Theorem poimirlem21 35725

Description: Lemma for poimir 35737 stating that, given a face not on a back face of the cube and a simplex in which it's opposite the final point of the walk, there exists exactly one other simplex containing it. (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)))
poimirlem22.2	⊢ (𝜑 → 𝑇 ∈ 𝑆)
poimirlem22.3	⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 0)
poimirlem21.4	⊢ (𝜑 → (2nd ‘𝑇) = 𝑁)

Assertion

Ref	Expression
poimirlem21	⊢ (𝜑 → ∃!𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)

Distinct variable groups:   𝑓,𝑗,𝑛,𝑝,𝑡,𝑦,𝑧   𝜑,𝑗,𝑛,𝑦   𝑗,𝐹,𝑛,𝑦   𝑗,𝑁,𝑛,𝑦   𝑇,𝑗,𝑛,𝑦   𝜑,𝑝,𝑡   𝑓,𝐾,𝑗,𝑛,𝑝,𝑡   𝑓,𝑁,𝑝,𝑡   𝑇,𝑓,𝑝   𝜑,𝑧   𝑓,𝐹,𝑝,𝑡,𝑧   𝑧,𝐾   𝑧,𝑁   𝑡,𝑇,𝑧   𝑆,𝑗,𝑛,𝑝,𝑡,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑓)   𝑆(𝑓)   𝐾(𝑦)

Proof of Theorem poimirlem21

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		poimir.0	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ)
2		poimirlem22.s	. . 3 ⊢ 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}))))}
3		poimirlem22.1	. . 3 ⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))
4		poimirlem22.2	. . 3 ⊢ (𝜑 → 𝑇 ∈ 𝑆)
5		poimirlem22.3	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 0)
6		poimirlem21.4	. . 3 ⊢ (𝜑 → (2nd ‘𝑇) = 𝑁)
7	1, 2, 3, 4, 5, 6	poimirlem20 35724	. 2 ⊢ (𝜑 → ∃𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)
8	6	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (2nd ‘𝑇) = 𝑁)
9	1	nnred 11918	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℝ)
10	9	ltm1d 11837	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 − 1) < 𝑁)
11		nnm1nn0 12204	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
12	1, 11	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ0)
13	12	nn0red 12224	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁 − 1) ∈ ℝ)
14	13, 9	ltnled 11052	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑁 − 1) < 𝑁 ↔ ¬ 𝑁 ≤ (𝑁 − 1)))
15	10, 14	mpbid 231	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ 𝑁 ≤ (𝑁 − 1))
16		elfzle2 13189	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (1...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1))
17	15, 16	nsyl 140	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 𝑁 ∈ (1...(𝑁 − 1)))
18		eleq1 2826	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝑧) = 𝑁 → ((2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ↔ 𝑁 ∈ (1...(𝑁 − 1))))
19	18	notbid 317	. . . . . . . . . . . . 13 ⊢ ((2nd ‘𝑧) = 𝑁 → (¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ↔ ¬ 𝑁 ∈ (1...(𝑁 − 1))))
20	17, 19	syl5ibrcom 246	. . . . . . . . . . . 12 ⊢ (𝜑 → ((2nd ‘𝑧) = 𝑁 → ¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))))
21	20	necon2ad 2957	. . . . . . . . . . 11 ⊢ (𝜑 → ((2nd ‘𝑧) ∈ (1...(𝑁 − 1)) → (2nd ‘𝑧) ≠ 𝑁))
22	21	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ((2nd ‘𝑧) ∈ (1...(𝑁 − 1)) → (2nd ‘𝑧) ≠ 𝑁))
23	1	ad2antrr 722	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → 𝑁 ∈ ℕ)
24		fveq2 6756	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑧 → (2nd ‘𝑡) = (2nd ‘𝑧))
25	24	breq2d 5082	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = 𝑧 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑧)))
26	25	ifbid 4479	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑧 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑧), 𝑦, (𝑦 + 1)))
27	26	csbeq1d 3832	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑧 → ⦋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}))) = ⦋if(𝑦 < (2nd ‘𝑧), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))
28		2fveq3 6761	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = 𝑧 → (1st ‘(1st ‘𝑡)) = (1st ‘(1st ‘𝑧)))
29		2fveq3 6761	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑡 = 𝑧 → (2nd ‘(1st ‘𝑡)) = (2nd ‘(1st ‘𝑧)))
30	29	imaeq1d 5957	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑧 → ((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑧)) “ (1...𝑗)))
