poimirlem13 - Mathbox for Brendan Leahy

	Mathbox for Brendan Leahy		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > poimirlem13		Structured version Visualization version GIF version

Theorem poimirlem13 36091

Description: Lemma for poimir 36111- 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 724	. . . . . . . 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 724	. . . . . . . 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 775	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → 𝑧 ∈ 𝑆)
7		simprl 769	. . . . . . . 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 36088	. . . . . . 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 776	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → 𝑘 ∈ 𝑆)
10		simprr 771	. . . . . . . 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 36088	. . . . . . 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 2778	. . . . . 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 3639	. . . . . . . . . . . . . 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 2856	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
15		xp1st 7953	. . . . . . . . . . . . 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 7954	. . . . . . . . . . . 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 6855	. . . . . . . . . . . 12 ⊢ (2^nd ‘(1^st ‘𝑧)) ∈ V
20		f1oeq1 6772	. . . . . . . . . . . 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 3630	. . . . . . . . . . 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 217	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑆 → (2^nd ‘(1^st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
23		f1ofn 6785	. . . . . . . . . 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 725	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → (2^nd ‘(1^st ‘𝑧)) Fn (1...𝑁))
27		elrabi 3639	. . . . . . . . . . . . . 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 2856	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝑆 → 𝑘 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
29		xp1st 7953	. . . . . . . . . . . . 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 7954	. . . . . . . . . . . 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 6855	. . . . . . . . . . . 12 ⊢ (2^nd ‘(1^st ‘𝑘)) ∈ V
34		f1oeq1 6772	. . . . . . . . . . . 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 3630	. . . . . . . . . . 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 217	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝑆 → (2^nd ‘(1^st ‘𝑘)):(1...𝑁)–1-1-onto→(1...𝑁))
37		f1ofn 6785	. . . . . . . . . 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 725	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → (2^nd ‘(1^st ‘𝑘)) Fn (1...𝑁))
41		eleq1 2825	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → (𝑚 ∈ (1...𝑁) ↔ 𝑛 ∈ (1...𝑁)))
42	41	anbi2d 629	. . . . . . . . . . . 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 7365	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
44	43	imaeq2d 6013	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → ((2^nd ‘(1^st ‘𝑧)) “ (1...𝑚)) = ((2^nd ‘(1^st ‘𝑧)) “ (1...𝑛)))
45	43	imaeq2d 6013	. . . . . . . . . . . . 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 728	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑁 ∈ ℕ)
49	4	ad3antrrr 728	. . . . . . . . . . . . 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 483	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → 𝑧 ∈ 𝑆)
51	50	ad3antlr 729	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑧 ∈ 𝑆)
52		simplrl 775	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → (2^nd ‘𝑧) = 0)
53		simpr 485	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → 𝑘 ∈ 𝑆)
54	53	ad3antlr 729	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑘 ∈ 𝑆)
55		simplrr 776	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → (2^nd ‘𝑘) = 0)
