Theorem poimirlem14 37658

Description: Lemma for poimir 37677- for at most one simplex associated with a shared face is the opposite vertex last 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 < (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
poimirlem14	⊢ (𝜑 → ∃*𝑧 ∈ 𝑆 (2nd ‘𝑧) = 𝑁)

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 poimirlem14

Dummy variables 𝑘 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		poimir.0	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ)
2	1	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) → 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 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
4		simplrl 776	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) → 𝑧 ∈ 𝑆)
5	1	nngt0d 12289	. . . . . . . . . 10 ⊢ (𝜑 → 0 < 𝑁)
6		breq2 5123	. . . . . . . . . . 11 ⊢ ((2nd ‘𝑧) = 𝑁 → (0 < (2nd ‘𝑧) ↔ 0 < 𝑁))
7	6	biimparc 479	. . . . . . . . . 10 ⊢ ((0 < 𝑁 ∧ (2nd ‘𝑧) = 𝑁) → 0 < (2nd ‘𝑧))
8	5, 7	sylan 580	. . . . . . . . 9 ⊢ ((𝜑 ∧ (2nd ‘𝑧) = 𝑁) → 0 < (2nd ‘𝑧))
9	8	ad2ant2r 747	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) → 0 < (2nd ‘𝑧))
10	2, 3, 4, 9	poimirlem5 37649	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) → (𝐹‘0) = (1st ‘(1st ‘𝑧)))
11		simplrr 777	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) → 𝑘 ∈ 𝑆)
12		breq2 5123	. . . . . . . . . . 11 ⊢ ((2nd ‘𝑘) = 𝑁 → (0 < (2nd ‘𝑘) ↔ 0 < 𝑁))
13	12	biimparc 479	. . . . . . . . . 10 ⊢ ((0 < 𝑁 ∧ (2nd ‘𝑘) = 𝑁) → 0 < (2nd ‘𝑘))
14	5, 13	sylan 580	. . . . . . . . 9 ⊢ ((𝜑 ∧ (2nd ‘𝑘) = 𝑁) → 0 < (2nd ‘𝑘))
15	14	ad2ant2rl 749	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) → 0 < (2nd ‘𝑘))
16	2, 3, 11, 15	poimirlem5 37649	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) → (𝐹‘0) = (1st ‘(1st ‘𝑘)))
17	10, 16	eqtr3d 2772	. . . . . 6 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) → (1st ‘(1st ‘𝑧)) = (1st ‘(1st ‘𝑘)))
18		elrabi 3666	. . . . . . . . . . . . 13 ⊢ (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 < (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)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
19	18, 3	eleq2s 2852	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
20		xp1st 8020	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st ‘𝑧) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
21		xp2nd 8021	. . . . . . . . . . . 12 ⊢ ((1st ‘𝑧) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘(1st ‘𝑧)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
22	19, 20, 21	3syl 18	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑆 → (2nd ‘(1st ‘𝑧)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
23		fvex 6889	. . . . . . . . . . . 12 ⊢ (2nd ‘(1st ‘𝑧)) ∈ V
24		f1oeq1 6806	. . . . . . . . . . . 12 ⊢ (𝑓 = (2nd ‘(1st ‘𝑧)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁)))
25	23, 24	elab 3658	. . . . . . . . . . 11 ⊢ ((2nd ‘(1st ‘𝑧)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
26	22, 25	sylib 218	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑆 → (2nd ‘(1st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
27		f1ofn 6819	. . . . . . . . . 10 ⊢ ((2nd ‘(1st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st ‘𝑧)) Fn (1...𝑁))
28	26, 27	syl 17	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑆 → (2nd ‘(1st ‘𝑧)) Fn (1...𝑁))
