Theorem poimirlem18 33851

Description: Lemma for poimir 33866 stating that, given a face not on a front face of the main cube and a simplex in which it's opposite the first vertex on 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)
poimirlem18.3	⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 𝐾)
poimirlem18.4	⊢ (𝜑 → (2nd ‘𝑇) = 0)

Assertion

Ref	Expression
poimirlem18	⊢ (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 poimirlem18

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		poimirlem18.3	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 𝐾)
6		poimirlem18.4	. . 3 ⊢ (𝜑 → (2nd ‘𝑇) = 0)
7	1, 2, 3, 4, 5, 6	poimirlem17 33850	. 2 ⊢ (𝜑 → ∃𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)
8	6	adantr 472	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (2nd ‘𝑇) = 0)
9		0nnn 11309	. . . . . . . . . . . . 13 ⊢ ¬ 0 ∈ ℕ
10		elfznn 12577	. . . . . . . . . . . . 13 ⊢ (0 ∈ (1...(𝑁 − 1)) → 0 ∈ ℕ)
11	9, 10	mto 188	. . . . . . . . . . . 12 ⊢ ¬ 0 ∈ (1...(𝑁 − 1))
12		eleq1 2832	. . . . . . . . . . . 12 ⊢ ((2nd ‘𝑧) = 0 → ((2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ↔ 0 ∈ (1...(𝑁 − 1))))
13	11, 12	mtbiri 318	. . . . . . . . . . 11 ⊢ ((2nd ‘𝑧) = 0 → ¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)))
14	13	necon2ai 2966	. . . . . . . . . 10 ⊢ ((2nd ‘𝑧) ∈ (1...(𝑁 − 1)) → (2nd ‘𝑧) ≠ 0)
15	1	ad2antrr 717	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → 𝑁 ∈ ℕ)
16		fveq2 6375	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑧 → (2nd ‘𝑡) = (2nd ‘𝑧))
17	16	breq2d 4821	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = 𝑧 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑧)))
18	17	ifbid 4265	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑧 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑧), 𝑦, (𝑦 + 1)))
19	18	csbeq1d 3698	. . . . . . . . . . . . . . . . . . . 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}))))
20		2fveq3 6380	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = 𝑧 → (1st ‘(1st ‘𝑡)) = (1st ‘(1st ‘𝑧)))
21		2fveq3 6380	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑡 = 𝑧 → (2nd ‘(1st ‘𝑡)) = (2nd ‘(1st ‘𝑧)))
22	21	imaeq1d 5647	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑧 → ((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑧)) “ (1...𝑗)))
23	22	xpeq1d 5306	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑧 → (((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st ‘𝑧)) “ (1...𝑗)) × {1}))
24	21	imaeq1d 5647	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑧 → ((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑧)) “ ((𝑗 + 1)...𝑁)))
25	24	xpeq1d 5306	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑧 → (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))
26	23, 25	uneq12d 3930	. . . . . . . . . . . . . . . . . . . . . 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})))
27	20, 26	oveq12d 6860	. . . . . . . . . . . . . . . . . . . . 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}))))
28	27	csbeq2dv 4153	. . . . . . . . . . . . . . . . . . . 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}))))
29	19, 28	eqtrd 2799	. . . . . . . . . . . . . . . . . . 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}))))
30	29	mpteq2dv 4904	. . . . . . . . . . . . . . . . . 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})))))
31	30	eqeq2d 2775	. . . . . . . . . . . . . . . . 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}))))))
32	31, 2	elrab2 3523	. . . . . . . . . . . . . . . 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}))))))
33	32	simprbi 490	. . . . . . . . . . . . . . 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})))))
34	33	ad2antlr 718	. . . . . . . . . . . . . 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})))))
35		elrabi 3514	. . . . . . . . . . . . . . . . . . . 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)))