56		simpr 485	. . . . . . . . . . . . 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 36089	. . . . . . . . . . . 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 36089	. . . . . . . . . . . 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 3961	. . . . . . . . . . 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 2002	. . . . . . . . . 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 765	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → 𝜑)
62		elfznn 13470	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℕ)
63		nnm1nn0 12454	. . . . . . . . . . . . . . . . . . 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 12169	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℂ)
67		ax-1cn 11109	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℂ
68		subeq0 11427	. . . . . . . . . . . . . . . . . . . 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 2980	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) ≠ 0 ↔ ¬ 𝑛 = 1))
71	70	biimpar 478	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1) → (𝑛 − 1) ≠ 0)
72		elnnne0 12427	. . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → (𝑛 − 1) ∈ ℕ)
75	64	nn0red 12474	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ∈ ℝ)
76	75	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ∈ ℝ)
77	62	nnred 12168	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℝ)
78	77	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℝ)
79	1	nnred 12168	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℝ)
80	79	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → 𝑁 ∈ ℝ)
81	77	lem1d 12088	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ≤ 𝑛)
82	81	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ≤ 𝑛)
83		elfzle2 13445	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ≤ 𝑁)
84	83	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ≤ 𝑁)
85	76, 78, 80, 82, 84	letrd 11312	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ≤ 𝑁)
86	85	adantrr 715	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → (𝑛 − 1) ≤ 𝑁)
87	1	nnzd 12526	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ ℤ)
88		fznn 13509	. . . . . . . . . . . . . . . . 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 711	. . . . . . . . . . . . . 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 7390	. . . . . . . . . . . . . 14 ⊢ (𝑛 − 1) ∈ V
94		eleq1 2825	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (𝑛 − 1) → (𝑚 ∈ (1...𝑁) ↔ (𝑛 − 1) ∈ (1...𝑁)))
95	94	anbi2d 629	. . . . . . . . . . . . . . 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 7365	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = (𝑛 − 1) → (1...𝑚) = (1...(𝑛 − 1)))
97	96	imaeq2d 6013	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (𝑛 − 1) → ((2^nd ‘(1^st ‘𝑧)) “ (1...𝑚)) = ((2^nd ‘(1^st ‘𝑧)) “ (1...(𝑛 − 1))))
98	96	imaeq2d 6013	. . . . . . . . . . . . . . . 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 3518	. . . . . . . . . . . . 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 457	. . . . . . . . . . 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 6029	. . . . . . . . . . . . 13 ⊢ ((2^nd ‘(1^st ‘𝑧)) “ ∅) = ∅
105		ima0 6029	. . . . . . . . . . . . 13 ⊢ ((2^nd ‘(1^st ‘𝑘)) “ ∅) = ∅
106	104, 105	eqtr4i 2767	. . . . . . . . . . . 12 ⊢ ((2^nd ‘(1^st ‘𝑧)) “ ∅) = ((2^nd ‘(1^st ‘𝑘)) “ ∅)
107		oveq1 7364	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 1 → (𝑛 − 1) = (1 − 1))
108		1m1e0 12225	. . . . . . . . . . . . . . . 16 ⊢ (1 − 1) = 0
109	107, 108	eqtrdi 2792	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 1 → (𝑛 − 1) = 0)
110	109	oveq2d 7373	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 1 → (1...(𝑛 − 1)) = (1...0))
111		fz10 13462	. . . . . . . . . . . . . 14 ⊢ (1...0) = ∅
112	110, 111	eqtrdi 2792	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1 → (1...(𝑛 − 1)) = ∅)
113	112	imaeq2d 6013	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → ((2^nd ‘(1^st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2^nd ‘(1^st ‘𝑧)) “ ∅))
114	112	imaeq2d 6013	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → ((2^nd ‘(1^st ‘𝑘)) “ (1...(𝑛 − 1))) = ((2^nd ‘(1^st ‘𝑘)) “ ∅))