31	30	xpeq1d 5609	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑧 → (((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st ‘𝑧)) “ (1...𝑗)) × {1}))
32	29	imaeq1d 5957	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑧 → ((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑧)) “ ((𝑗 + 1)...𝑁)))
33	32	xpeq1d 5609	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑧 → (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))
34	31, 33	uneq12d 4094	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = 𝑧 → ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st ‘𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0})))
35	28, 34	oveq12d 7273	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑧 → ((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st ‘𝑧)) ∘f + ((((2nd ‘(1st ‘𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))))
36	35	csbeq2dv 3835	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑧 → ⦋if(𝑦 < (2nd ‘𝑧), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd ‘𝑧), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑧)) ∘f + ((((2nd ‘(1st ‘𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))))
37	27, 36	eqtrd 2778	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑧 → ⦋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}))) = ⦋if(𝑦 < (2nd ‘𝑧), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑧)) ∘f + ((((2nd ‘(1st ‘𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))))
38	37	mpteq2dv 5172	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑧 → (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})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑧), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑧)) ∘f + ((((2nd ‘(1st ‘𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0})))))
39	38	eqeq2d 2749	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑧 → (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})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑧), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑧)) ∘f + ((((2nd ‘(1st ‘𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
40	39, 2	elrab2 3620	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑆 ↔ (𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑧), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑧)) ∘f + ((((2nd ‘(1st ‘𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
41	40	simprbi 496	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑆 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑧), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑧)) ∘f + ((((2nd ‘(1st ‘𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0})))))
42	41	ad2antlr 723	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑧), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑧)) ∘f + ((((2nd ‘(1st ‘𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0})))))
43		elrabi 3611	. . . . . . . . . . . . . . . . . . . 20 ⊢ (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)))
44	43, 2	eleq2s 2857	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
45		xp1st 7836	. . . . . . . . . . . . . . . . . . 19 ⊢ (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)}))
46	44, 45	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ 𝑆 → (1st ‘𝑧) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
47		xp1st 7836	. . . . . . . . . . . . . . . . . 18 ⊢ ((1st ‘𝑧) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st ‘𝑧)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
48	46, 47	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ 𝑆 → (1st ‘(1st ‘𝑧)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
49		elmapi 8595	. . . . . . . . . . . . . . . . 17 ⊢ ((1st ‘(1st ‘𝑧)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st ‘(1st ‘𝑧)):(1...𝑁)⟶(0..^𝐾))
50	48, 49	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑆 → (1st ‘(1st ‘𝑧)):(1...𝑁)⟶(0..^𝐾))
51		elfzoelz 13316	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (0..^𝐾) → 𝑛 ∈ ℤ)
52	51	ssriv 3921	. . . . . . . . . . . . . . . 16 ⊢ (0..^𝐾) ⊆ ℤ