29	28	adantr 480	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → (2nd ‘(1st ‘𝑧)) Fn (1...𝑁))
30	29	ad2antlr 727	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) → (2nd ‘(1st ‘𝑧)) Fn (1...𝑁))
31		elrabi 3666	. . . . . . . . . . . . 13 ⊢ (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 < (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)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
32	31, 3	eleq2s 2852	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝑆 → 𝑘 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
33		xp1st 8020	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st ‘𝑘) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
34		xp2nd 8021	. . . . . . . . . . . 12 ⊢ ((1st ‘𝑘) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘(1st ‘𝑘)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
35	32, 33, 34	3syl 18	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝑆 → (2nd ‘(1st ‘𝑘)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
36		fvex 6889	. . . . . . . . . . . 12 ⊢ (2nd ‘(1st ‘𝑘)) ∈ V
37		f1oeq1 6806	. . . . . . . . . . . 12 ⊢ (𝑓 = (2nd ‘(1st ‘𝑘)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st ‘𝑘)):(1...𝑁)–1-1-onto→(1...𝑁)))
38	36, 37	elab 3658	. . . . . . . . . . 11 ⊢ ((2nd ‘(1st ‘𝑘)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st ‘𝑘)):(1...𝑁)–1-1-onto→(1...𝑁))
39	35, 38	sylib 218	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝑆 → (2nd ‘(1st ‘𝑘)):(1...𝑁)–1-1-onto→(1...𝑁))
40		f1ofn 6819	. . . . . . . . . 10 ⊢ ((2nd ‘(1st ‘𝑘)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st ‘𝑘)) Fn (1...𝑁))
41	39, 40	syl 17	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝑆 → (2nd ‘(1st ‘𝑘)) Fn (1...𝑁))
42	41	adantl 481	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → (2nd ‘(1st ‘𝑘)) Fn (1...𝑁))
43	42	ad2antlr 727	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) → (2nd ‘(1st ‘𝑘)) Fn (1...𝑁))
44		simpllr 775	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆))
45		oveq2 7413	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
46	45	imaeq2d 6047	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑁 → ((2nd ‘(1st ‘𝑧)) “ (1...𝑛)) = ((2nd ‘(1st ‘𝑧)) “ (1...𝑁)))
47		f1ofo 6825	. . . . . . . . . . . . . . . 16 ⊢ ((2nd ‘(1st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st ‘𝑧)):(1...𝑁)–onto→(1...𝑁))
48		foima 6795	. . . . . . . . . . . . . . . 16 ⊢ ((2nd ‘(1st ‘𝑧)):(1...𝑁)–onto→(1...𝑁) → ((2nd ‘(1st ‘𝑧)) “ (1...𝑁)) = (1...𝑁))
49	26, 47, 48	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑆 → ((2nd ‘(1st ‘𝑧)) “ (1...𝑁)) = (1...𝑁))
50	46, 49	sylan9eqr 2792	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 = 𝑁) → ((2nd ‘(1st ‘𝑧)) “ (1...𝑛)) = (1...𝑁))
51	50	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) ∧ 𝑛 = 𝑁) → ((2nd ‘(1st ‘𝑧)) “ (1...𝑛)) = (1...𝑁))
52	45	imaeq2d 6047	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑁 → ((2nd ‘(1st ‘𝑘)) “ (1...𝑛)) = ((2nd ‘(1st ‘𝑘)) “ (1...𝑁)))
53		f1ofo 6825	. . . . . . . . . . . . . . . 16 ⊢ ((2nd ‘(1st ‘𝑘)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st ‘𝑘)):(1...𝑁)–onto→(1...𝑁))
54		foima 6795	. . . . . . . . . . . . . . . 16 ⊢ ((2nd ‘(1st ‘𝑘)):(1...𝑁)–onto→(1...𝑁) → ((2nd ‘(1st ‘𝑘)) “ (1...𝑁)) = (1...𝑁))
