Theorem poimirlem13 38742

Description: Lemma for poimir 38762- 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)⟶1-1onto(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
poimirlem22.1	⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))

Assertion

Ref	Expression
poimirlem13	⊢ (𝜑 → ∃*𝑧 ∈ 𝑆 (2nd ‘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)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) → 𝑁 ∈ ℕ)
3		poimirlem22.s	. . . . . . . 8 ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘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)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))
6		simplrl 789	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) → 𝑧 ∈ 𝑆)
7		simprl 783	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) → (2nd ‘𝑧) = 0)
8	2, 3, 5, 6, 7	poimirlem10 38739	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) → ((𝐹‘(𝑁 − 1)) ∘f − ((1...𝑁) × {1})) = (1st ‘(1st ‘𝑧)))
9		simplrr 790	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) → 𝑘 ∈ 𝑆)
10		simprr 785	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) → (2nd ‘𝑘) = 0)
11	2, 3, 5, 9, 10	poimirlem10 38739	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) → ((𝐹‘(𝑁 − 1)) ∘f − ((1...𝑁) × {1})) = (1st ‘(1st ‘𝑘)))
12	8, 11	eqtr3d 2800	. . . . . 6 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) → (1st ‘(1st ‘𝑧)) = (1st ‘(1st ‘𝑘)))
13		elrabi 3644	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}) × (0...𝑁)))
14	13, 3	eleq2s 2882	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}) × (0...𝑁)))
15		xp1st 8054	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}) × (0...𝑁)) → (1st ‘𝑧) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}))
16	14, 15	syl 18	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑆 → (1st ‘𝑧) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}))
17		xp2nd 8055	. . . . . . . . . . . 12 ⊢ ((1st ‘𝑧) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}) → (2nd ‘(1st ‘𝑧)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)})
18	16, 17	syl 18	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑆 → (2nd ‘(1st ‘𝑧)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)})
19		fvex 6918	. . . . . . . . . . . 12 ⊢ (2nd ‘(1st ‘𝑧)) ∈ V
20		f1oeq1 6831	. . . . . . . . . . . 12 ⊢ (𝑓 = (2nd ‘(1st ‘𝑧)) → (𝑓:(1...𝑁)⟶1-1onto(1...𝑁) ↔ (2nd ‘(1st ‘𝑧)):(1...𝑁)⟶1-1onto(1...𝑁)))
21	19, 20	elab 3636	. . . . . . . . . . 11 ⊢ ((2nd ‘(1st ‘𝑧)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)} ↔ (2nd ‘(1st ‘𝑧)):(1...𝑁)⟶1-1onto(1...𝑁))
22	18, 21	sylib 221	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑆 → (2nd ‘(1st ‘𝑧)):(1...𝑁)⟶1-1onto(1...𝑁))
23		f1ofn 6844	. . . . . . . . . 10 ⊢ ((2nd ‘(1st ‘𝑧)):(1...𝑁)⟶1-1onto(1...𝑁) → (2nd ‘(1st ‘𝑧)) Fn (1...𝑁))
24	22, 23	syl 18	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑆 → (2nd ‘(1st ‘𝑧)) Fn (1...𝑁))
25	24	adantr 486	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → (2nd ‘(1st ‘𝑧)) Fn (1...𝑁))
26	25	ad2antlr 740	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) → (2nd ‘(1st ‘𝑧)) Fn (1...𝑁))
27		elrabi 3644	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 𝑘 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}) × (0...𝑁)))
28	27, 3	eleq2s 2882	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝑆 → 𝑘 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}) × (0...𝑁)))