36	35, 2	eleq2s 2862	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
37		xp1st 7398	. . . . . . . . . . . . . . . . . . 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)}))
38	36, 37	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ 𝑆 → (1st ‘𝑧) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
39		xp1st 7398	. . . . . . . . . . . . . . . . . 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)))
40	38, 39	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ 𝑆 → (1st ‘(1st ‘𝑧)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
41		elmapi 8082	. . . . . . . . . . . . . . . . 17 ⊢ ((1st ‘(1st ‘𝑧)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st ‘(1st ‘𝑧)):(1...𝑁)⟶(0..^𝐾))
42	40, 41	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑆 → (1st ‘(1st ‘𝑧)):(1...𝑁)⟶(0..^𝐾))
43		elfzoelz 12678	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (0..^𝐾) → 𝑛 ∈ ℤ)
44	43	ssriv 3765	. . . . . . . . . . . . . . . 16 ⊢ (0..^𝐾) ⊆ ℤ
45		fss 6236	. . . . . . . . . . . . . . . 16 ⊢ (((1st ‘(1st ‘𝑧)):(1...𝑁)⟶(0..^𝐾) ∧ (0..^𝐾) ⊆ ℤ) → (1st ‘(1st ‘𝑧)):(1...𝑁)⟶ℤ)
46	42, 44, 45	sylancl 580	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑆 → (1st ‘(1st ‘𝑧)):(1...𝑁)⟶ℤ)
47	46	ad2antlr 718	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → (1st ‘(1st ‘𝑧)):(1...𝑁)⟶ℤ)
48		xp2nd 7399	. . . . . . . . . . . . . . . . 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)})
49	38, 48	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑆 → (2nd ‘(1st ‘𝑧)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
50		fvex 6388	. . . . . . . . . . . . . . . . 17 ⊢ (2nd ‘(1st ‘𝑧)) ∈ V
51		f1oeq1 6310	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (2nd ‘(1st ‘𝑧)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁)))
52	50, 51	elab 3504	. . . . . . . . . . . . . . . 16 ⊢ ((2nd ‘(1st ‘𝑧)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
53	49, 52	sylib 209	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑆 → (2nd ‘(1st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
54	53	ad2antlr 718	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → (2nd ‘(1st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
55		simpr 477	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → (2nd ‘𝑧) ∈ (1...(𝑁 − 1)))
56	15, 34, 47, 54, 55	poimirlem1 33834	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd ‘𝑧) − 1))‘𝑛) ≠ ((𝐹‘(2nd ‘𝑧))‘𝑛))
57	1	ad2antrr 717	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)) → 𝑁 ∈ ℕ)
58		fveq2 6375	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 = 𝑇 → (2nd ‘𝑡) = (2nd ‘𝑇))
59	58	breq2d 4821	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 = 𝑇 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑇)))
60	59	ifbid 4265	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑡 = 𝑇 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)))
61	60	csbeq1d 3698	. . . . . . . . . . . . . . . . . . . . . . . 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}))))
62		2fveq3 6380	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 = 𝑇 → (1st ‘(1st ‘𝑡)) = (1st ‘(1st ‘𝑇)))
63		2fveq3 6380	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑡 = 𝑇 → (2nd ‘(1st ‘𝑡)) = (2nd ‘(1st ‘𝑇)))
64	63	imaeq1d 5647	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑡 = 𝑇 → ((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑇)) “ (1...𝑗)))
65	64	xpeq1d 5306	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 = 𝑇 → (((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}))
66	63	imaeq1d 5647	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑡 = 𝑇 → ((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)))
67	66	xpeq1d 5306	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 = 𝑇 → (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
68	65, 67	uneq12d 3930	. . . . . . . . . . . . . . . . . . . . . . . . . 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})))