55	39, 53, 54	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝑆 → ((2nd ‘(1st ‘𝑘)) “ (1...𝑁)) = (1...𝑁))
56	52, 55	sylan9eqr 2792	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ 𝑆 ∧ 𝑛 = 𝑁) → ((2nd ‘(1st ‘𝑘)) “ (1...𝑛)) = (1...𝑁))
57	56	adantll 714	. . . . . . . . . . . . 13 ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) ∧ 𝑛 = 𝑁) → ((2nd ‘(1st ‘𝑘)) “ (1...𝑛)) = (1...𝑁))
58	51, 57	eqtr4d 2773	. . . . . . . . . . . 12 ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) ∧ 𝑛 = 𝑁) → ((2nd ‘(1st ‘𝑧)) “ (1...𝑛)) = ((2nd ‘(1st ‘𝑘)) “ (1...𝑛)))
59	44, 58	sylan 580	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑛 = 𝑁) → ((2nd ‘(1st ‘𝑧)) “ (1...𝑛)) = ((2nd ‘(1st ‘𝑘)) “ (1...𝑛)))
60		simpll 766	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) → 𝜑)
61		elnnuz 12896	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ≥‘1))
62	1, 61	sylib 218	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘1))
63		fzm1 13624	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ (ℤ≥‘1) → (𝑛 ∈ (1...𝑁) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁)))
64	62, 63	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁)))
65	64	anbi1d 631	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ 𝑁) ↔ ((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁) ∧ 𝑛 ≠ 𝑁)))
66	65	biimpa 476	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ 𝑁)) → ((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁) ∧ 𝑛 ≠ 𝑁))
67		df-ne 2933	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ≠ 𝑁 ↔ ¬ 𝑛 = 𝑁)
68	67	anbi2i 623	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁) ∧ 𝑛 ≠ 𝑁) ↔ ((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁) ∧ ¬ 𝑛 = 𝑁))
69		pm5.61 1002	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁) ∧ ¬ 𝑛 = 𝑁) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∧ ¬ 𝑛 = 𝑁))
70	68, 69	bitri 275	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁) ∧ 𝑛 ≠ 𝑁) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∧ ¬ 𝑛 = 𝑁))
71	66, 70	sylib 218	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ 𝑁)) → (𝑛 ∈ (1...(𝑁 − 1)) ∧ ¬ 𝑛 = 𝑁))
72		fz1ssfz0 13640	. . . . . . . . . . . . . . . . 17 ⊢ (1...(𝑁 − 1)) ⊆ (0...(𝑁 − 1))
73	72	sseli 3954	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ (0...(𝑁 − 1)))
74	73	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ (1...(𝑁 − 1)) ∧ ¬ 𝑛 = 𝑁) → 𝑛 ∈ (0...(𝑁 − 1)))
75	71, 74	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ 𝑁)) → 𝑛 ∈ (0...(𝑁 − 1)))
76	60, 75	sylan 580	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ 𝑁)) → 𝑛 ∈ (0...(𝑁 − 1)))
77		eleq1 2822	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑛 → (𝑚 ∈ (0...(𝑁 − 1)) ↔ 𝑛 ∈ (0...(𝑁 − 1))))
78	77	anbi2d 630	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑛 → ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) ↔ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 1)))))
79		oveq2 7413	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
80	79	imaeq2d 6047	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑛 → ((2nd ‘(1st ‘𝑧)) “ (1...𝑚)) = ((2nd ‘(1st ‘𝑧)) “ (1...𝑛)))
81	79	imaeq2d 6047	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑛 → ((2nd ‘(1st ‘𝑘)) “ (1...𝑚)) = ((2nd ‘(1st ‘𝑘)) “ (1...𝑛)))