115	106, 113, 114	3eqtr4a 2802	. . . . . . . . . . 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 4083	. . . . . . . . 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 6920	. . . . . . . . . . . 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 482	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℕ)
121		uncom 4113	. . . . . . . . . . . . . . . 16 ⊢ ((1...(𝑛 − 1)) ∪ {𝑛}) = ({𝑛} ∪ (1...(𝑛 − 1)))
122	121	difeq1i 4078	. . . . . . . . . . . . . . 15 ⊢ (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))) = (({𝑛} ∪ (1...(𝑛 − 1))) ∖ (1...(𝑛 − 1)))
123		difun2 4440	. . . . . . . . . . . . . . 15 ⊢ (({𝑛} ∪ (1...(𝑛 − 1))) ∖ (1...(𝑛 − 1))) = ({𝑛} ∖ (1...(𝑛 − 1)))
124	122, 123	eqtri 2764	. . . . . . . . . . . . . 14 ⊢ (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))) = ({𝑛} ∖ (1...(𝑛 − 1)))
125		nncn 12161	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
126		npcan1 11580	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℂ → ((𝑛 − 1) + 1) = 𝑛)
127	125, 126	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) = 𝑛)
128		elnnuz 12807	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ_≥‘1))
129	128	biimpi 215	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ_≥‘1))
130	127, 129	eqeltrd 2838	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) ∈ (ℤ_≥‘1))
131	63	nn0zd 12525	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℤ)
132		uzid 12778	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 − 1) ∈ ℤ → (𝑛 − 1) ∈ (ℤ_≥‘(𝑛 − 1)))
133	131, 132	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ (ℤ_≥‘(𝑛 − 1)))
134		peano2uz 12826	. . . . . . . . . . . . . . . . . . 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 2839	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ_≥‘(𝑛 − 1)))
137		fzsplit2 13466	. . . . . . . . . . . . . . . . 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 7372	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → (((𝑛 − 1) + 1)...𝑛) = (𝑛...𝑛))
140		nnz 12520	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
141		fzsn 13483	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℤ → (𝑛...𝑛) = {𝑛})
142	140, 141	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → (𝑛...𝑛) = {𝑛})
143	139, 142	eqtrd 2776	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → (((𝑛 − 1) + 1)...𝑛) = {𝑛})
144	143	uneq2d 4123	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)) = ((1...(𝑛 − 1)) ∪ {𝑛}))
145	138, 144	eqtrd 2776	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → (1...𝑛) = ((1...(𝑛 − 1)) ∪ {𝑛}))
146	145	difeq1d 4081	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → ((1...𝑛) ∖ (1...(𝑛 − 1))) = (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))))
147		nnre 12160	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
148		ltm1 11997	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℝ → (𝑛 − 1) < 𝑛)
149		peano2rem 11468	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℝ → (𝑛 − 1) ∈ ℝ)
150		ltnle 11234	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑛 − 1) ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((𝑛 − 1) < 𝑛 ↔ ¬ 𝑛 ≤ (𝑛 − 1)))
151	149, 150	mpancom 686	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℝ → ((𝑛 − 1) < 𝑛 ↔ ¬ 𝑛 ≤ (𝑛 − 1)))
152	148, 151	mpbid 231	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℝ → ¬ 𝑛 ≤ (𝑛 − 1))
153		elfzle2 13445	. . . . . . . . . . . . . . . . 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 4161	. . . . . . . . . . . . . . . . 17 ⊢ ((1...(𝑛 − 1)) ∩ {𝑛}) = ({𝑛} ∩ (1...(𝑛 − 1)))
157	156	eqeq1i 2741	. . . . . . . . . . . . . . . 16 ⊢ (((1...(𝑛 − 1)) ∩ {𝑛}) = ∅ ↔ ({𝑛} ∩ (1...(𝑛 − 1))) = ∅)
158		disjsn 4672	. . . . . . . . . . . . . . . 16 ⊢ (((1...(𝑛 − 1)) ∩ {𝑛}) = ∅ ↔ ¬ 𝑛 ∈ (1...(𝑛 − 1)))
159		disj3 4413	. . . . . . . . . . . . . . . 16 ⊢ (({𝑛} ∩ (1...(𝑛 − 1))) = ∅ ↔ {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1))))
160	157, 158, 159	3bitr3i 300	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑛 ∈ (1...(𝑛 − 1)) ↔ {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1))))