53		fss 6601	. . . . . . . . . . . . . . . 16 ⊢ (((1st ‘(1st ‘𝑧)):(1...𝑁)⟶(0..^𝐾) ∧ (0..^𝐾) ⊆ ℤ) → (1st ‘(1st ‘𝑧)):(1...𝑁)⟶ℤ)
54	50, 52, 53	sylancl 585	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑆 → (1st ‘(1st ‘𝑧)):(1...𝑁)⟶ℤ)
55	54	ad2antlr 723	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → (1st ‘(1st ‘𝑧)):(1...𝑁)⟶ℤ)
56		xp2nd 7837	. . . . . . . . . . . . . . . . 17 ⊢ ((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)})
57	46, 56	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑆 → (2nd ‘(1st ‘𝑧)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
58		fvex 6769	. . . . . . . . . . . . . . . . 17 ⊢ (2nd ‘(1st ‘𝑧)) ∈ V
59		f1oeq1 6688	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (2nd ‘(1st ‘𝑧)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁)))
60	58, 59	elab 3602	. . . . . . . . . . . . . . . 16 ⊢ ((2nd ‘(1st ‘𝑧)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
61	57, 60	sylib 217	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑆 → (2nd ‘(1st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
62	61	ad2antlr 723	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → (2nd ‘(1st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
63		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → (2nd ‘𝑧) ∈ (1...(𝑁 − 1)))
64	23, 42, 55, 62, 63	poimirlem1 35705	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd ‘𝑧) − 1))‘𝑛) ≠ ((𝐹‘(2nd ‘𝑧))‘𝑛))
65	1	ad2antrr 722	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)) → 𝑁 ∈ ℕ)
66		fveq2 6756	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 = 𝑇 → (2nd ‘𝑡) = (2nd ‘𝑇))
67	66	breq2d 5082	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 = 𝑇 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑇)))
68	67	ifbid 4479	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑡 = 𝑇 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)))
69	68	csbeq1d 3832	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑇 → ⦋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}))) = ⦋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}))))
70		2fveq3 6761	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 = 𝑇 → (1st ‘(1st ‘𝑡)) = (1st ‘(1st ‘𝑇)))
71		2fveq3 6761	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑡 = 𝑇 → (2nd ‘(1st ‘𝑡)) = (2nd ‘(1st ‘𝑇)))
72	71	imaeq1d 5957	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑡 = 𝑇 → ((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑇)) “ (1...𝑗)))
73	72	xpeq1d 5609	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 = 𝑇 → (((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}))
74	71	imaeq1d 5957	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑡 = 𝑇 → ((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)))
75	74	xpeq1d 5609	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 = 𝑇 → (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
76	73, 75	uneq12d 4094	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 = 𝑇 → ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))
77	70, 76	oveq12d 7273	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑡 = 𝑇 → ((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st ‘𝑇)) ∘f + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
78	77	csbeq2dv 3835	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑇 → ⦋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}))) = ⦋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}))))
79	69, 78	eqtrd 2778	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑇 → ⦋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}))) = ⦋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}))))
80	79	mpteq2dv 5172	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = 𝑇 → (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})))) = (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})))))
81	80	eqeq2d 2749	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑇 → (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})))) ↔ 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}))))))
