Theorem poimirlem18 34380

Description: Lemma for poimir 34395 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 34379	. 2 ⊢ (𝜑 → ∃𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)
8	6	adantr 473	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (2nd ‘𝑇) = 0)
9		0nnn 11474	. . . . . . . . . . . . 13 ⊢ ¬ 0 ∈ ℕ
10		elfznn 12750	. . . . . . . . . . . . 13 ⊢ (0 ∈ (1...(𝑁 − 1)) → 0 ∈ ℕ)
11	9, 10	mto 189	. . . . . . . . . . . 12 ⊢ ¬ 0 ∈ (1...(𝑁 − 1))
12		eleq1 2847	. . . . . . . . . . . 12 ⊢ ((2nd ‘𝑧) = 0 → ((2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ↔ 0 ∈ (1...(𝑁 − 1))))
13	11, 12	mtbiri 319	. . . . . . . . . . 11 ⊢ ((2nd ‘𝑧) = 0 → ¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)))
14	13	necon2ai 2990	. . . . . . . . . 10 ⊢ ((2nd ‘𝑧) ∈ (1...(𝑁 − 1)) → (2nd ‘𝑧) ≠ 0)
15	1	ad2antrr 713	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → 𝑁 ∈ ℕ)
16		fveq2 6496	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑧 → (2nd ‘𝑡) = (2nd ‘𝑧))
17	16	breq2d 4937	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = 𝑧 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑧)))
18	17	ifbid 4366	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑧 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑧), 𝑦, (𝑦 + 1)))
19	18	csbeq1d 3787	. . . . . . . . . . . . . . . . . . . 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 6501	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = 𝑧 → (1st ‘(1st ‘𝑡)) = (1st ‘(1st ‘𝑧)))
21		2fveq3 6501	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑡 = 𝑧 → (2nd ‘(1st ‘𝑡)) = (2nd ‘(1st ‘𝑧)))
22	21	imaeq1d 5766	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑧 → ((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑧)) “ (1...𝑗)))
23	22	xpeq1d 5432	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑧 → (((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st ‘𝑧)) “ (1...𝑗)) × {1}))
24	21	imaeq1d 5766	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑧 → ((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑧)) “ ((𝑗 + 1)...𝑁)))
25	24	xpeq1d 5432	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑧 → (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))
26	23, 25	uneq12d 4023	. . . . . . . . . . . . . . . . . . . . . 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 6992	. . . . . . . . . . . . . . . . . . . . 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 4250	. . . . . . . . . . . . . . . . . . . 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 2808	. . . . . . . . . . . . . . . . . . 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 5019	. . . . . . . . . . . . . . . . . 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 2782	. . . . . . . . . . . . . . . . 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 3593	. . . . . . . . . . . . . . . 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 489	. . . . . . . . . . . . . . 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 714	. . . . . . . . . . . . . 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 3584	. . . . . . . . . . . . . . . . . . . 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 2878	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
37		xp1st 7531	. . . . . . . . . . . . . . . . . . 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 7531	. . . . . . . . . . . . . . . . . 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 8226	. . . . . . . . . . . . . . . . 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 12852	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (0..^𝐾) → 𝑛 ∈ ℤ)
44	43	ssriv 3856	. . . . . . . . . . . . . . . 16 ⊢ (0..^𝐾) ⊆ ℤ
45		fss 6354	. . . . . . . . . . . . . . . 16 ⊢ (((1st ‘(1st ‘𝑧)):(1...𝑁)⟶(0..^𝐾) ∧ (0..^𝐾) ⊆ ℤ) → (1st ‘(1st ‘𝑧)):(1...𝑁)⟶ℤ)
46	42, 44, 45	sylancl 577	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑆 → (1st ‘(1st ‘𝑧)):(1...𝑁)⟶ℤ)
47	46	ad2antlr 714	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → (1st ‘(1st ‘𝑧)):(1...𝑁)⟶ℤ)
48		xp2nd 7532	. . . . . . . . . . . . . . . . 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 6509	. . . . . . . . . . . . . . . . 17 ⊢ (2nd ‘(1st ‘𝑧)) ∈ V