29		xp1st 8054	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}) × (0...𝑁)) → (1st ‘𝑘) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}))
30	28, 29	syl 18	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝑆 → (1st ‘𝑘) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}))
31		xp2nd 8055	. . . . . . . . . . . 12 ⊢ ((1st ‘𝑘) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}) → (2nd ‘(1st ‘𝑘)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)})
32	30, 31	syl 18	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝑆 → (2nd ‘(1st ‘𝑘)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)})
33		fvex 6918	. . . . . . . . . . . 12 ⊢ (2nd ‘(1st ‘𝑘)) ∈ V
34		f1oeq1 6831	. . . . . . . . . . . 12 ⊢ (𝑓 = (2nd ‘(1st ‘𝑘)) → (𝑓:(1...𝑁)⟶1-1onto(1...𝑁) ↔ (2nd ‘(1st ‘𝑘)):(1...𝑁)⟶1-1onto(1...𝑁)))
35	33, 34	elab 3636	. . . . . . . . . . 11 ⊢ ((2nd ‘(1st ‘𝑘)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)} ↔ (2nd ‘(1st ‘𝑘)):(1...𝑁)⟶1-1onto(1...𝑁))
36	32, 35	sylib 221	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝑆 → (2nd ‘(1st ‘𝑘)):(1...𝑁)⟶1-1onto(1...𝑁))
37		f1ofn 6844	. . . . . . . . . 10 ⊢ ((2nd ‘(1st ‘𝑘)):(1...𝑁)⟶1-1onto(1...𝑁) → (2nd ‘(1st ‘𝑘)) Fn (1...𝑁))
38	36, 37	syl 18	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝑆 → (2nd ‘(1st ‘𝑘)) Fn (1...𝑁))
39	38	adantl 487	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → (2nd ‘(1st ‘𝑘)) Fn (1...𝑁))
40	39	ad2antlr 740	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) → (2nd ‘(1st ‘𝑘)) Fn (1...𝑁))
41		eleq1 2852	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → (𝑚 ∈ (1...𝑁) ↔ 𝑛 ∈ (1...𝑁)))
42	41	anbi2d 642	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) ↔ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁))))
43		oveq2 7448	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
44	43	imaeq2d 6063	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → ((2nd ‘(1st ‘𝑧)) “ (1...𝑚)) = ((2nd ‘(1st ‘𝑧)) “ (1...𝑛)))
45	43	imaeq2d 6063	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → ((2nd ‘(1st ‘𝑘)) “ (1...𝑚)) = ((2nd ‘(1st ‘𝑘)) “ (1...𝑛)))
46	44, 45	eqeq12d 2779	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (((2nd ‘(1st ‘𝑧)) “ (1...𝑚)) = ((2nd ‘(1st ‘𝑘)) “ (1...𝑚)) ↔ ((2nd ‘(1st ‘𝑧)) “ (1...𝑛)) = ((2nd ‘(1st ‘𝑘)) “ (1...𝑛))))
47	42, 46	imbi12d 347	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → ((2nd ‘(1st ‘𝑧)) “ (1...𝑚)) = ((2nd ‘(1st ‘𝑘)) “ (1...𝑚))) ↔ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st ‘𝑧)) “ (1...𝑛)) = ((2nd ‘(1st ‘𝑘)) “ (1...𝑛)))))
48	1	ad3antrrr 743	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑁 ∈ ℕ)
49	4	ad3antrrr 743	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘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)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑧 ∈ 𝑆)
52		simplrl 789	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → (2nd ‘𝑧) = 0)
53		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → 𝑘 ∈ 𝑆)
54	53	ad3antlr 744	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑘 ∈ 𝑆)
55		simplrr 790	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → (2nd ‘𝑘) = 0)
56		simpr 490	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑚 ∈ (1...𝑁))
57	48, 3, 49, 51, 52, 54, 55, 56	poimirlem11 38740	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → ((2nd ‘(1st ‘𝑧)) “ (1...𝑚)) ⊆ ((2nd ‘(1st ‘𝑘)) “ (1...𝑚)))