82	81, 2	elrab2 3620	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑇 ∈ 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}))))))
83	82	simprbi 496	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑇 ∈ 𝑆 → 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})))))
84	4, 83	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 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})))))
85	84	ad2antrr 722	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)) → 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})))))
86		elrabi 3611	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑇 ∈ {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}))))} → 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
87	86, 2	eleq2s 2857	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑇 ∈ 𝑆 → 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
88	4, 87	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
89		xp1st 7836	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st ‘𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
90	88, 89	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (1st ‘𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
91		xp1st 7836	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1st ‘𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
92	90, 91	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
93		elmapi 8595	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
94	92, 93	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
95		fss 6601	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1st ‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾) ∧ (0..^𝐾) ⊆ ℤ) → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶ℤ)
96	94, 52, 95	sylancl 585	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶ℤ)
97	96	ad2antrr 722	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)) → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶ℤ)
98		xp2nd 7837	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1st ‘𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
99	90, 98	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (2nd ‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
100		fvex 6769	. . . . . . . . . . . . . . . . . . . 20 ⊢ (2nd ‘(1st ‘𝑇)) ∈ V
101		f1oeq1 6688	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = (2nd ‘(1st ‘𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
102	100, 101	elab 3602	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2nd ‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
103	99, 102	sylib 217	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
104	103	ad2antrr 722	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)) → (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
105		simplr 765	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)) → (2nd ‘𝑧) ∈ (1...(𝑁 − 1)))
106		xp2nd 7837	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd ‘𝑇) ∈ (0...𝑁))
107	88, 106	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (2nd ‘𝑇) ∈ (0...𝑁))
108	107	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → (2nd ‘𝑇) ∈ (0...𝑁))
109		eldifsn 4717	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2nd ‘𝑇) ∈ ((0...𝑁) ∖ {(2nd ‘𝑧)}) ↔ ((2nd ‘𝑇) ∈ (0...𝑁) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)))
110	109	biimpri 227	. . . . . . . . . . . . . . . . . 18 ⊢ (((2nd ‘𝑇) ∈ (0...𝑁) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)) → (2nd ‘𝑇) ∈ ((0...𝑁) ∖ {(2nd ‘𝑧)}))
111	108, 110	sylan 579	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)) → (2nd ‘𝑇) ∈ ((0...𝑁) ∖ {(2nd ‘𝑧)}))
112	65, 85, 97, 104, 105, 111	poimirlem2 35706	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)) → ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd ‘𝑧) − 1))‘𝑛) ≠ ((𝐹‘(2nd ‘𝑧))‘𝑛))