161	155, 160	sylib 217	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1))))
162	124, 146, 161	3eqtr4a 2802	. . . . . . . . . . . . 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 6013	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → ((2^nd ‘(1^st ‘𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = ((2^nd ‘(1^st ‘𝑧)) “ {𝑛}))
165		dff1o3 6790	. . . . . . . . . . . . . . 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 497	. . . . . . . . . . . . . 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 6585	. . . . . . . . . . . . 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 2782	. . . . . . . . . 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 2825	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑘 → (𝑧 ∈ 𝑆 ↔ 𝑘 ∈ 𝑆))
174	173	anbi1d 630	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑘 → ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) ↔ (𝑘 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁))))
175		2fveq3 6847	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑘 → (2^nd ‘(1^st ‘𝑧)) = (2^nd ‘(1^st ‘𝑘)))
176	175	fveq1d 6844	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑘 → ((2^nd ‘(1^st ‘𝑧))‘𝑛) = ((2^nd ‘(1^st ‘𝑘))‘𝑛))
177	176	sneqd 4598	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑘 → {((2^nd ‘(1^st ‘𝑧))‘𝑛)} = {((2^nd ‘(1^st ‘𝑘))‘𝑛)})
178	175	imaeq1d 6012	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑘 → ((2^nd ‘(1^st ‘𝑧)) “ (1...𝑛)) = ((2^nd ‘(1^st ‘𝑘)) “ (1...𝑛)))
179	175	imaeq1d 6012	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑘 → ((2^nd ‘(1^st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2^nd ‘(1^st ‘𝑘)) “ (1...(𝑛 − 1))))
180	178, 179	difeq12d 4083	. . . . . . . . . . . . 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 2002	. . . . . . . . . 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 2786	. . . . . . . 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 6855	. . . . . . . . 9 ⊢ ((2^nd ‘(1^st ‘𝑧))‘𝑛) ∈ V
187	186	sneqr 4798	. . . . . . . 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 6985	. . . . . 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 7962	. . . . . . . 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 725	. . . . . 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 710	. . . . 5 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → (1^st ‘𝑧) = (1^st ‘𝑘))
194		eqtr3 2762	. . . . . 6 ⊢ (((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0) → (2^nd ‘𝑧) = (2^nd ‘𝑘))
195	194	adantl 482	. . . . 5 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → (2^nd ‘𝑧) = (2^nd ‘𝑘))
196		xpopth 7962	. . . . . . 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 725	. . . . 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 710	. . . 4 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0)) → 𝑧 = 𝑘)
200	199	ex 413	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) → (((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0) → 𝑧 = 𝑘))
201	200	ralrimivva 3197	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 ∀𝑘 ∈ 𝑆 (((2^nd ‘𝑧) = 0 ∧ (2^nd ‘𝑘) = 0) → 𝑧 = 𝑘))
202		fveqeq2 6851	. . 3 ⊢ (𝑧 = 𝑘 → ((2^nd ‘𝑧) = 0 ↔ (2^nd ‘𝑘) = 0))
203	202	rmo4 3688	. 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 233	1 ⊢ (𝜑 → ∃*𝑧 ∈ 𝑆 (2^nd ‘𝑧) = 0)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {cab 2713 ≠ wne 2943 ∀wral 3064 ∃*wrmo 3352 {crab 3407 ⦋csb 3855 ∖ cdif 3907 ∪ cun 3908 ∩ cin 3909 ∅c0 4282 ifcif 4486 {csn 4586 class class class wbr 5105 ↦ cmpt 5188 × cxp 5631 ^◡ccnv 5632 “ cima 5636 Fun wfun 6490 Fn wfn 6491 ⟶wf 6492 –onto→wfo 6494 –1-1-onto→wf1o 6495 ‘cfv 6496 (class class class)co 7357 ∘_f cof 7615 1^st c1st 7919 2^nd c2nd 7920 ↑_m cmap 8765 ℂcc 11049 ℝcr 11050 0cc0 11051 1c1 11052 + caddc 11054 < clt 11189 ≤ cle 11190 − cmin 11385 ℕcn 12153 ℕ₀cn0 12413 ℤcz 12499 ℤ_≥cuz 12763 ...cfz 13424 ..^cfzo 13567

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-n0 12414 df-z 12500 df-uz 12764 df-fz 13425 df-fzo 13568

This theorem is referenced by: poimirlem18 36096 poimirlem21 36099

W3C validator