58	48, 3, 49, 54, 55, 51, 52, 56	poimirlem11 38740	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → ((2nd ‘(1st ‘𝑘)) “ (1...𝑚)) ⊆ ((2nd ‘(1st ‘𝑧)) “ (1...𝑚)))
59	57, 58	eqssd 3930	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → ((2nd ‘(1st ‘𝑧)) “ (1...𝑚)) = ((2nd ‘(1st ‘𝑘)) “ (1...𝑚)))
60	47, 59	chvarvv 2024	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st ‘𝑧)) “ (1...𝑛)) = ((2nd ‘(1st ‘𝑘)) “ (1...𝑛)))
61		simpll 779	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) → 𝜑)
62		elfznn 13711	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℕ)
63		nnm1nn0 12671	. . . . . . . . . . . . . . . . . . 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 12375	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℂ)
67		ax-1cn 11282	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℂ
68		subeq0 11608	. . . . . . . . . . . . . . . . . . . 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 2995	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) ≠ 0 ↔ ¬ 𝑛 = 1))
71	70	biimpar 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1) → (𝑛 − 1) ≠ 0)
72		elnnne0 12644	. . . . . . . . . . . . . . . . 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 12692	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ∈ ℝ)
76	75	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ∈ ℝ)
77	62	nnred 12374	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℝ)
78	77	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℝ)
79	1	nnred 12374	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℝ)
80	79	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → 𝑁 ∈ ℝ)
81	77	lem1d 12274	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ≤ 𝑛)
82	81	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ≤ 𝑛)
83		elfzle2 13685	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ≤ 𝑁)
84	83	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ≤ 𝑁)
85	76, 78, 80, 82, 84	letrd 11491	. . . . . . . . . . . . . . . 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 12743	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ ℤ)
88		fznn 13750	. . . . . . . . . . . . . . . . 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)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → (𝑛 − 1) ∈ (1...𝑁))
93		ovex 7473	. . . . . . . . . . . . . 14 ⊢ (𝑛 − 1) ∈ V
94		eleq1 2852	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (𝑛 − 1) → (𝑚 ∈ (1...𝑁) ↔ (𝑛 − 1) ∈ (1...𝑁)))
95	94	anbi2d 642	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (𝑛 − 1) → ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) ↔ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) ∧ (𝑛 − 1) ∈ (1...𝑁))))
96		oveq2 7448	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = (𝑛 − 1) → (1...𝑚) = (1...(𝑛 − 1)))
97	96	imaeq2d 6063	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (𝑛 − 1) → ((2nd ‘(1st ‘𝑧)) “ (1...𝑚)) = ((2nd ‘(1st ‘𝑧)) “ (1...(𝑛 − 1))))
98	96	imaeq2d 6063	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (𝑛 − 1) → ((2nd ‘(1st ‘𝑘)) “ (1...𝑚)) = ((2nd ‘(1st ‘𝑘)) “ (1...(𝑛 − 1))))
99	97, 98	eqeq12d 2779	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (𝑛 − 1) → (((2nd ‘(1st ‘𝑧)) “ (1...𝑚)) = ((2nd ‘(1st ‘𝑘)) “ (1...𝑚)) ↔ ((2nd ‘(1st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st ‘𝑘)) “ (1...(𝑛 − 1)))))