82	80, 81	eqeq12d 2751	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑛 → (((2nd ‘(1st ‘𝑧)) “ (1...𝑚)) = ((2nd ‘(1st ‘𝑘)) “ (1...𝑚)) ↔ ((2nd ‘(1st ‘𝑧)) “ (1...𝑛)) = ((2nd ‘(1st ‘𝑘)) “ (1...𝑛))))
83	78, 82	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑧)) “ (1...𝑚)) = ((2nd ‘(1st ‘𝑘)) “ (1...𝑚))) ↔ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑧)) “ (1...𝑛)) = ((2nd ‘(1st ‘𝑘)) “ (1...𝑛)))))
84	1	ad3antrrr 730	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℕ)
85		poimirlem22.1	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))
86	85	ad3antrrr 730	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))
87		simpl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → 𝑧 ∈ 𝑆)
88	87	ad3antlr 731	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → 𝑧 ∈ 𝑆)
89		simplrl 776	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → (2nd ‘𝑧) = 𝑁)
90		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → 𝑘 ∈ 𝑆)
91	90	ad3antlr 731	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ 𝑆)
92		simplrr 777	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → (2nd ‘𝑘) = 𝑁)
93		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → 𝑚 ∈ (0...(𝑁 − 1)))
94	84, 3, 86, 88, 89, 91, 92, 93	poimirlem12 37656	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑧)) “ (1...𝑚)) ⊆ ((2nd ‘(1st ‘𝑘)) “ (1...𝑚)))
95	84, 3, 86, 91, 92, 88, 89, 93	poimirlem12 37656	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑘)) “ (1...𝑚)) ⊆ ((2nd ‘(1st ‘𝑧)) “ (1...𝑚)))
96	94, 95	eqssd 3976	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑧)) “ (1...𝑚)) = ((2nd ‘(1st ‘𝑘)) “ (1...𝑚)))
97	83, 96	chvarvv 1998	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑧)) “ (1...𝑛)) = ((2nd ‘(1st ‘𝑘)) “ (1...𝑛)))
98	76, 97	syldan 591	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ 𝑁)) → ((2nd ‘(1st ‘𝑧)) “ (1...𝑛)) = ((2nd ‘(1st ‘𝑘)) “ (1...𝑛)))
99	98	anassrs 467	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑛 ≠ 𝑁) → ((2nd ‘(1st ‘𝑧)) “ (1...𝑛)) = ((2nd ‘(1st ‘𝑘)) “ (1...𝑛)))
100	59, 99	pm2.61dane 3019	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st ‘𝑧)) “ (1...𝑛)) = ((2nd ‘(1st ‘𝑘)) “ (1...𝑛)))
101		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → 𝑛 ∈ (1...𝑁))
102		elfzelz 13541	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℤ)
103	1	nnzd 12615	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℤ)
104		elfzm1b 13619	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑛 ∈ (1...𝑁) ↔ (𝑛 − 1) ∈ (0...(𝑁 − 1))))
105	102, 103, 104	syl2anr 597	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 ∈ (1...𝑁) ↔ (𝑛 − 1) ∈ (0...(𝑁 − 1))))
106	101, 105	mpbid 232	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ∈ (0...(𝑁 − 1)))
107	60, 106	sylan 580	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ∈ (0...(𝑁 − 1)))
108		ovex 7438	. . . . . . . . . . . 12 ⊢ (𝑛 − 1) ∈ V
109		eleq1 2822	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑛 − 1) → (𝑚 ∈ (0...(𝑁 − 1)) ↔ (𝑛 − 1) ∈ (0...(𝑁 − 1))))
110	109	anbi2d 630	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑛 − 1) → ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) ↔ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ (𝑛 − 1) ∈ (0...(𝑁 − 1)))))
111		oveq2 7413	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (𝑛 − 1) → (1...𝑚) = (1...(𝑛 − 1)))