69	62, 68	oveq12d 6860	. . . . . . . . . . . . . . . . . . . . . . . . 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}))))
70	69	csbeq2dv 4153	. . . . . . . . . . . . . . . . . . . . . . . 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}))))
71	61, 70	eqtrd 2799	. . . . . . . . . . . . . . . . . . . . . . 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}))))
72	71	mpteq2dv 4904	. . . . . . . . . . . . . . . . . . . . . 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})))))
73	72	eqeq2d 2775	. . . . . . . . . . . . . . . . . . . . 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}))))))
74	73, 2	elrab2 3523	. . . . . . . . . . . . . . . . . . . 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}))))))
75	74	simprbi 490	. . . . . . . . . . . . . . . . . . 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})))))
76	4, 75	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})))))
77	76	ad2antrr 717	. . . . . . . . . . . . . . . . 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})))))
78		elrabi 3514	. . . . . . . . . . . . . . . . . . . . . . . 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)))
79	78, 2	eleq2s 2862	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑇 ∈ 𝑆 → 𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
80	4, 79	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
81		xp1st 7398	. . . . . . . . . . . . . . . . . . . . . 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)}))
82	80, 81	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (1st ‘𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
83		xp1st 7398	. . . . . . . . . . . . . . . . . . . . 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)))
84	82, 83	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
85		elmapi 8082	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
86	84, 85	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
87		fss 6236	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1st ‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾) ∧ (0..^𝐾) ⊆ ℤ) → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶ℤ)
88	86, 44, 87	sylancl 580	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶ℤ)
89	88	ad2antrr 717	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)) → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶ℤ)
90		xp2nd 7399	. . . . . . . . . . . . . . . . . . . 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)})
91	82, 90	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (2nd ‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
92		fvex 6388	. . . . . . . . . . . . . . . . . . . 20 ⊢ (2nd ‘(1st ‘𝑇)) ∈ V
93		f1oeq1 6310	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = (2nd ‘(1st ‘𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
94	92, 93	elab 3504	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2nd ‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
95	91, 94	sylib 209	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
96	95	ad2antrr 717	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)) → (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
97		simplr 785	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)) → (2nd ‘𝑧) ∈ (1...(𝑁 − 1)))
98		xp2nd 7399	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd ‘𝑇) ∈ (0...𝑁))
99	80, 98	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (2nd ‘𝑇) ∈ (0...𝑁))
100	99	adantr 472	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → (2nd ‘𝑇) ∈ (0...𝑁))
101		eldifsn 4472	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2nd ‘𝑇) ∈ ((0...𝑁) ∖ {(2nd ‘𝑧)}) ↔ ((2nd ‘𝑇) ∈ (0...𝑁) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)))
102	101	biimpri 219	. . . . . . . . . . . . . . . . . 18 ⊢ (((2nd ‘𝑇) ∈ (0...𝑁) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)) → (2nd ‘𝑇) ∈ ((0...𝑁) ∖ {(2nd ‘𝑧)}))
