poimirlem13 - Mathbox for Brendan Leahy

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Theorem poimirlem13 33422

Description: Lemma for poimir 33442- 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 762	. . . . . . . 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 762	. . . . . . . 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 800	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → 𝑧 ∈ 𝑆)
7		simprl 794	. . . . . . . 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 33419	. . . . . . 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 801	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → 𝑘 ∈ 𝑆)
10		simprr 796	. . . . . . . 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 33419	. . . . . . 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 2658	. . . . . 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 3359	. . . . . . . . . . . . . 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 2719	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ∈ ((((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
15		xp1st 7198	. . . . . . . . . . . . 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 7199	. . . . . . . . . . . 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 6201	. . . . . . . . . . . 12 ⊢ (2^nd ‘(1^st ‘𝑧)) ∈ V
20		f1oeq1 6127	. . . . . . . . . . . 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 3350	. . . . . . . . . . 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 208	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑆 → (2^nd ‘(1^st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
23		f1ofn 6138	. . . . . . . . . 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 481	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → (2^nd ‘(1^st ‘𝑧)) Fn (1...𝑁))
26	25	ad2antlr 763	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → (2^nd ‘(1^st ‘𝑧)) Fn (1...𝑁))
27		elrabi 3359	. . . . . . . . . . . . . 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 2719	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝑆 → 𝑘 ∈ ((((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
29		xp1st 7198	. . . . . . . . . . . . 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 7199	. . . . . . . . . . . 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 6201	. . . . . . . . . . . 12 ⊢ (2^nd ‘(1^st ‘𝑘)) ∈ V
34		f1oeq1 6127	. . . . . . . . . . . 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 3350	. . . . . . . . . . 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 208	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝑆 → (2^nd ‘(1^st ‘𝑘)):(1...𝑁)–1-1-onto→(1...𝑁))
37		f1ofn 6138	. . . . . . . . . 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 482	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → (2^nd ‘(1^st ‘𝑘)) Fn (1...𝑁))
40	39	ad2antlr 763	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → (2^nd ‘(1^st ‘𝑘)) Fn (1...𝑁))
41		eleq1 2689	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → (𝑚 ∈ (1...𝑁) ↔ 𝑛 ∈ (1...𝑁)))
42	41	anbi2d 740	. . . . . . . . . . . 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 6658	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
44	43	imaeq2d 5466	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → ((2^nd ‘(1^st ‘𝑧)) “ (1...𝑚)) = ((2^nd ‘(1^st ‘𝑧)) “ (1...𝑛)))
45	43	imaeq2d 5466	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → ((2^nd ‘(1^st ‘𝑘)) “ (1...𝑚)) = ((2^nd ‘(1^st ‘𝑘)) “ (1...𝑛)))
46	44, 45	eqeq12d 2637	. . . . . . . . . . . 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 334	. . . . . . . . . . 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 766	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑁 ∈ ℕ)
49	4	ad3antrrr 766	. . . . . . . . . . . . 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 473	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → 𝑧 ∈ 𝑆)
51	50	ad3antlr 767	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑧 ∈ 𝑆)
52		simplrl 800	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → (2^nd ‘𝑧) = 0)
53		simpr 477	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → 𝑘 ∈ 𝑆)
54	53	ad3antlr 767	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑘 ∈ 𝑆)
55		simplrr 801	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → (2^nd ‘𝑘) = 0)
56		simpr 477	. . . . . . . . . . . . 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 33420	. . . . . . . . . . . 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 33420	. . . . . . . . . . . 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 3620	. . . . . . . . . . 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	chvarv 2263	. . . . . . . . . 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 790	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → 𝜑)
62		elfznn 12370	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℕ)