113	112	ex 412	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → ((2nd ‘𝑇) ≠ (2nd ‘𝑧) → ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd ‘𝑧) − 1))‘𝑛) ≠ ((𝐹‘(2nd ‘𝑧))‘𝑛)))
114	113	necon1bd 2960	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → (¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd ‘𝑧) − 1))‘𝑛) ≠ ((𝐹‘(2nd ‘𝑧))‘𝑛) → (2nd ‘𝑇) = (2nd ‘𝑧)))
115	114	adantlr 711	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → (¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd ‘𝑧) − 1))‘𝑛) ≠ ((𝐹‘(2nd ‘𝑧))‘𝑛) → (2nd ‘𝑇) = (2nd ‘𝑧)))
116	64, 115	mpd 15	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → (2nd ‘𝑇) = (2nd ‘𝑧))
117	116	neeq1d 3002	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → ((2nd ‘𝑇) ≠ 𝑁 ↔ (2nd ‘𝑧) ≠ 𝑁))
118	117	exbiri 807	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ((2nd ‘𝑧) ∈ (1...(𝑁 − 1)) → ((2nd ‘𝑧) ≠ 𝑁 → (2nd ‘𝑇) ≠ 𝑁)))
119	22, 118	mpdd 43	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ((2nd ‘𝑧) ∈ (1...(𝑁 − 1)) → (2nd ‘𝑇) ≠ 𝑁))
120	119	necon2bd 2958	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ((2nd ‘𝑇) = 𝑁 → ¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))))
121	8, 120	mpd 15	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)))
122		xp2nd 7837	. . . . . . . . 9 ⊢ (𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd ‘𝑧) ∈ (0...𝑁))
123	44, 122	syl 17	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑆 → (2nd ‘𝑧) ∈ (0...𝑁))
124		nn0uz 12549	. . . . . . . . . . . . . . . . . 18 ⊢ ℕ0 = (ℤ≥‘0)
125	12, 124	eleqtrdi 2849	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑁 − 1) ∈ (ℤ≥‘0))
126		fzpred 13233	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 − 1) ∈ (ℤ≥‘0) → (0...(𝑁 − 1)) = ({0} ∪ ((0 + 1)...(𝑁 − 1))))
127	125, 126	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0...(𝑁 − 1)) = ({0} ∪ ((0 + 1)...(𝑁 − 1))))
128		0p1e1 12025	. . . . . . . . . . . . . . . . . 18 ⊢ (0 + 1) = 1
129	128	oveq1i 7265	. . . . . . . . . . . . . . . . 17 ⊢ ((0 + 1)...(𝑁 − 1)) = (1...(𝑁 − 1))
130	129	uneq2i 4090	. . . . . . . . . . . . . . . 16 ⊢ ({0} ∪ ((0 + 1)...(𝑁 − 1))) = ({0} ∪ (1...(𝑁 − 1)))
131	127, 130	eqtrdi 2795	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0...(𝑁 − 1)) = ({0} ∪ (1...(𝑁 − 1))))
132	131	eleq2d 2824	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2nd ‘𝑧) ∈ (0...(𝑁 − 1)) ↔ (2nd ‘𝑧) ∈ ({0} ∪ (1...(𝑁 − 1)))))
133	132	notbid 317	. . . . . . . . . . . . 13 ⊢ (𝜑 → (¬ (2nd ‘𝑧) ∈ (0...(𝑁 − 1)) ↔ ¬ (2nd ‘𝑧) ∈ ({0} ∪ (1...(𝑁 − 1)))))
134		ioran 980	. . . . . . . . . . . . . 14 ⊢ (¬ ((2nd ‘𝑧) = 0 ∨ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) ↔ (¬ (2nd ‘𝑧) = 0 ∧ ¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))))
135		elun 4079	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘𝑧) ∈ ({0} ∪ (1...(𝑁 − 1))) ↔ ((2nd ‘𝑧) ∈ {0} ∨ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))))
136		fvex 6769	. . . . . . . . . . . . . . . . 17 ⊢ (2nd ‘𝑧) ∈ V
137	136	elsn 4573	. . . . . . . . . . . . . . . 16 ⊢ ((2nd ‘𝑧) ∈ {0} ↔ (2nd ‘𝑧) = 0)
138	137	orbi1i 910	. . . . . . . . . . . . . . 15 ⊢ (((2nd ‘𝑧) ∈ {0} ∨ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) ↔ ((2nd ‘𝑧) = 0 ∨ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))))
139	135, 138	bitri 274	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝑧) ∈ ({0} ∪ (1...(𝑁 − 1))) ↔ ((2nd ‘𝑧) = 0 ∨ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))))
140	134, 139	xchnxbir 332	. . . . . . . . . . . . 13 ⊢ (¬ (2nd ‘𝑧) ∈ ({0} ∪ (1...(𝑁 − 1))) ↔ (¬ (2nd ‘𝑧) = 0 ∧ ¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))))
141	133, 140	bitrdi 286	. . . . . . . . . . . 12 ⊢ (𝜑 → (¬ (2nd ‘𝑧) ∈ (0...(𝑁 − 1)) ↔ (¬ (2nd ‘𝑧) = 0 ∧ ¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)))))
142	141	anbi2d 628	. . . . . . . . . . 11 ⊢ (𝜑 → (((2nd ‘𝑧) ∈ (0...𝑁) ∧ ¬ (2nd ‘𝑧) ∈ (0...(𝑁 − 1))) ↔ ((2nd ‘𝑧) ∈ (0...𝑁) ∧ (¬ (2nd ‘𝑧) = 0 ∧ ¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))))))
143		uncom 4083	. . . . . . . . . . . . . . . 16 ⊢ ((0...(𝑁 − 1)) ∪ {𝑁}) = ({𝑁} ∪ (0...(𝑁 − 1)))