112	111	imaeq2d 6047	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑛 − 1) → ((2nd ‘(1st ‘𝑧)) “ (1...𝑚)) = ((2nd ‘(1st ‘𝑧)) “ (1...(𝑛 − 1))))
113	111	imaeq2d 6047	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑛 − 1) → ((2nd ‘(1st ‘𝑘)) “ (1...𝑚)) = ((2nd ‘(1st ‘𝑘)) “ (1...(𝑛 − 1))))
114	112, 113	eqeq12d 2751	. . . . . . . . . . . . 13 ⊢ (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)))))
115	110, 114	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 − 1) → (((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑧)) “ (1...𝑚)) = ((2nd ‘(1st ‘𝑘)) “ (1...𝑚))) ↔ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ (𝑛 − 1) ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st ‘𝑘)) “ (1...(𝑛 − 1))))))
116	108, 115, 96	vtocl 3537	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ (𝑛 − 1) ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st ‘𝑘)) “ (1...(𝑛 − 1))))
117	107, 116	syldan 591	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st ‘𝑘)) “ (1...(𝑛 − 1))))
118	100, 117	difeq12d 4102	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 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)))))
119		fnsnfv 6958	. . . . . . . . . . . 12 ⊢ (((2nd ‘(1st ‘𝑧)) Fn (1...𝑁) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st ‘𝑧))‘𝑛)} = ((2nd ‘(1st ‘𝑧)) “ {𝑛}))
120	28, 119	sylan 580	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st ‘𝑧))‘𝑛)} = ((2nd ‘(1st ‘𝑧)) “ {𝑛}))
121		elfznn 13570	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℕ)
122		uncom 4133	. . . . . . . . . . . . . . . . 17 ⊢ ((1...(𝑛 − 1)) ∪ {𝑛}) = ({𝑛} ∪ (1...(𝑛 − 1)))
123	122	difeq1i 4097	. . . . . . . . . . . . . . . 16 ⊢ (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))) = (({𝑛} ∪ (1...(𝑛 − 1))) ∖ (1...(𝑛 − 1)))
124		difun2 4456	. . . . . . . . . . . . . . . 16 ⊢ (({𝑛} ∪ (1...(𝑛 − 1))) ∖ (1...(𝑛 − 1))) = ({𝑛} ∖ (1...(𝑛 − 1)))
125	123, 124	eqtri 2758	. . . . . . . . . . . . . . 15 ⊢ (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))) = ({𝑛} ∖ (1...(𝑛 − 1)))
126		nncn 12248	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
127		npcan1 11662	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℂ → ((𝑛 − 1) + 1) = 𝑛)
128	126, 127	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) = 𝑛)
129		elnnuz 12896	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ≥‘1))
130	129	biimpi 216	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ≥‘1))
131	128, 130	eqeltrd 2834	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) ∈ (ℤ≥‘1))
132		nnm1nn0 12542	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
133	132	nn0zd 12614	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℤ)
134		uzid 12867	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 − 1) ∈ ℤ → (𝑛 − 1) ∈ (ℤ≥‘(𝑛 − 1)))
135		peano2uz 12917	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 − 1) ∈ (ℤ≥‘(𝑛 − 1)) → ((𝑛 − 1) + 1) ∈ (ℤ≥‘(𝑛 − 1)))
136	133, 134, 135	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) ∈ (ℤ≥‘(𝑛 − 1)))
137	128, 136	eqeltrrd 2835	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ≥‘(𝑛 − 1)))
138		fzsplit2 13566	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑛 − 1) + 1) ∈ (ℤ≥‘1) ∧ 𝑛 ∈ (ℤ≥‘(𝑛 − 1))) → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)))
139	131, 137, 138	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)))