51		f1oeq1 6430	. . . . . . . . . . . . . . . . 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 3576	. . . . . . . . . . . . . . . 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 210	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑆 → (2nd ‘(1st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
54	53	ad2antlr 714	. . . . . . . . . . . . . 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 34363	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd ‘𝑧) − 1))‘𝑛) ≠ ((𝐹‘(2nd ‘𝑧))‘𝑛))
57	1	ad2antrr 713	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)) → 𝑁 ∈ ℕ)
58		fveq2 6496	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 = 𝑇 → (2nd ‘𝑡) = (2nd ‘𝑇))
59	58	breq2d 4937	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 = 𝑇 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑇)))
60	59	ifbid 4366	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑡 = 𝑇 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)))
61	60	csbeq1d 3787	. . . . . . . . . . . . . . . . . . . . . . . 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 6501	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 = 𝑇 → (1st ‘(1st ‘𝑡)) = (1st ‘(1st ‘𝑇)))
63		2fveq3 6501	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑡 = 𝑇 → (2nd ‘(1st ‘𝑡)) = (2nd ‘(1st ‘𝑇)))
64	63	imaeq1d 5766	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑡 = 𝑇 → ((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑇)) “ (1...𝑗)))
65	64	xpeq1d 5432	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 = 𝑇 → (((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}))
66	63	imaeq1d 5766	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑡 = 𝑇 → ((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)))
67	66	xpeq1d 5432	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 = 𝑇 → (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
68	65, 67	uneq12d 4023	. . . . . . . . . . . . . . . . . . . . . . . . . 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 6992	. . . . . . . . . . . . . . . . . . . . . . . . 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 4250	. . . . . . . . . . . . . . . . . . . . . . . 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 2808	. . . . . . . . . . . . . . . . . . . . . . 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 5019	. . . . . . . . . . . . . . . . . . . . . 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 2782	. . . . . . . . . . . . . . . . . . . . 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 3593	. . . . . . . . . . . . . . . . . . . 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 489	. . . . . . . . . . . . . . . . . . 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 713	. . . . . . . . . . . . . . . . 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 3584	. . . . . . . . . . . . . . . . . . . . . . . 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 2878	. . . . . . . . . . . . . . . . . . . . . . 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 7531	. . . . . . . . . . . . . . . . . . . . . 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 7531	. . . . . . . . . . . . . . . . . . . . 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 8226	. . . . . . . . . . . . . . . . . . . 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 6354	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1st ‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾) ∧ (0..^𝐾) ⊆ ℤ) → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶ℤ)
88	86, 44, 87	sylancl 577	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶ℤ)
89	88	ad2antrr 713	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)) → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶ℤ)
90		xp2nd 7532	. . . . . . . . . . . . . . . . . . . 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 6509	. . . . . . . . . . . . . . . . . . . 20 ⊢ (2nd ‘(1st ‘𝑇)) ∈ V
93		f1oeq1 6430	. . . . . . . . . . . . . . . . . . . 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 3576	. . . . . . . . . . . . . . . . . . 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 210	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