100	95, 99	imbi12d 347	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑛 − 1) → (((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → ((2nd ‘(1st ‘𝑧)) “ (1...𝑚)) = ((2nd ‘(1st ‘𝑘)) “ (1...𝑚))) ↔ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) ∧ (𝑛 − 1) ∈ (1...𝑁)) → ((2nd ‘(1st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st ‘𝑘)) “ (1...(𝑛 − 1))))))
101	93, 100, 59	vtocl 3524	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) ∧ (𝑛 − 1) ∈ (1...𝑁)) → ((2nd ‘(1st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st ‘𝑘)) “ (1...(𝑛 − 1))))
102	92, 101	syldan 603	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → ((2nd ‘(1st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st ‘𝑘)) “ (1...(𝑛 − 1))))
103	102	expr 462	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → (¬ 𝑛 = 1 → ((2nd ‘(1st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st ‘𝑘)) “ (1...(𝑛 − 1)))))
104		ima0 6086	. . . . . . . . . . . . 13 ⊢ ((2nd ‘(1st ‘𝑧)) “ ∅) = ∅
105		ima0 6086	. . . . . . . . . . . . 13 ⊢ ((2nd ‘(1st ‘𝑘)) “ ∅) = ∅
106	104, 105	eqtr4i 2789	. . . . . . . . . . . 12 ⊢ ((2nd ‘(1st ‘𝑧)) “ ∅) = ((2nd ‘(1st ‘𝑘)) “ ∅)
107		oveq1 7447	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 1 → (𝑛 − 1) = (1 − 1))
108		1m1e0 12439	. . . . . . . . . . . . . . . 16 ⊢ (1 − 1) = 0
109	107, 108	eqtrdi 2814	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 1 → (𝑛 − 1) = 0)
110	109	oveq2d 7456	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 1 → (1...(𝑛 − 1)) = (1...0))
111		fz10 13702	. . . . . . . . . . . . . 14 ⊢ (1...0) = ∅
112	110, 111	eqtrdi 2814	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1 → (1...(𝑛 − 1)) = ∅)
113	112	imaeq2d 6063	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → ((2nd ‘(1st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st ‘𝑧)) “ ∅))
114	112	imaeq2d 6063	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → ((2nd ‘(1st ‘𝑘)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st ‘𝑘)) “ ∅))
115	106, 113, 114	3eqtr4a 2824	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → ((2nd ‘(1st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st ‘𝑘)) “ (1...(𝑛 − 1))))
116	103, 115	pm2.61d2 183	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st ‘𝑘)) “ (1...(𝑛 − 1))))
117	60, 116	difeq12d 4067	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → (((2nd ‘(1st ‘𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st ‘𝑧)) “ (1...(𝑛 − 1)))) = (((2nd ‘(1st ‘𝑘)) “ (1...𝑛)) ∖ ((2nd ‘(1st ‘𝑘)) “ (1...(𝑛 − 1)))))
118		fnsnfv 6984	. . . . . . . . . . . 12 ⊢ (((2nd ‘(1st ‘𝑧)) Fn (1...𝑁) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st ‘𝑧))‘𝑛)} = ((2nd ‘(1st ‘𝑧)) “ {𝑛}))
119	24, 118	sylan 592	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st ‘𝑧))‘𝑛)} = ((2nd ‘(1st ‘𝑧)) “ {𝑛}))
120	62	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℕ)
121		uncom 4160	. . . . . . . . . . . . . . . 16 ⊢ ((1...(𝑛 − 1)) ∪ {𝑛}) = ({𝑛} ∪ (1...(𝑛 − 1)))
122	121	difeq1i 4062	. . . . . . . . . . . . . . 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 2786	. . . . . . . . . . . . . 14 ⊢ (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))) = ({𝑛} ∖ (1...(𝑛 − 1)))
125		nncn 12367	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
126		npcan1 11765	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℂ → ((𝑛 − 1) + 1) = 𝑛)