140	128	oveq1d 7420	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → (((𝑛 − 1) + 1)...𝑛) = (𝑛...𝑛))
141		nnz 12609	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
142		fzsn 13583	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℤ → (𝑛...𝑛) = {𝑛})
143	141, 142	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → (𝑛...𝑛) = {𝑛})
144	140, 143	eqtrd 2770	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → (((𝑛 − 1) + 1)...𝑛) = {𝑛})
145	144	uneq2d 4143	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)) = ((1...(𝑛 − 1)) ∪ {𝑛}))
146	139, 145	eqtrd 2770	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → (1...𝑛) = ((1...(𝑛 − 1)) ∪ {𝑛}))
147	146	difeq1d 4100	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → ((1...𝑛) ∖ (1...(𝑛 − 1))) = (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))))
148		nnre 12247	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
149		ltm1 12083	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℝ → (𝑛 − 1) < 𝑛)
150		peano2rem 11550	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℝ → (𝑛 − 1) ∈ ℝ)
151		ltnle 11314	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 − 1) ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((𝑛 − 1) < 𝑛 ↔ ¬ 𝑛 ≤ (𝑛 − 1)))
152	150, 151	mpancom 688	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℝ → ((𝑛 − 1) < 𝑛 ↔ ¬ 𝑛 ≤ (𝑛 − 1)))
153	149, 152	mpbid 232	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℝ → ¬ 𝑛 ≤ (𝑛 − 1))
154		elfzle2 13545	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...(𝑛 − 1)) → 𝑛 ≤ (𝑛 − 1))
155	153, 154	nsyl 140	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℝ → ¬ 𝑛 ∈ (1...(𝑛 − 1)))
156	148, 155	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → ¬ 𝑛 ∈ (1...(𝑛 − 1)))
157		incom 4184	. . . . . . . . . . . . . . . . . 18 ⊢ ((1...(𝑛 − 1)) ∩ {𝑛}) = ({𝑛} ∩ (1...(𝑛 − 1)))
158	157	eqeq1i 2740	. . . . . . . . . . . . . . . . 17 ⊢ (((1...(𝑛 − 1)) ∩ {𝑛}) = ∅ ↔ ({𝑛} ∩ (1...(𝑛 − 1))) = ∅)
159		disjsn 4687	. . . . . . . . . . . . . . . . 17 ⊢ (((1...(𝑛 − 1)) ∩ {𝑛}) = ∅ ↔ ¬ 𝑛 ∈ (1...(𝑛 − 1)))
160		disj3 4429	. . . . . . . . . . . . . . . . 17 ⊢ (({𝑛} ∩ (1...(𝑛 − 1))) = ∅ ↔ {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1))))
161	158, 159, 160	3bitr3i 301	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑛 ∈ (1...(𝑛 − 1)) ↔ {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1))))
162	156, 161	sylib 218	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1))))
163	125, 147, 162	3eqtr4a 2796	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → ((1...𝑛) ∖ (1...(𝑛 − 1))) = {𝑛})
164	121, 163	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...𝑁) → ((1...𝑛) ∖ (1...(𝑛 − 1))) = {𝑛})
165	164	imaeq2d 6047	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1...𝑁) → ((2nd ‘(1st ‘𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = ((2nd ‘(1st ‘𝑧)) “ {𝑛}))
166	165	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st ‘𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = ((2nd ‘(1st ‘𝑧)) “ {𝑛}))
167		dff1o3 6824	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘(1st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd ‘(1st ‘𝑧)):(1...𝑁)–onto→(1...𝑁) ∧ Fun ◡(2nd ‘(1st ‘𝑧))))
168	167	simprbi 496	. . . . . . . . . . . . 13 ⊢ ((2nd ‘(1st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ◡(2nd ‘(1st ‘𝑧)))