103	100, 102	sylan 575	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)) → (2nd ‘𝑇) ∈ ((0...𝑁) ∖ {(2nd ‘𝑧)}))
104	57, 77, 89, 96, 97, 103	poimirlem2 33835	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)) → ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd ‘𝑧) − 1))‘𝑛) ≠ ((𝐹‘(2nd ‘𝑧))‘𝑛))
105	104	ex 401	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → ((2nd ‘𝑇) ≠ (2nd ‘𝑧) → ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd ‘𝑧) − 1))‘𝑛) ≠ ((𝐹‘(2nd ‘𝑧))‘𝑛)))
106	105	necon1bd 2955	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → (¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd ‘𝑧) − 1))‘𝑛) ≠ ((𝐹‘(2nd ‘𝑧))‘𝑛) → (2nd ‘𝑇) = (2nd ‘𝑧)))
107	106	adantlr 706	. . . . . . . . . . . . 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)))
108	56, 107	mpd 15	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → (2nd ‘𝑇) = (2nd ‘𝑧))
109	108	neeq1d 2996	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → ((2nd ‘𝑇) ≠ 0 ↔ (2nd ‘𝑧) ≠ 0))
110	109	exbiri 845	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ((2nd ‘𝑧) ∈ (1...(𝑁 − 1)) → ((2nd ‘𝑧) ≠ 0 → (2nd ‘𝑇) ≠ 0)))
111	14, 110	mpdi 45	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ((2nd ‘𝑧) ∈ (1...(𝑁 − 1)) → (2nd ‘𝑇) ≠ 0))
112	111	necon2bd 2953	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ((2nd ‘𝑇) = 0 → ¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))))
113	8, 112	mpd 15	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)))
114		xp2nd 7399	. . . . . . . . 9 ⊢ (𝑧 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd ‘𝑧) ∈ (0...𝑁))
115	36, 114	syl 17	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑆 → (2nd ‘𝑧) ∈ (0...𝑁))
116	1	nncnd 11292	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℂ)
117		npcan1 10709	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
118	116, 117	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
119		nnuz 11923	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ = (ℤ≥‘1)
120	1, 119	syl6eleq 2854	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘1))
121	118, 120	eqeltrd 2844	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ≥‘1))
122	1	nnzd 11728	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑁 ∈ ℤ)
123		peano2zm 11667	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
124	122, 123	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑁 − 1) ∈ ℤ)
125		uzid 11901	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ≥‘(𝑁 − 1)))
126		peano2uz 11941	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 − 1) ∈ (ℤ≥‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ≥‘(𝑁 − 1)))
127	124, 125, 126	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ≥‘(𝑁 − 1)))
128	118, 127	eqeltrrd 2845	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑁 − 1)))
129		fzsplit2 12573	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 − 1) + 1) ∈ (ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘(𝑁 − 1))) → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
130	121, 128, 129	syl2anc 579	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
131	118	oveq1d 6857	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = (𝑁...𝑁))
132		fzsn 12590	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
133	122, 132	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑁...𝑁) = {𝑁})
134	131, 133	eqtrd 2799	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = {𝑁})
135	134	uneq2d 3929	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = ((1...(𝑁 − 1)) ∪ {𝑁}))
136	130, 135	eqtrd 2799	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁}))
137	136	eleq2d 2830	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2nd ‘𝑧) ∈ (1...𝑁) ↔ (2nd ‘𝑧) ∈ ((1...(𝑁 − 1)) ∪ {𝑁})))
138	137	notbid 309	. . . . . . . . . . . . 13 ⊢ (𝜑 → (¬ (2nd ‘𝑧) ∈ (1...𝑁) ↔ ¬ (2nd ‘𝑧) ∈ ((1...(𝑁 − 1)) ∪ {𝑁})))