63		nnm1nn0 11334	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ₀)
64	62, 63	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ∈ ℕ₀)
65	64	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1) → (𝑛 − 1) ∈ ℕ₀)
66	62	nncnd 11036	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℂ)
67		ax-1cn 9994	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℂ
68		subeq0 10307	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) = 0 ↔ 𝑛 = 1))
69	66, 67, 68	sylancl 694	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) = 0 ↔ 𝑛 = 1))
70	69	necon3abid 2830	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) ≠ 0 ↔ ¬ 𝑛 = 1))
71	70	biimpar 502	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1) → (𝑛 − 1) ≠ 0)
72		elnnne0 11306	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 − 1) ∈ ℕ ↔ ((𝑛 − 1) ∈ ℕ₀ ∧ (𝑛 − 1) ≠ 0))
73	65, 71, 72	sylanbrc 698	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1) → (𝑛 − 1) ∈ ℕ)
74	73	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → (𝑛 − 1) ∈ ℕ)
75	64	nn0red 11352	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ∈ ℝ)
76	75	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ∈ ℝ)
77	62	nnred 11035	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℝ)
78	77	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℝ)
79	1	nnred 11035	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℝ)
80	79	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → 𝑁 ∈ ℝ)
81	77	lem1d 10957	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ≤ 𝑛)
82	81	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ≤ 𝑛)
83		elfzle2 12345	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ≤ 𝑁)
84	83	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ≤ 𝑁)
85	76, 78, 80, 82, 84	letrd 10194	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ≤ 𝑁)
86	85	adantrr 753	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → (𝑛 − 1) ≤ 𝑁)
87	1	nnzd 11481	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ ℤ)
88		fznn 12408	. . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → ((𝑛 − 1) ∈ (1...𝑁) ↔ ((𝑛 − 1) ∈ ℕ ∧ (𝑛 − 1) ≤ 𝑁)))
91	74, 86, 90	mpbir2and 957	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → (𝑛 − 1) ∈ (1...𝑁))
92	61, 91	sylan 488	. . . . . . . . . . . . 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 6678	. . . . . . . . . . . . . 14 ⊢ (𝑛 − 1) ∈ V
94		eleq1 2689	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (𝑛 − 1) → (𝑚 ∈ (1...𝑁) ↔ (𝑛 − 1) ∈ (1...𝑁)))
95	94	anbi2d 740	. . . . . . . . . . . . . . 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 6658	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = (𝑛 − 1) → (1...𝑚) = (1...(𝑛 − 1)))
97	96	imaeq2d 5466	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (𝑛 − 1) → ((2^nd ‘(1^st ‘𝑧)) “ (1...𝑚)) = ((2^nd ‘(1^st ‘𝑧)) “ (1...(𝑛 − 1))))
98	96	imaeq2d 5466	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (𝑛 − 1) → ((2^nd ‘(1^st ‘𝑘)) “ (1...𝑚)) = ((2^nd ‘(1^st ‘𝑘)) “ (1...(𝑛 − 1))))
99	97, 98	eqeq12d 2637	. . . . . . . . . . . . . . 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 334	. . . . . . . . . . . . . 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 3259	. . . . . . . . . . . . 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 487	. . . . . . . . . . . 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 643	. . . . . . . . . . 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 5481	. . . . . . . . . . . . 13 ⊢ ((2^nd ‘(1^st ‘𝑧)) “ ∅) = ∅
105		ima0 5481	. . . . . . . . . . . . 13 ⊢ ((2^nd ‘(1^st ‘𝑘)) “ ∅) = ∅
106	104, 105	eqtr4i 2647	. . . . . . . . . . . 12 ⊢ ((2^nd ‘(1^st ‘𝑧)) “ ∅) = ((2^nd ‘(1^st ‘𝑘)) “ ∅)
107		oveq1 6657	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 1 → (𝑛 − 1) = (1 − 1))
108		1m1e0 11089	. . . . . . . . . . . . . . . 16 ⊢ (1 − 1) = 0
109	107, 108	syl6eq 2672	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 1 → (𝑛 − 1) = 0)
110	109	oveq2d 6666	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 1 → (1...(𝑛 − 1)) = (1...0))
111		fz10 12362	. . . . . . . . . . . . . 14 ⊢ (1...0) = ∅
112	110, 111	syl6eq 2672	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1 → (1...(𝑛 − 1)) = ∅)
113	112	imaeq2d 5466	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → ((2^nd ‘(1^st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2^nd ‘(1^st ‘𝑧)) “ ∅))
114	112	imaeq2d 5466	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → ((2^nd ‘(1^st ‘𝑘)) “ (1...(𝑛 − 1))) = ((2^nd ‘(1^st ‘𝑘)) “ ∅))
115	106, 113, 114	3eqtr4a 2682	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → ((2^nd ‘(1^st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2^nd ‘(1^st ‘𝑘)) “ (1...(𝑛 − 1))))