144	143	difeq1i 4049	. . . . . . . . . . . . . . 15 ⊢ (((0...(𝑁 − 1)) ∪ {𝑁}) ∖ (0...(𝑁 − 1))) = (({𝑁} ∪ (0...(𝑁 − 1))) ∖ (0...(𝑁 − 1)))
145		difun2 4411	. . . . . . . . . . . . . . 15 ⊢ (({𝑁} ∪ (0...(𝑁 − 1))) ∖ (0...(𝑁 − 1))) = ({𝑁} ∖ (0...(𝑁 − 1)))
146	144, 145	eqtri 2766	. . . . . . . . . . . . . 14 ⊢ (((0...(𝑁 − 1)) ∪ {𝑁}) ∖ (0...(𝑁 − 1))) = ({𝑁} ∖ (0...(𝑁 − 1)))
147	1	nncnd 11919	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℂ)
148		npcan1 11330	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
149	147, 148	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
150	1	nnnn0d 12223	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
151	150, 124	eleqtrdi 2849	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0))
152	149, 151	eqeltrd 2839	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ≥‘0))
153	12	nn0zd 12353	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑁 − 1) ∈ ℤ)
154		uzid 12526	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ≥‘(𝑁 − 1)))
155		peano2uz 12570	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 − 1) ∈ (ℤ≥‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ≥‘(𝑁 − 1)))
156	153, 154, 155	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ≥‘(𝑁 − 1)))
157	149, 156	eqeltrrd 2840	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑁 − 1)))
158		fzsplit2 13210	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 − 1) + 1) ∈ (ℤ≥‘0) ∧ 𝑁 ∈ (ℤ≥‘(𝑁 − 1))) → (0...𝑁) = ((0...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
159	152, 157, 158	syl2anc 583	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0...𝑁) = ((0...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
160	149	oveq1d 7270	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = (𝑁...𝑁))
161	1	nnzd 12354	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℤ)
162		fzsn 13227	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
163	161, 162	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑁...𝑁) = {𝑁})
164	160, 163	eqtrd 2778	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = {𝑁})
165	164	uneq2d 4093	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((0...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = ((0...(𝑁 − 1)) ∪ {𝑁}))
166	159, 165	eqtrd 2778	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0...𝑁) = ((0...(𝑁 − 1)) ∪ {𝑁}))
167	166	difeq1d 4052	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((0...𝑁) ∖ (0...(𝑁 − 1))) = (((0...(𝑁 − 1)) ∪ {𝑁}) ∖ (0...(𝑁 − 1))))
168		elfzle2 13189	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (0...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1))
169	15, 168	nsyl 140	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ¬ 𝑁 ∈ (0...(𝑁 − 1)))
170		incom 4131	. . . . . . . . . . . . . . . . 17 ⊢ ((0...(𝑁 − 1)) ∩ {𝑁}) = ({𝑁} ∩ (0...(𝑁 − 1)))
171	170	eqeq1i 2743	. . . . . . . . . . . . . . . 16 ⊢ (((0...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ({𝑁} ∩ (0...(𝑁 − 1))) = ∅)
172		disjsn 4644	. . . . . . . . . . . . . . . 16 ⊢ (((0...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (0...(𝑁 − 1)))
173		disj3 4384	. . . . . . . . . . . . . . . 16 ⊢ (({𝑁} ∩ (0...(𝑁 − 1))) = ∅ ↔ {𝑁} = ({𝑁} ∖ (0...(𝑁 − 1))))
174	171, 172, 173	3bitr3i 300	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑁 ∈ (0...(𝑁 − 1)) ↔ {𝑁} = ({𝑁} ∖ (0...(𝑁 − 1))))
175	169, 174	sylib 217	. . . . . . . . . . . . . 14 ⊢ (𝜑 → {𝑁} = ({𝑁} ∖ (0...(𝑁 − 1))))
176	146, 167, 175	3eqtr4a 2805	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((0...𝑁) ∖ (0...(𝑁 − 1))) = {𝑁})
177	176	eleq2d 2824	. . . . . . . . . . . 12 ⊢ (𝜑 → ((2nd ‘𝑧) ∈ ((0...𝑁) ∖ (0...(𝑁 − 1))) ↔ (2nd ‘𝑧) ∈ {𝑁}))
178		eldif 3893	. . . . . . . . . . . 12 ⊢ ((2nd ‘𝑧) ∈ ((0...𝑁) ∖ (0...(𝑁 − 1))) ↔ ((2nd ‘𝑧) ∈ (0...𝑁) ∧ ¬ (2nd ‘𝑧) ∈ (0...(𝑁 − 1))))
179	136	elsn 4573	. . . . . . . . . . . 12 ⊢ ((2nd ‘𝑧) ∈ {𝑁} ↔ (2nd ‘𝑧) = 𝑁)
180	177, 178, 179	3bitr3g 312	. . . . . . . . . . 11 ⊢ (𝜑 → (((2nd ‘𝑧) ∈ (0...𝑁) ∧ ¬ (2nd ‘𝑧) ∈ (0...(𝑁 − 1))) ↔ (2nd ‘𝑧) = 𝑁))