169		imadif 6620	. . . . . . . . . . . . 13 ⊢ (Fun ◡(2nd ‘(1st ‘𝑧)) → ((2nd ‘(1st ‘𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = (((2nd ‘(1st ‘𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st ‘𝑧)) “ (1...(𝑛 − 1)))))
170	26, 168, 169	3syl 18	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑆 → ((2nd ‘(1st ‘𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = (((2nd ‘(1st ‘𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st ‘𝑧)) “ (1...(𝑛 − 1)))))
171	170	adantr 480	. . . . . . . . . . 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)))))
172	120, 166, 171	3eqtr2d 2776	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st ‘𝑧))‘𝑛)} = (((2nd ‘(1st ‘𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st ‘𝑧)) “ (1...(𝑛 − 1)))))
173	4, 172	sylan 580	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st ‘𝑧))‘𝑛)} = (((2nd ‘(1st ‘𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st ‘𝑧)) “ (1...(𝑛 − 1)))))
174		eleq1 2822	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑘 → (𝑧 ∈ 𝑆 ↔ 𝑘 ∈ 𝑆))
175	174	anbi1d 631	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑘 → ((𝑧 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) ↔ (𝑘 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁))))
176		2fveq3 6881	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑘 → (2nd ‘(1st ‘𝑧)) = (2nd ‘(1st ‘𝑘)))
177	176	fveq1d 6878	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑘 → ((2nd ‘(1st ‘𝑧))‘𝑛) = ((2nd ‘(1st ‘𝑘))‘𝑛))
178	177	sneqd 4613	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑘 → {((2nd ‘(1st ‘𝑧))‘𝑛)} = {((2nd ‘(1st ‘𝑘))‘𝑛)})
179	176	imaeq1d 6046	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑘 → ((2nd ‘(1st ‘𝑧)) “ (1...𝑛)) = ((2nd ‘(1st ‘𝑘)) “ (1...𝑛)))
180	176	imaeq1d 6046	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑘 → ((2nd ‘(1st ‘𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st ‘𝑘)) “ (1...(𝑛 − 1))))
181	179, 180	difeq12d 4102	. . . . . . . . . . . . 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)))))
182	178, 181	eqeq12d 2751	. . . . . . . . . . . 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))))))
183	175, 182	imbi12d 344	. . . . . . . . . . 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)))))))
184	183, 172	chvarvv 1998	. . . . . . . . . 10 ⊢ ((𝑘 ∈ 𝑆 ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st ‘𝑘))‘𝑛)} = (((2nd ‘(1st ‘𝑘)) “ (1...𝑛)) ∖ ((2nd ‘(1st ‘𝑘)) “ (1...(𝑛 − 1)))))
185	11, 184	sylan 580	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st ‘𝑘))‘𝑛)} = (((2nd ‘(1st ‘𝑘)) “ (1...𝑛)) ∖ ((2nd ‘(1st ‘𝑘)) “ (1...(𝑛 − 1)))))
186	118, 173, 185	3eqtr4d 2780	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st ‘𝑧))‘𝑛)} = {((2nd ‘(1st ‘𝑘))‘𝑛)})
187		fvex 6889	. . . . . . . . 9 ⊢ ((2nd ‘(1st ‘𝑧))‘𝑛) ∈ V
188	187	sneqr 4816	. . . . . . . 8 ⊢ ({((2nd ‘(1st ‘𝑧))‘𝑛)} = {((2nd ‘(1st ‘𝑘))‘𝑛)} → ((2nd ‘(1st ‘𝑧))‘𝑛) = ((2nd ‘(1st ‘𝑘))‘𝑛))
189	186, 188	syl 17	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st ‘𝑧))‘𝑛) = ((2nd ‘(1st ‘𝑘))‘𝑛))
190	30, 43, 189	eqfnfvd 7024	. . . . . 6 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) → (2nd ‘(1st ‘𝑧)) = (2nd ‘(1st ‘𝑘)))
191	19, 20	syl 17	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑆 → (1st ‘𝑧) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