127	125, 126	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) = 𝑛)
128		elnnuz 13029	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ≥‘1))
129	128	biimpi 219	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ≥‘1))
130	127, 129	eqeltrd 2864	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) ∈ (ℤ≥‘1))
131	63	nn0zd 12742	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℤ)
132		uzid 13004	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 − 1) ∈ ℤ → (𝑛 − 1) ∈ (ℤ≥‘(𝑛 − 1)))
133	131, 132	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ (ℤ≥‘(𝑛 − 1)))
134		peano2uz 13052	. . . . . . . . . . . . . . . . . . 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 2865	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ≥‘(𝑛 − 1)))
137		fzsplit2 13707	. . . . . . . . . . . . . . . . 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 7455	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → (((𝑛 − 1) + 1)...𝑛) = (𝑛...𝑛))
140		nnz 12738	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
141		fzsn 13724	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℤ → (𝑛...𝑛) = {𝑛})
142	140, 141	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → (𝑛...𝑛) = {𝑛})
143	139, 142	eqtrd 2798	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → (((𝑛 − 1) + 1)...𝑛) = {𝑛})
144	143	uneq2d 4170	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)) = ((1...(𝑛 − 1)) ∪ {𝑛}))
145	138, 144	eqtrd 2798	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → (1...𝑛) = ((1...(𝑛 − 1)) ∪ {𝑛}))
146	145	difeq1d 4065	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → ((1...𝑛) ∖ (1...(𝑛 − 1))) = (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))))
147		nnre 12366	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
148		ltm1 12183	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℝ → (𝑛 − 1) < 𝑛)
149		peano2rem 11649	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℝ → (𝑛 − 1) ∈ ℝ)
150		ltnle 11413	. . . . . . . . . . . . . . . . . . 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 13685	. . . . . . . . . . . . . . . . 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 4109	. . . . . . . . . . . . . . . . 17 ⊢ ((1...(𝑛 − 1)) ∩ {𝑛}) = ({𝑛} ∩ (1...(𝑛 − 1)))
157	156	eqeq1i 2768	. . . . . . . . . . . . . . . 16 ⊢ (((1...(𝑛 − 1)) ∩ {𝑛}) = ∅ ↔ ({𝑛} ∩ (1...(𝑛 − 1))) = ∅)
158		disjsn 4672	. . . . . . . . . . . . . . . 16 ⊢ (((1...(𝑛 − 1)) ∩ {𝑛}) = ∅ ↔ ¬ 𝑛 ∈ (1...(𝑛 − 1)))
159		disj3 4410	. . . . . . . . . . . . . . . 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 2824	. . . . . . . . . . . . 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 6063	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st ‘𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = ((2nd ‘(1st ‘𝑧)) “ {𝑛}))
165		dff1o3 6850	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘(1st ‘𝑧)):(1...𝑁)⟶1-1onto(1...𝑁) ↔ ((2nd ‘(1st ‘𝑧)):(1...𝑁)⟶onto(1...𝑁) ∧ Fun ◡(2nd ‘(1st ‘𝑧))))
166	165	simprbi 503	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘(1st ‘𝑧)):(1...𝑁)⟶1-1onto(1...𝑁) → Fun ◡(2nd ‘(1st ‘𝑧)))
167	22, 166	syl 18	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑆 → Fun ◡(2nd ‘(1st ‘𝑧)))
168		imadif 6643	. . . . . . . . . . . . 13 ⊢ (Fun ◡(2nd ‘(1st ‘𝑧)) → ((2nd ‘(1st ‘𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = (((2nd ‘(1st ‘𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st ‘𝑧)) “ (1...(𝑛 − 1)))))
