛 ≤ (𝑛 − 1))
154	152, 153	nsyl 141	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℝ → ¬ 𝑛 ∈ (1...(𝑛 − 1)))
155	147, 154	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → ¬ 𝑛 ∈ (1...(𝑛 − 1)))
156		incom 4106	. . . . . . . . . . . . . . . . 17 ⊢ ((1...(𝑛 − 1)) ∩ {𝑛}) = ({𝑛} ∩ (1...(𝑛 − 1)))
157	156	eqeq1i 2765	. . . . . . . . . . . . . . . 16 ⊢ (((1...(𝑛 − 1)) ∩ {𝑛}) = ∅ ↔ ({𝑛} ∩ (1...(𝑛 − 1))) = ∅)
158		disjsn 4671	. . . . . . . . . . . . . . . 16 ⊢ (((1...(𝑛 − 1)) ∩ {𝑛}) = ∅ ↔ ¬ 𝑛 ∈ (1...(𝑛 − 1)))
159		disj3 4406	. . . . . . . . . . . . . . . 16 ⊢ (({𝑛} ∩ (1...(𝑛 − 1))) = ∅ ↔ {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1))))
160	157, 158, 159	3bitr3i 304	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑛 ∈ (1...(𝑛 − 1)) ↔ {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1))))
161	155, 160	sylib 221	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1))))
162	124, 146, 161	3eqtr4a 2821	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → ((1...𝑛) ∖ (1...(𝑛 − 1))) = {𝑛})
163	120, 162	syl 18	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → ((1...𝑛) ∖ (1...(𝑛 − 1))) = {𝑛})
164	163	imaeq2d 6051	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → ((2^nd ‘(1^st ‘𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = ((2^nd ‘(1^st ‘𝑧)) “ {𝑛}))
165		dff1o3 6831	. . . . . . . . . . . . . . 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 503	. . . . . . . . . . . . . 14 ⊢ ((2^nd ‘(1^st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ^◡(2^nd ‘(1^st ‘𝑧)))
167	22, 166	syl 18	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑆 → Fun ^◡(2^nd ‘(1^st ‘𝑧)))
168		imadif 6624	. . . . . . . . . . . . 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 18	. . . . . . . . . . . 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 486	. . . . . . . . . . 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 2801	. . . . . . . . . 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 592	. . . . . . . . 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 2848	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑘 → (𝑧 ∈ 𝑆 ↔ 𝑘 ∈ 𝑆))
174	173	anbi1d 643	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑘 → ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) ↔ (𝑘 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁))))
175		2fveq3 6890	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑘 → (2^nd ‘(1^st ‘𝑧)) = (2^nd ‘(1^st ‘𝑘)))
176	175	fveq1d 6887	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑘 → ((2^nd ‘(1^st ‘𝑧))‘𝑛) = ((2^nd ‘(1^st ‘𝑘))‘𝑛))
177	176	sneqd 4595	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑘 → {((2^nd ‘(1^st ‘𝑧))‘𝑛)} = {((2^nd ‘(1^st ‘𝑘))‘𝑛)})
178	175	imaeq1d 6050	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑘 → ((2^nd ‘(1^st ‘𝑧)) “ (1...𝑛)) = ((2^nd ‘(1^st ‘𝑘)) “ (1...𝑛)))
179	175	imaeq1d 6050	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑘 → ((2^nd ‘(1^st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2^nd ‘(1^st ‘𝑘)) “ (1...(𝑛 − 1))))
180	178, 179	difeq12d 4064	. . . . . . . . . . . . 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 2776	. . . . . . . . . . . 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 347	. . . . . . . . . . 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 2021	. . . . . . . . . 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 592	. . . . . . . . 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 2805	. . . . . . . 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 6898	. . . . . . . . 9 ⊢ ((2^nd ‘(1^st ‘𝑧))‘𝑛) ∈ V
187	186	sneqr 4799	. . . . . . . 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 18	. . . . . . 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 7032	. . . . . 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 8042	. . . . . . . 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 608	. . . . . . 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 740	. . . . . 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 725	. . . . 5 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → (1^st ‘𝑧) = (1^st ‘𝑘))
194		eqtr3 2782	. . . . . 6 ⊢ (((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0) → (2^nd ‘𝑧) = (2^nd ‘𝑘))
195	194	adantl 487	. . . . 5 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → (2^nd ‘𝑧) = (2^nd ‘𝑘))
196		xpopth 8042	. . . . . . 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 608	. . . . . 6 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → (((1^st ‘𝑧) = (1^st ‘𝑘) ∧ (2^nd ‘𝑧) = (2^nd ‘𝑘)) ↔ 𝑧 = 𝑘))
198	197	ad2antlr 740	. . . . 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 725	. . . 4 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → 𝑧 = 𝑘)
200	199	ex 418	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) → (((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0) → 𝑧 = 𝑘))
201	200	ralrimivva 3205	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 ∀𝑘 ∈ 𝑆 (((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0) → 𝑧 = 𝑘))
202		fveqeq2 6894	. . 3 ⊢ (𝑧 = 𝑘 → ((2^nd ‘𝑧) = 0 ↔ (2^nd ‘𝑘) = 0))
203	202	rmo4 3687	. 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 237	1 ⊢ (𝜑 → ∃*𝑧 ∈ 𝑆 (2^nd ‘𝑧) = 0)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1569 ∈ wcel 2144 {cab 2738 ≠ wne 2955 ∀wral 3076 ∃*wrmo 3364 {crab 3412 ⦋csb 3846 ∖ cdif 4045 ∩ cin 4089 ∪ cun 4149 ∅c0 4278 ifcif 4481 {csn 4583 class class class wbr 5102 ↦ cmpt 5185 × cxp 5648 ^◡ccnv 5649 “ cima 5653 Fun wfun 6532 Fn wfn 6533 ⟶wf 6534 –onto→wfo 6536 –1-1-onto→wf1o 6537 ‘cfv 6538 (class class class)co 7420 ∘_f cof 7691 1^st c1st 7999 2^nd c2nd 8000 ↑_m cmap 8847 ℂcc 11198 ℝcr 11199 0cc0 11200 1c1 11201 + caddc 11203 < clt 11343 ≤ cle 11344 − cmin 11541 ℕcn 12335 ℕ₀cn0 12606 ℤcz 12693 ℤ_≥cuz 12965 ...cfz 13639 ..^cfzo 13788

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-er 8717 df-map 8849 df-en 8974 df-dom 8975 df-sdom 8976 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-nn 12336 df-n0 12607 df-z 12694 df-uz 12966 df-fz 13640 df-fzo 13789

This theorem is used by: poimirlem18 38557 poimirlem21 38560

W3C validator