181	142, 180	bitr3d 280	. . . . . . . . . 10 ⊢ (𝜑 → (((2nd ‘𝑧) ∈ (0...𝑁) ∧ (¬ (2nd ‘𝑧) = 0 ∧ ¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)))) ↔ (2nd ‘𝑧) = 𝑁))
182	181	biimpd 228	. . . . . . . . 9 ⊢ (𝜑 → (((2nd ‘𝑧) ∈ (0...𝑁) ∧ (¬ (2nd ‘𝑧) = 0 ∧ ¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)))) → (2nd ‘𝑧) = 𝑁))
183	182	expdimp 452	. . . . . . . 8 ⊢ ((𝜑 ∧ (2nd ‘𝑧) ∈ (0...𝑁)) → ((¬ (2nd ‘𝑧) = 0 ∧ ¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → (2nd ‘𝑧) = 𝑁))
184	123, 183	sylan2 592	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ((¬ (2nd ‘𝑧) = 0 ∧ ¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → (2nd ‘𝑧) = 𝑁))
185	121, 184	mpan2d 690	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (¬ (2nd ‘𝑧) = 0 → (2nd ‘𝑧) = 𝑁))
186	1, 2, 3	poimirlem14 35718	. . . . . . . . . 10 ⊢ (𝜑 → ∃*𝑧 ∈ 𝑆 (2nd ‘𝑧) = 𝑁)
187		fveqeq2 6765	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑠 → ((2nd ‘𝑧) = 𝑁 ↔ (2nd ‘𝑠) = 𝑁))
188	187	rmo4 3660	. . . . . . . . . 10 ⊢ (∃*𝑧 ∈ 𝑆 (2nd ‘𝑧) = 𝑁 ↔ ∀𝑧 ∈ 𝑆 ∀𝑠 ∈ 𝑆 (((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑠) = 𝑁) → 𝑧 = 𝑠))
189	186, 188	sylib 217	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 ∀𝑠 ∈ 𝑆 (((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑠) = 𝑁) → 𝑧 = 𝑠))
190	189	r19.21bi 3132	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ∀𝑠 ∈ 𝑆 (((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑠) = 𝑁) → 𝑧 = 𝑠))
191	4	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝑇 ∈ 𝑆)
192		fveqeq2 6765	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑇 → ((2nd ‘𝑠) = 𝑁 ↔ (2nd ‘𝑇) = 𝑁))
193	192	anbi2d 628	. . . . . . . . . 10 ⊢ (𝑠 = 𝑇 → (((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑠) = 𝑁) ↔ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑇) = 𝑁)))
194		eqeq2 2750	. . . . . . . . . 10 ⊢ (𝑠 = 𝑇 → (𝑧 = 𝑠 ↔ 𝑧 = 𝑇))
195	193, 194	imbi12d 344	. . . . . . . . 9 ⊢ (𝑠 = 𝑇 → ((((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑠) = 𝑁) → 𝑧 = 𝑠) ↔ (((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑇) = 𝑁) → 𝑧 = 𝑇)))
196	195	rspccv 3549	. . . . . . . 8 ⊢ (∀𝑠 ∈ 𝑆 (((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑠) = 𝑁) → 𝑧 = 𝑠) → (𝑇 ∈ 𝑆 → (((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑇) = 𝑁) → 𝑧 = 𝑇)))
197	190, 191, 196	sylc 65	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑇) = 𝑁) → 𝑧 = 𝑇))
198	8, 197	mpan2d 690	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ((2nd ‘𝑧) = 𝑁 → 𝑧 = 𝑇))
199	185, 198	syld 47	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (¬ (2nd ‘𝑧) = 0 → 𝑧 = 𝑇))
200	199	necon1ad 2959	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝑧 ≠ 𝑇 → (2nd ‘𝑧) = 0))
201	200	ralrimiva 3107	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 (𝑧 ≠ 𝑇 → (2nd ‘𝑧) = 0))
202	1, 2, 3	poimirlem13 35717	. . 3 ⊢ (𝜑 → ∃*𝑧 ∈ 𝑆 (2nd ‘𝑧) = 0)
203		rmoim 3670	. . 3 ⊢ (∀𝑧 ∈ 𝑆 (𝑧 ≠ 𝑇 → (2nd ‘𝑧) = 0) → (∃*𝑧 ∈ 𝑆 (2nd ‘𝑧) = 0 → ∃*𝑧 ∈ 𝑆 𝑧 ≠ 𝑇))
204	201, 202, 203	sylc 65	. 2 ⊢ (𝜑 → ∃*𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)
205		reu5 3351	. 2 ⊢ (∃!𝑧 ∈ 𝑆 𝑧 ≠ 𝑇 ↔ (∃𝑧 ∈ 𝑆 𝑧 ≠ 𝑇 ∧ ∃*𝑧 ∈ 𝑆 𝑧 ≠ 𝑇))
206	7, 204, 205	sylanbrc 582	1 ⊢ (𝜑 → ∃!𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 843   = wceq 1539   ∈ wcel 2108  {cab 2715   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  ∃!wreu 3065  ∃*wrmo 3066  {crab 3067  ⦋csb 3828   ∖ cdif 3880   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  ifcif 4456  {csn 4558   class class class wbr 5070   ↦ cmpt 5153   × cxp 5578  ran crn 5581   “ cima 5583  ⟶wf 6414  –1-1-onto→wf1o 6417  ‘cfv 6418  (class class class)co 7255   ∘_f cof 7509  1^st c1st 7802  2^nd c2nd 7803   ↑_m cmap 8573  ℂcc 10800  0cc0 10802  1c1 10803   + caddc 10805   < clt 10940   ≤ cle 10941   − cmin 11135  ℕcn 11903  ℕ₀cn0 12163  ℤcz 12249  ℤ_≥cuz 12511  ...cfz 13168  ..^cfzo 13311

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169  df-fzo 13312

This theorem is referenced by:  poimirlem22  35726