116	103, 115	pm2.61d2 172	. . . . . . . . . 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 3729	. . . . . . . . 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 6258	. . . . . . . . . . . 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 488	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → {((2^nd ‘(1^st ‘𝑧))‘𝑛)} = ((2^nd ‘(1^st ‘𝑧)) “ {𝑛}))
120	62	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℕ)
121		uncom 3757	. . . . . . . . . . . . . . . 16 ⊢ ((1...(𝑛 − 1)) ∪ {𝑛}) = ({𝑛} ∪ (1...(𝑛 − 1)))
122	121	difeq1i 3724	. . . . . . . . . . . . . . 15 ⊢ (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))) = (({𝑛} ∪ (1...(𝑛 − 1))) ∖ (1...(𝑛 − 1)))
123		difun2 4048	. . . . . . . . . . . . . . 15 ⊢ (({𝑛} ∪ (1...(𝑛 − 1))) ∖ (1...(𝑛 − 1))) = ({𝑛} ∖ (1...(𝑛 − 1)))
124	122, 123	eqtri 2644	. . . . . . . . . . . . . 14 ⊢ (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))) = ({𝑛} ∖ (1...(𝑛 − 1)))
125		nncn 11028	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
126		npcan1 10455	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℂ → ((𝑛 − 1) + 1) = 𝑛)
127	125, 126	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) = 𝑛)
128		elnnuz 11724	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ_≥‘1))
129	128	biimpi 206	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ_≥‘1))
130	127, 129	eqeltrd 2701	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) ∈ (ℤ_≥‘1))
131	63	nn0zd 11480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℤ)
132		uzid 11702	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 − 1) ∈ ℤ → (𝑛 − 1) ∈ (ℤ_≥‘(𝑛 − 1)))
133	131, 132	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ (ℤ_≥‘(𝑛 − 1)))
134		peano2uz 11741	. . . . . . . . . . . . . . . . . . 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 2702	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ_≥‘(𝑛 − 1)))
137		fzsplit2 12366	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 − 1) + 1) ∈ (ℤ_≥‘1) ∧ 𝑛 ∈ (ℤ_≥‘(𝑛 − 1))) → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)))
138	130, 136, 137	syl2anc 693	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)))
139	127	oveq1d 6665	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → (((𝑛 − 1) + 1)...𝑛) = (𝑛...𝑛))
140		nnz 11399	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
141		fzsn 12383	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℤ → (𝑛...𝑛) = {𝑛})
142	140, 141	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → (𝑛...𝑛) = {𝑛})
143	139, 142	eqtrd 2656	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → (((𝑛 − 1) + 1)...𝑛) = {𝑛})
144	143	uneq2d 3767	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)) = ((1...(𝑛 − 1)) ∪ {𝑛}))
145	138, 144	eqtrd 2656	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → (1...𝑛) = ((1...(𝑛 − 1)) ∪ {𝑛}))
146	145	difeq1d 3727	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → ((1...𝑛) ∖ (1...(𝑛 − 1))) = (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))))
147		nnre 11027	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
148		ltm1 10863	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℝ → (𝑛 − 1) < 𝑛)
149		peano2rem 10348	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℝ → (𝑛 − 1) ∈ ℝ)
150		ltnle 10117	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 − 1) ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((𝑛 − 1) < 𝑛 ↔ ¬ 𝑛 ≤ (𝑛 − 1)))
151	149, 150	mpancom 703	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℝ → ((𝑛 − 1) < 𝑛 ↔ ¬ 𝑛 ≤ (𝑛 − 1)))
152	148, 151	mpbid 222	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℝ → ¬ 𝑛 ≤ (𝑛 − 1))
153		elfzle2 12345	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (1...(𝑛 − 1)) → 𝑛 ≤ (𝑛 − 1))
154	152, 153	nsyl 135	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℝ → ¬ 𝑛 ∈ (1...(𝑛 − 1)))
155	147, 154	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → ¬ 𝑛 ∈ (1...(𝑛 − 1)))
156		incom 3805	. . . . . . . . . . . . . . . . 17 ⊢ ((1...(𝑛 − 1)) ∩ {𝑛}) = ({𝑛} ∩ (1...(𝑛 − 1)))
157	156	eqeq1i 2627	. . . . . . . . . . . . . . . 16 ⊢ (((1...(𝑛 − 1)) ∩ {𝑛}) = ∅ ↔ ({𝑛} ∩ (1...(𝑛 − 1))) = ∅)
158		disjsn 4246	. . . . . . . . . . . . . . . 16 ⊢ (((1...(𝑛 − 1)) ∩ {𝑛}) = ∅ ↔ ¬ 𝑛 ∈ (1...(𝑛 − 1)))
159		disj3 4021	. . . . . . . . . . . . . . . 16 ⊢ (({𝑛} ∩ (1...(𝑛 − 1))) = ∅ ↔ {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1))))
160	157, 158, 159	3bitr3i 290	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑛 ∈ (1...(𝑛 − 1)) ↔ {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1))))
161	155, 160	sylib 208	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1))))
162	124, 146, 161	3eqtr4a 2682	. . . . . . . . . . . . 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 5466	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → ((2^nd ‘(1^st ‘𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = ((2^nd ‘(1^st ‘𝑧)) “ {𝑛}))