192	32, 33	syl 17	. . . . . . . 8 ⊢ (𝑘 ∈ 𝑆 → (1st ‘𝑘) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
193		xpopth 8029	. . . . . . . 8 ⊢ (((1st ‘𝑧) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (1st ‘𝑘) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) → (((1st ‘(1st ‘𝑧)) = (1st ‘(1st ‘𝑘)) ∧ (2nd ‘(1st ‘𝑧)) = (2nd ‘(1st ‘𝑘))) ↔ (1st ‘𝑧) = (1st ‘𝑘)))
194	191, 192, 193	syl2an 596	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → (((1st ‘(1st ‘𝑧)) = (1st ‘(1st ‘𝑘)) ∧ (2nd ‘(1st ‘𝑧)) = (2nd ‘(1st ‘𝑘))) ↔ (1st ‘𝑧) = (1st ‘𝑘)))
195	194	ad2antlr 727	. . . . . 6 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) → (((1st ‘(1st ‘𝑧)) = (1st ‘(1st ‘𝑘)) ∧ (2nd ‘(1st ‘𝑧)) = (2nd ‘(1st ‘𝑘))) ↔ (1st ‘𝑧) = (1st ‘𝑘)))
196	17, 190, 195	mpbi2and 712	. . . . 5 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) → (1st ‘𝑧) = (1st ‘𝑘))
197		eqtr3 2757	. . . . . 6 ⊢ (((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁) → (2nd ‘𝑧) = (2nd ‘𝑘))
198	197	adantl 481	. . . . 5 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) → (2nd ‘𝑧) = (2nd ‘𝑘))
199		xpopth 8029	. . . . . . 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))) → (((1st ‘𝑧) = (1st ‘𝑘) ∧ (2nd ‘𝑧) = (2nd ‘𝑘)) ↔ 𝑧 = 𝑘))
200	19, 32, 199	syl2an 596	. . . . . 6 ⊢ ((𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) → (((1st ‘𝑧) = (1st ‘𝑘) ∧ (2nd ‘𝑧) = (2nd ‘𝑘)) ↔ 𝑧 = 𝑘))
201	200	ad2antlr 727	. . . . 5 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) → (((1st ‘𝑧) = (1st ‘𝑘) ∧ (2nd ‘𝑧) = (2nd ‘𝑘)) ↔ 𝑧 = 𝑘))
202	196, 198, 201	mpbi2and 712	. . . 4 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) ∧ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁)) → 𝑧 = 𝑘)
203	202	ex 412	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) → (((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁) → 𝑧 = 𝑘))
204	203	ralrimivva 3187	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 ∀𝑘 ∈ 𝑆 (((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁) → 𝑧 = 𝑘))
205		fveqeq2 6885	. . 3 ⊢ (𝑧 = 𝑘 → ((2nd ‘𝑧) = 𝑁 ↔ (2nd ‘𝑘) = 𝑁))
206	205	rmo4 3713	. 2 ⊢ (∃*𝑧 ∈ 𝑆 (2nd ‘𝑧) = 𝑁 ↔ ∀𝑧 ∈ 𝑆 ∀𝑘 ∈ 𝑆 (((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑘) = 𝑁) → 𝑧 = 𝑘))
207	204, 206	sylibr 234	1 ⊢ (𝜑 → ∃*𝑧 ∈ 𝑆 (2nd ‘𝑧) = 𝑁)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2108  {cab 2713   ≠ wne 2932  ∀wral 3051  ∃*wrmo 3358  {crab 3415  ⦋csb 3874   ∖ cdif 3923   ∪ cun 3924   ∩ cin 3925  ∅c0 4308  ifcif 4500  {csn 4601   class class class wbr 5119   ↦ cmpt 5201   × cxp 5652  ^◡ccnv 5653   “ cima 5657  Fun wfun 6525   Fn wfn 6526  ⟶wf 6527  –onto→wfo 6529  –1-1-onto→wf1o 6530  ‘cfv 6531  (class class class)co 7405   ∘_f cof 7669  1^st c1st 7986  2^nd c2nd 7987   ↑_m cmap 8840  ℂcc 11127  ℝcr 11128  0cc0 11129  1c1 11130   + caddc 11132   < clt 11269   ≤ cle 11270   − cmin 11466  ℕcn 12240  ℤcz 12588  ℤ_≥cuz 12852  ...cfz 13524  ..^cfzo 13671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7671  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8719  df-map 8842  df-en 8960  df-dom 8961  df-sdom 8962  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-n0 12502  df-z 12589  df-uz 12853  df-fz 13525  df-fzo 13672

This theorem is referenced by:  poimirlem18  37662  poimirlem21  37665