96	95	ad2antrr 713	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)) → (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
97		simplr 756	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)) → (2nd ‘𝑧) ∈ (1...(𝑁 − 1)))
98		xp2nd 7532	. . . . . . . . . . . . . . . . . . . 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 473	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → (2nd ‘𝑇) ∈ (0...𝑁))
101		eldifsn 4589	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2nd ‘𝑇) ∈ ((0...𝑁) ∖ {(2nd ‘𝑧)}) ↔ ((2nd ‘𝑇) ∈ (0...𝑁) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)))
102	101	biimpri 220	. . . . . . . . . . . . . . . . . 18 ⊢ (((2nd ‘𝑇) ∈ (0...𝑁) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)) → (2nd ‘𝑇) ∈ ((0...𝑁) ∖ {(2nd ‘𝑧)}))
103	100, 102	sylan 572	. . . . . . . . . . . . . . . . 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 34364	. . . . . . . . . . . . . . . 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 405	. . . . . . . . . . . . . . 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 2979	. . . . . . . . . . . . . 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 702	. . . . . . . . . . . . 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 3020	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → ((2nd ‘𝑇) ≠ 0 ↔ (2nd ‘𝑧) ≠ 0))
110	109	exbiri 798	. . . . . . . . . 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 2977	. . . . . . . 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 7532	. . . . . . . . 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 11455	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℂ)
117		npcan1 10864	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
118	116, 117	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
119		nnuz 12093	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ = (ℤ≥‘1)
120	1, 119	syl6eleq 2870	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘1))
121	118, 120	eqeltrd 2860	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ≥‘1))
122	1	nnzd 11897	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑁 ∈ ℤ)
123		peano2zm 11836	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
124	122, 123	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑁 − 1) ∈ ℤ)
125		uzid 12071	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ≥‘(𝑁 − 1)))
126		peano2uz 12113	. . . . . . . . . . . . . . . . . . 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 2861	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑁 − 1)))
129		fzsplit2 12746	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 − 1) + 1) ∈ (ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘(𝑁 − 1))) → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
130	121, 128, 129	syl2anc 576	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
131	118	oveq1d 6989	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = (𝑁...𝑁))
132		fzsn 12763	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
133	122, 132	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑁...𝑁) = {𝑁})
134	131, 133	eqtrd 2808	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = {𝑁})
135	134	uneq2d 4022	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = ((1...(𝑁 − 1)) ∪ {𝑁}))
136	130, 135	eqtrd 2808	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁}))
137	136	eleq2d 2845	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2nd ‘𝑧) ∈ (1...𝑁) ↔ (2nd ‘𝑧) ∈ ((1...(𝑁 − 1)) ∪ {𝑁})))
138	137	notbid 310	. . . . . . . . . . . . 13 ⊢ (𝜑 → (¬ (2nd ‘𝑧) ∈ (1...𝑁) ↔ ¬ (2nd ‘𝑧) ∈ ((1...(𝑁 − 1)) ∪ {𝑁})))
139		ioran 966	. . . . . . . . . . . . . 14 ⊢ (¬ ((2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∨ (2nd ‘𝑧) = 𝑁) ↔ (¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd ‘𝑧) = 𝑁))
140		elun 4008	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘𝑧) ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ ((2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∨ (2nd ‘𝑧) ∈ {𝑁}))
141		fvex 6509	. . . . . . . . . . . . . . . . 17 ⊢ (2nd ‘𝑧) ∈ V