165		dff1o3 6143	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 5973	. . . . . . . . . . . . 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 481	. . . . . . . . . . 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 2662	. . . . . . . . . 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 488	. . . . . . . . 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 2689	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑘 → (𝑧 ∈ 𝑆 ↔ 𝑘 ∈ 𝑆))
174	173	anbi1d 741	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑘 → ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) ↔ (𝑘 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁))))
175		fveq2 6191	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑘 → (1^st ‘𝑧) = (1^st ‘𝑘))
176	175	fveq2d 6195	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑘 → (2^nd ‘(1^st ‘𝑧)) = (2^nd ‘(1^st ‘𝑘)))
177	176	fveq1d 6193	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑘 → ((2^nd ‘(1^st ‘𝑧))‘𝑛) = ((2^nd ‘(1^st ‘𝑘))‘𝑛))
178	177	sneqd 4189	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑘 → {((2^nd ‘(1^st ‘𝑧))‘𝑛)} = {((2^nd ‘(1^st ‘𝑘))‘𝑛)})
179	176	imaeq1d 5465	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑘 → ((2^nd ‘(1^st ‘𝑧)) “ (1...𝑛)) = ((2^nd ‘(1^st ‘𝑘)) “ (1...𝑛)))
180	176	imaeq1d 5465	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑘 → ((2^nd ‘(1^st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2^nd ‘(1^st ‘𝑘)) “ (1...(𝑛 − 1))))
181	179, 180	difeq12d 3729	. . . . . . . . . . . . 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)))))
182	178, 181	eqeq12d 2637	. . . . . . . . . . . 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))))))
183	174, 182	imbi12d 334	. . . . . . . . . . 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)))))))
184	183, 171	chvarv 2263	. . . . . . . . . 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)))))
185	9, 184	sylan 488	. . . . . . . . 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)))))
186	117, 172, 185	3eqtr4d 2666	. . . . . . . 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)})
187		fvex 6201	. . . . . . . . 9 ⊢ ((2^nd ‘(1^st ‘𝑧))‘𝑛) ∈ V
188	187	sneqr 4371	. . . . . . . 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))
189	186, 188	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))
190	26, 40, 189	eqfnfvd 6314	. . . . . 6 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → (2^nd ‘(1^st ‘𝑧)) = (2^nd ‘(1^st ‘𝑘)))
191		xpopth 7207	. . . . . . . 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)))
192	16, 30, 191	syl2an 494	. . . . . . 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)))
193	192	ad2antlr 763	. . . . . 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)))
194	12, 190, 193	mpbi2and 956	. . . . 5 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → (1^st ‘𝑧) = (1^st ‘𝑘))
195		eqtr3 2643	. . . . . 6 ⊢ (((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0) → (2^nd ‘𝑧) = (2^nd ‘𝑘))
196	195	adantl 482	. . . . 5 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → (2^nd ‘𝑧) = (2^nd ‘𝑘))
197		xpopth 7207	. . . . . . 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))
198	14, 28, 197	syl2an 494	. . . . . 6 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → (((1^st ‘𝑧) = (1^st ‘𝑘) ∧ (2^nd ‘𝑧) = (2^nd ‘𝑘)) ↔ 𝑧 = 𝑘))
199	198	ad2antlr 763	. . . . 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))
200	194, 196, 199	mpbi2and 956	. . . 4 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → 𝑧 = 𝑘)
201	200	ex 450	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) → (((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0) → 𝑧 = 𝑘))
202	201	ralrimivva 2971	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 ∀𝑘 ∈ 𝑆 (((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0) → 𝑧 = 𝑘))
203		fveq2 6191	. . . 4 ⊢ (𝑧 = 𝑘 → (2^nd ‘𝑧) = (2^nd ‘𝑘))
204	203	eqeq1d 2624	. . 3 ⊢ (𝑧 = 𝑘 → ((2^nd ‘𝑧) = 0 ↔ (2^nd ‘𝑘) = 0))
205	204	rmo4 3399	. 2 ⊢ (∃*𝑧 ∈ 𝑆 (2^nd ‘𝑧) = 0 ↔ ∀𝑧 ∈ 𝑆 ∀𝑘 ∈ 𝑆 (((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0) → 𝑧 = 𝑘))
206	202, 205	sylibr 224	1 ⊢ (𝜑 → ∃*𝑧 ∈ 𝑆 (2^nd ‘𝑧) = 0)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {cab 2608 ≠ wne 2794 ∀wral 2912 ∃*wrmo 2915 {crab 2916 ⦋csb 3533 ∖ cdif 3571 ∪ cun 3572 ∩ cin 3573 ∅c0 3915 ifcif 4086 {csn 4177 class class class wbr 4653 ↦ cmpt 4729 × cxp 5112 ^◡ccnv 5113 “ cima 5117 Fun wfun 5882 Fn wfn 5883 ⟶wf 5884 –onto→wfo 5886 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 ∘_𝑓 cof 6895 1^st c1st 7166 2^nd c2nd 7167 ↑_𝑚 cmap 7857 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 < clt 10074 ≤ cle 10075 − cmin 10266 ℕcn 11020 ℕ₀cn0 11292 ℤcz 11377 ℤ_≥cuz 11687 ...cfz 12326 ..^cfzo 12465

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466

This theorem is referenced by: poimirlem18 33427 poimirlem21 33430

W3C validator