169	167, 168	syl 18	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑆 → ((2nd ‘(1st ‘𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = (((2nd ‘(1st ‘𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st ‘𝑧)) “ (1...(𝑛 − 1)))))
170	169	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st ‘𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = (((2nd ‘(1st ‘𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st ‘𝑧)) “ (1...(𝑛 − 1)))))
171	119, 164, 170	3eqtr2d 2804	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st ‘𝑧))‘𝑛)} = (((2nd ‘(1st ‘𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st ‘𝑧)) “ (1...(𝑛 − 1)))))
172	6, 171	sylan 592	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st ‘𝑧))‘𝑛)} = (((2nd ‘(1st ‘𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st ‘𝑧)) “ (1...(𝑛 − 1)))))
173		eleq1 2852	. . . . . . . . . . . . 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 6910	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑘 → (2nd ‘(1st ‘𝑧)) = (2nd ‘(1st ‘𝑘)))
176	175	fveq1d 6907	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑘 → ((2nd ‘(1st ‘𝑧))‘𝑛) = ((2nd ‘(1st ‘𝑘))‘𝑛))
177	176	sneqd 4596	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑘 → {((2nd ‘(1st ‘𝑧))‘𝑛)} = {((2nd ‘(1st ‘𝑘))‘𝑛)})
178	175	imaeq1d 6062	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑘 → ((2nd ‘(1st ‘𝑧)) “ (1...𝑛)) = ((2nd ‘(1st ‘𝑘)) “ (1...𝑛)))
179	175	imaeq1d 6062	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑘 → ((2nd ‘(1st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st ‘𝑘)) “ (1...(𝑛 − 1))))
180	178, 179	difeq12d 4067	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑘 → (((2nd ‘(1st ‘𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st ‘𝑧)) “ (1...(𝑛 − 1)))) = (((2nd ‘(1st ‘𝑘)) “ (1...𝑛)) ∖ ((2nd ‘(1st ‘𝑘)) “ (1...(𝑛 − 1)))))
181	177, 180	eqeq12d 2779	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑘 → ({((2nd ‘(1st ‘𝑧))‘𝑛)} = (((2nd ‘(1st ‘𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st ‘𝑧)) “ (1...(𝑛 − 1)))) ↔ {((2nd ‘(1st ‘𝑘))‘𝑛)} = (((2nd ‘(1st ‘𝑘)) “ (1...𝑛)) ∖ ((2nd ‘(1st ‘𝑘)) “ (1...(𝑛 − 1))))))
182	174, 181	imbi12d 347	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑘 → (((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st ‘𝑧))‘𝑛)} = (((2nd ‘(1st ‘𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st ‘𝑧)) “ (1...(𝑛 − 1))))) ↔ ((𝑘 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st ‘𝑘))‘𝑛)} = (((2nd ‘(1st ‘𝑘)) “ (1...𝑛)) ∖ ((2nd ‘(1st ‘𝑘)) “ (1...(𝑛 − 1)))))))
183	182, 171	chvarvv 2024	. . . . . . . . . 10 ⊢ ((𝑘 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st ‘𝑘))‘𝑛)} = (((2nd ‘(1st ‘𝑘)) “ (1...𝑛)) ∖ ((2nd ‘(1st ‘𝑘)) “ (1...(𝑛 − 1)))))
184	9, 183	sylan 592	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st ‘𝑘))‘𝑛)} = (((2nd ‘(1st ‘𝑘)) “ (1...𝑛)) ∖ ((2nd ‘(1st ‘𝑘)) “ (1...(𝑛 − 1)))))
185	117, 172, 184	3eqtr4d 2808	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st ‘𝑧))‘𝑛)} = {((2nd ‘(1st ‘𝑘))‘𝑛)})
186		fvex 6918	. . . . . . . . 9 ⊢ ((2nd ‘(1st ‘𝑧))‘𝑛) ∈ V
187	186	sneqr 4798	. . . . . . . 8 ⊢ ({((2nd ‘(1st ‘𝑧))‘𝑛)} = {((2nd ‘(1st ‘𝑘))‘𝑛)} → ((2nd ‘(1st ‘𝑧))‘𝑛) = ((2nd ‘(1st ‘𝑘))‘𝑛))
188	185, 187	syl 18	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st ‘𝑧))‘𝑛) = ((2nd ‘(1st ‘𝑘))‘𝑛))