139		ioran 1006	. . . . . . . . . . . . . 14 ⊢ (¬ ((2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∨ (2nd ‘𝑧) = 𝑁) ↔ (¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd ‘𝑧) = 𝑁))
140		elun 3915	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘𝑧) ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ ((2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∨ (2nd ‘𝑧) ∈ {𝑁}))
141		fvex 6388	. . . . . . . . . . . . . . . . 17 ⊢ (2nd ‘𝑧) ∈ V
142	141	elsn 4349	. . . . . . . . . . . . . . . 16 ⊢ ((2nd ‘𝑧) ∈ {𝑁} ↔ (2nd ‘𝑧) = 𝑁)
143	142	orbi2i 936	. . . . . . . . . . . . . . 15 ⊢ (((2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∨ (2nd ‘𝑧) ∈ {𝑁}) ↔ ((2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∨ (2nd ‘𝑧) = 𝑁))
144	140, 143	bitri 266	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝑧) ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ ((2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∨ (2nd ‘𝑧) = 𝑁))
145	139, 144	xchnxbir 324	. . . . . . . . . . . . 13 ⊢ (¬ (2nd ‘𝑧) ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ (¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd ‘𝑧) = 𝑁))
146	138, 145	syl6bb 278	. . . . . . . . . . . 12 ⊢ (𝜑 → (¬ (2nd ‘𝑧) ∈ (1...𝑁) ↔ (¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd ‘𝑧) = 𝑁)))
147	146	anbi2d 622	. . . . . . . . . . 11 ⊢ (𝜑 → (((2nd ‘𝑧) ∈ (0...𝑁) ∧ ¬ (2nd ‘𝑧) ∈ (1...𝑁)) ↔ ((2nd ‘𝑧) ∈ (0...𝑁) ∧ (¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd ‘𝑧) = 𝑁))))
148	1	nnnn0d 11598	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
149		nn0uz 11922	. . . . . . . . . . . . . . . . 17 ⊢ ℕ0 = (ℤ≥‘0)
150	148, 149	syl6eleq 2854	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0))
151		fzpred 12596	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ≥‘0) → (0...𝑁) = ({0} ∪ ((0 + 1)...𝑁)))
152	150, 151	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0...𝑁) = ({0} ∪ ((0 + 1)...𝑁)))
153	152	difeq1d 3889	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((0...𝑁) ∖ (1...𝑁)) = (({0} ∪ ((0 + 1)...𝑁)) ∖ (1...𝑁)))
154		difun2 4208	. . . . . . . . . . . . . . 15 ⊢ (({0} ∪ (1...𝑁)) ∖ (1...𝑁)) = ({0} ∖ (1...𝑁))
155		0p1e1 11401	. . . . . . . . . . . . . . . . . 18 ⊢ (0 + 1) = 1
156	155	oveq1i 6852	. . . . . . . . . . . . . . . . 17 ⊢ ((0 + 1)...𝑁) = (1...𝑁)
157	156	uneq2i 3926	. . . . . . . . . . . . . . . 16 ⊢ ({0} ∪ ((0 + 1)...𝑁)) = ({0} ∪ (1...𝑁))
158	157	difeq1i 3886	. . . . . . . . . . . . . . 15 ⊢ (({0} ∪ ((0 + 1)...𝑁)) ∖ (1...𝑁)) = (({0} ∪ (1...𝑁)) ∖ (1...𝑁))
159		incom 3967	. . . . . . . . . . . . . . . . 17 ⊢ ({0} ∩ (1...𝑁)) = ((1...𝑁) ∩ {0})
160		elfznn 12577	. . . . . . . . . . . . . . . . . . 19 ⊢ (0 ∈ (1...𝑁) → 0 ∈ ℕ)
161	9, 160	mto 188	. . . . . . . . . . . . . . . . . 18 ⊢ ¬ 0 ∈ (1...𝑁)
162		disjsn 4402	. . . . . . . . . . . . . . . . . 18 ⊢ (((1...𝑁) ∩ {0}) = ∅ ↔ ¬ 0 ∈ (1...𝑁))
163	161, 162	mpbir 222	. . . . . . . . . . . . . . . . 17 ⊢ ((1...𝑁) ∩ {0}) = ∅
164	159, 163	eqtri 2787	. . . . . . . . . . . . . . . 16 ⊢ ({0} ∩ (1...𝑁)) = ∅
165		disj3 4182	. . . . . . . . . . . . . . . 16 ⊢ (({0} ∩ (1...𝑁)) = ∅ ↔ {0} = ({0} ∖ (1...𝑁)))
166	164, 165	mpbi 221	. . . . . . . . . . . . . . 15 ⊢ {0} = ({0} ∖ (1...𝑁))
167	154, 158, 166	3eqtr4i 2797	. . . . . . . . . . . . . 14 ⊢ (({0} ∪ ((0 + 1)...𝑁)) ∖ (1...𝑁)) = {0}
168	153, 167	syl6eq 2815	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((0...𝑁) ∖ (1...𝑁)) = {0})
169	168	eleq2d 2830	. . . . . . . . . . . 12 ⊢ (𝜑 → ((2nd ‘𝑧) ∈ ((0...𝑁) ∖ (1...𝑁)) ↔ (2nd ‘𝑧) ∈ {0}))
170		eldif 3742	. . . . . . . . . . . 12 ⊢ ((2nd ‘𝑧) ∈ ((0...𝑁) ∖ (1...𝑁)) ↔ ((2nd ‘𝑧) ∈ (0...𝑁) ∧ ¬ (2nd ‘𝑧) ∈ (1...𝑁)))
171	141	elsn 4349	. . . . . . . . . . . 12 ⊢ ((2nd ‘𝑧) ∈ {0} ↔ (2nd ‘𝑧) = 0)
172	169, 170, 171	3bitr3g 304	. . . . . . . . . . 11 ⊢ (𝜑 → (((2nd ‘𝑧) ∈ (0...𝑁) ∧ ¬ (2nd ‘𝑧) ∈ (1...𝑁)) ↔ (2nd ‘𝑧) = 0))
173	147, 172	bitr3d 272	. . . . . . . . . 10 ⊢ (𝜑 → (((2nd ‘𝑧) ∈ (0...𝑁) ∧ (¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd ‘𝑧) = 𝑁)) ↔ (2nd ‘𝑧) = 0))
174	173	biimpd 220	. . . . . . . . 9 ⊢ (𝜑 → (((2nd ‘𝑧) ∈ (0...𝑁) ∧ (¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd ‘𝑧) = 𝑁)) → (2nd ‘𝑧) = 0))
175	174	expdimp 444	. . . . . . . 8 ⊢ ((𝜑 ∧ (2nd ‘𝑧) ∈ (0...𝑁)) → ((¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd ‘𝑧) = 𝑁) → (2nd ‘𝑧) = 0))
176	115, 175	sylan2 586	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ((¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd ‘𝑧) = 𝑁) → (2nd ‘𝑧) = 0))
177	113, 176	mpand 686	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (¬ (2nd ‘𝑧) = 𝑁 → (2nd ‘𝑧) = 0))
178	1, 2, 3	poimirlem13 33846	. . . . . . . . . 10 ⊢ (𝜑 → ∃*𝑧 ∈ 𝑆 (2nd ‘𝑧) = 0)
179		fveqeq2 6384	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑠 → ((2nd ‘𝑧) = 0 ↔ (2nd ‘𝑠) = 0))
180	179	rmo4 3558	. . . . . . . . . 10 ⊢ (∃*𝑧 ∈ 𝑆 (2nd ‘𝑧) = 0 ↔ ∀𝑧 ∈ 𝑆 ∀𝑠 ∈ 𝑆 (((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑠) = 0) → 𝑧 = 𝑠))
181	178, 180	sylib 209	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 ∀𝑠 ∈ 𝑆 (((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑠) = 0) → 𝑧 = 𝑠))
182	181	r19.21bi 3079	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ∀𝑠 ∈ 𝑆 (((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑠) = 0) → 𝑧 = 𝑠))
183	4	adantr 472	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝑇 ∈ 𝑆)
184		fveqeq2 6384	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑇 → ((2nd ‘𝑠) = 0 ↔ (2nd ‘𝑇) = 0))
185	184	anbi2d 622	. . . . . . . . . 10 ⊢ (𝑠 = 𝑇 → (((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑠) = 0) ↔ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑇) = 0)))
186		eqeq2 2776	. . . . . . . . . 10 ⊢ (𝑠 = 𝑇 → (𝑧 = 𝑠 ↔ 𝑧 = 𝑇))
187	185, 186	imbi12d 335	. . . . . . . . 9 ⊢ (𝑠 = 𝑇 → ((((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑠) = 0) → 𝑧 = 𝑠) ↔ (((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑇) = 0) → 𝑧 = 𝑇)))
188	187	rspccv 3458	. . . . . . . 8 ⊢ (∀𝑠 ∈ 𝑆 (((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑠) = 0) → 𝑧 = 𝑠) → (𝑇 ∈ 𝑆 → (((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑇) = 0) → 𝑧 = 𝑇)))
189	182, 183, 188	sylc 65	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑇) = 0) → 𝑧 = 𝑇))
190	8, 189	mpan2d 685	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ((2nd ‘𝑧) = 0 → 𝑧 = 𝑇))
191	177, 190	syld 47	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (¬ (2nd ‘𝑧) = 𝑁 → 𝑧 = 𝑇))
192	191	necon1ad 2954	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝑧 ≠ 𝑇 → (2nd ‘𝑧) = 𝑁))
193	192	ralrimiva 3113	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 (𝑧 ≠ 𝑇 → (2nd ‘𝑧) = 𝑁))
194	1, 2, 3	poimirlem14 33847	. . 3 ⊢ (𝜑 → ∃*𝑧 ∈ 𝑆 (2nd ‘𝑧) = 𝑁)
195		rmoim 3568	. . 3 ⊢ (∀𝑧 ∈ 𝑆 (𝑧 ≠ 𝑇 → (2nd ‘𝑧) = 𝑁) → (∃*𝑧 ∈ 𝑆 (2nd ‘𝑧) = 𝑁 → ∃*𝑧 ∈ 𝑆 𝑧 ≠ 𝑇))
196	193, 194, 195	sylc 65	. 2 ⊢ (𝜑 → ∃*𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)
197		reu5 3307	. 2 ⊢ (∃!𝑧 ∈ 𝑆 𝑧 ≠ 𝑇 ↔ (∃𝑧 ∈ 𝑆 𝑧 ≠ 𝑇 ∧ ∃*𝑧 ∈ 𝑆 𝑧 ≠ 𝑇))
198	7, 196, 197	sylanbrc 578	1 ⊢ (𝜑 → ∃!𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   ∨ wo 873   = wceq 1652   ∈ wcel 2155  {cab 2751   ≠ wne 2937  ∀wral 3055  ∃wrex 3056  ∃!wreu 3057  ∃*wrmo 3058  {crab 3059  ⦋csb 3691   ∖ cdif 3729   ∪ cun 3730   ∩ cin 3731   ⊆ wss 3732  ∅c0 4079  ifcif 4243  {csn 4334   class class class wbr 4809   ↦ cmpt 4888   × cxp 5275  ran crn 5278   “ cima 5280  ⟶wf 6064  –1-1-onto→wf1o 6067  ‘cfv 6068  (class class class)co 6842   ∘_𝑓 cof 7093  1^st c1st 7364  2^nd c2nd 7365   ↑_𝑚 cmap 8060  ℂcc 10187  0cc0 10189  1c1 10190   + caddc 10192   < clt 10328   − cmin 10520  ℕcn 11274  ℕ₀cn0 11538  ℤcz 11624  ℤ_≥cuz 11886  ...cfz 12533  ..^cfzo 12673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-er 7947  df-map 8062  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-n0 11539  df-z 11625  df-uz 11887  df-fz 12534  df-fzo 12674

This theorem is referenced by:  poimirlem22  33855