189	26, 40, 188	eqfnfvd 7052	. . . . . 6 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) → (2nd ‘(1st ‘𝑧)) = (2nd ‘(1st ‘𝑘)))
190		xpopth 8063	. . . . . . . 8 ⊢ (((1st ‘𝑧) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}) ∧ (1st ‘𝑘) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)})) → (((1st ‘(1st ‘𝑧)) = (1st ‘(1st ‘𝑘)) ∧ (2nd ‘(1st ‘𝑧)) = (2nd ‘(1st ‘𝑘))) ↔ (1st ‘𝑧) = (1st ‘𝑘)))
191	16, 30, 190	syl2an 608	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → (((1st ‘(1st ‘𝑧)) = (1st ‘(1st ‘𝑘)) ∧ (2nd ‘(1st ‘𝑧)) = (2nd ‘(1st ‘𝑘))) ↔ (1st ‘𝑧) = (1st ‘𝑘)))
192	191	ad2antlr 740	. . . . . 6 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) → (((1st ‘(1st ‘𝑧)) = (1st ‘(1st ‘𝑘)) ∧ (2nd ‘(1st ‘𝑧)) = (2nd ‘(1st ‘𝑘))) ↔ (1st ‘𝑧) = (1st ‘𝑘)))
193	12, 189, 192	mpbi2and 725	. . . . 5 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) → (1st ‘𝑧) = (1st ‘𝑘))
194		eqtr3 2785	. . . . . 6 ⊢ (((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0) → (2nd ‘𝑧) = (2nd ‘𝑘))
195	194	adantl 487	. . . . 5 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) → (2nd ‘𝑧) = (2nd ‘𝑘))
196		xpopth 8063	. . . . . . 7 ⊢ ((𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}) × (0...𝑁)) ∧ 𝑘 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)⟶1-1onto(1...𝑁)}) × (0...𝑁))) → (((1st ‘𝑧) = (1st ‘𝑘) ∧ (2nd ‘𝑧) = (2nd ‘𝑘)) ↔ 𝑧 = 𝑘))
197	14, 28, 196	syl2an 608	. . . . . 6 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → (((1st ‘𝑧) = (1st ‘𝑘) ∧ (2nd ‘𝑧) = (2nd ‘𝑘)) ↔ 𝑧 = 𝑘))
198	197	ad2antlr 740	. . . . 5 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) → (((1st ‘𝑧) = (1st ‘𝑘) ∧ (2nd ‘𝑧) = (2nd ‘𝑘)) ↔ 𝑧 = 𝑘))
199	193, 195, 198	mpbi2and 725	. . . 4 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0)) → 𝑧 = 𝑘)
200	199	ex 418	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) → (((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0) → 𝑧 = 𝑘))
201	200	ralrimivva 3209	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 ∀𝑘 ∈ 𝑆 (((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0) → 𝑧 = 𝑘))
202		fveqeq2 6914	. . 3 ⊢ (𝑧 = 𝑘 → ((2nd ‘𝑧) = 0 ↔ (2nd ‘𝑘) = 0))
203	202	rmo4 3691	. 2 ⊢ (∃*𝑧 ∈ 𝑆 (2nd ‘𝑧) = 0 ↔ ∀𝑧 ∈ 𝑆 ∀𝑘 ∈ 𝑆 (((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑘) = 0) → 𝑧 = 𝑘))
204	201, 203	sylibr 237	1 ⊢ (𝜑 → ∃*𝑧 ∈ 𝑆 (2nd ‘𝑧) = 0)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {cab 2741   ≠ wne 2959  ∀wral 3080  ∃*wrmo 3368  {crab 3416  ⦋csb 3851   ∖ cdif 4048   ∩ cin 4092   ∪ cun 4152  ∅c0 4281  ifcif 4485  {csn 4588   class class class wbr 5106   ↦ cmpt 5189   × cxp 5653  ^◡ccnv 5654   “ cima 5658  Fun wfun 6549   Fn wfn 6550  ⟶wf 6551  ⟶ontowfo 6553  ⟶1-1ontowf1o 6554  ‘cfv 6555  (class class class)co 7440   ∘_f cof 7712  1^st c1st 8020  2^nd c2nd 8021   ↑_m cmap 8872  ℂcc 11222  ℝcr 11223  0cc0 11224  1c1 11225   + caddc 11227   < clt 11367   ≤ cle 11368   − cmin 11565  ℕcn 12359  ℕ₀cn0 12630  ℤcz 12717  ℤ_≥cuz 12989  ...cfz 13663  ..^cfzo 13812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-of 7714  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-er 8742  df-map 8874  df-en 8999  df-dom 9000  df-sdom 9001  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-nn 12360  df-n0 12631  df-z 12718  df-uz 12990  df-fz 13664  df-fzo 13813

This theorem is used by:  poimirlem18  38747  poimirlem21  38750