142	141	elsn 4450	. . . . . . . . . . . . . . . 16 ⊢ ((2nd ‘𝑧) ∈ {𝑁} ↔ (2nd ‘𝑧) = 𝑁)
143	142	orbi2i 896	. . . . . . . . . . . . . . 15 ⊢ (((2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∨ (2nd ‘𝑧) ∈ {𝑁}) ↔ ((2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∨ (2nd ‘𝑧) = 𝑁))
144	140, 143	bitri 267	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝑧) ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ ((2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∨ (2nd ‘𝑧) = 𝑁))
145	139, 144	xchnxbir 325	. . . . . . . . . . . . 13 ⊢ (¬ (2nd ‘𝑧) ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ (¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd ‘𝑧) = 𝑁))
146	138, 145	syl6bb 279	. . . . . . . . . . . 12 ⊢ (𝜑 → (¬ (2nd ‘𝑧) ∈ (1...𝑁) ↔ (¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd ‘𝑧) = 𝑁)))
147	146	anbi2d 619	. . . . . . . . . . 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 11765	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
149		nn0uz 12092	. . . . . . . . . . . . . . . . 17 ⊢ ℕ0 = (ℤ≥‘0)
150	148, 149	syl6eleq 2870	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0))
151		fzpred 12769	. . . . . . . . . . . . . . . 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 3982	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((0...𝑁) ∖ (1...𝑁)) = (({0} ∪ ((0 + 1)...𝑁)) ∖ (1...𝑁)))
154		difun2 4306	. . . . . . . . . . . . . . 15 ⊢ (({0} ∪ (1...𝑁)) ∖ (1...𝑁)) = ({0} ∖ (1...𝑁))
155		0p1e1 11567	. . . . . . . . . . . . . . . . . 18 ⊢ (0 + 1) = 1
156	155	oveq1i 6984	. . . . . . . . . . . . . . . . 17 ⊢ ((0 + 1)...𝑁) = (1...𝑁)
157	156	uneq2i 4019	. . . . . . . . . . . . . . . 16 ⊢ ({0} ∪ ((0 + 1)...𝑁)) = ({0} ∪ (1...𝑁))
158	157	difeq1i 3979	. . . . . . . . . . . . . . 15 ⊢ (({0} ∪ ((0 + 1)...𝑁)) ∖ (1...𝑁)) = (({0} ∪ (1...𝑁)) ∖ (1...𝑁))
159		incom 4060	. . . . . . . . . . . . . . . . 17 ⊢ ({0} ∩ (1...𝑁)) = ((1...𝑁) ∩ {0})
160		elfznn 12750	. . . . . . . . . . . . . . . . . . 19 ⊢ (0 ∈ (1...𝑁) → 0 ∈ ℕ)
161	9, 160	mto 189	. . . . . . . . . . . . . . . . . 18 ⊢ ¬ 0 ∈ (1...𝑁)
162		disjsn 4517	. . . . . . . . . . . . . . . . . 18 ⊢ (((1...𝑁) ∩ {0}) = ∅ ↔ ¬ 0 ∈ (1...𝑁))
163	161, 162	mpbir 223	. . . . . . . . . . . . . . . . 17 ⊢ ((1...𝑁) ∩ {0}) = ∅
164	159, 163	eqtri 2796	. . . . . . . . . . . . . . . 16 ⊢ ({0} ∩ (1...𝑁)) = ∅
165		disj3 4280	. . . . . . . . . . . . . . . 16 ⊢ (({0} ∩ (1...𝑁)) = ∅ ↔ {0} = ({0} ∖ (1...𝑁)))
166	164, 165	mpbi 222	. . . . . . . . . . . . . . 15 ⊢ {0} = ({0} ∖ (1...𝑁))
167	154, 158, 166	3eqtr4i 2806	. . . . . . . . . . . . . 14 ⊢ (({0} ∪ ((0 + 1)...𝑁)) ∖ (1...𝑁)) = {0}
168	153, 167	syl6eq 2824	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((0...𝑁) ∖ (1...𝑁)) = {0})
169	168	eleq2d 2845	. . . . . . . . . . . 12 ⊢ (𝜑 → ((2nd ‘𝑧) ∈ ((0...𝑁) ∖ (1...𝑁)) ↔ (2nd ‘𝑧) ∈ {0}))
170		eldif 3833	. . . . . . . . . . . 12 ⊢ ((2nd ‘𝑧) ∈ ((0...𝑁) ∖ (1...𝑁)) ↔ ((2nd ‘𝑧) ∈ (0...𝑁) ∧ ¬ (2nd ‘𝑧) ∈ (1...𝑁)))
171	141	elsn 4450	. . . . . . . . . . . 12 ⊢ ((2nd ‘𝑧) ∈ {0} ↔ (2nd ‘𝑧) = 0)
172	169, 170, 171	3bitr3g 305	. . . . . . . . . . 11 ⊢ (𝜑 → (((2nd ‘𝑧) ∈ (0...𝑁) ∧ ¬ (2nd ‘𝑧) ∈ (1...𝑁)) ↔ (2nd ‘𝑧) = 0))
173	147, 172	bitr3d 273	. . . . . . . . . 10 ⊢ (𝜑 → (((2nd ‘𝑧) ∈ (0...𝑁) ∧ (¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd ‘𝑧) = 𝑁)) ↔ (2nd ‘𝑧) = 0))
174	173	biimpd 221	. . . . . . . . 9 ⊢ (𝜑 → (((2nd ‘𝑧) ∈ (0...𝑁) ∧ (¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd ‘𝑧) = 𝑁)) → (2nd ‘𝑧) = 0))
175	174	expdimp 445	. . . . . . . 8 ⊢ ((𝜑 ∧ (2nd ‘𝑧) ∈ (0...𝑁)) → ((¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd ‘𝑧) = 𝑁) → (2nd ‘𝑧) = 0))
176	115, 175	sylan2 583	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ((¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd ‘𝑧) = 𝑁) → (2nd ‘𝑧) = 0))
177	113, 176	mpand 682	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (¬ (2nd ‘𝑧) = 𝑁 → (2nd ‘𝑧) = 0))
178	1, 2, 3	poimirlem13 34375	. . . . . . . . . 10 ⊢ (𝜑 → ∃*𝑧 ∈ 𝑆 (2nd ‘𝑧) = 0)
179		fveqeq2 6505	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑠 → ((2nd ‘𝑧) = 0 ↔ (2nd ‘𝑠) = 0))
180	179	rmo4 3627	. . . . . . . . . 10 ⊢ (∃*𝑧 ∈ 𝑆 (2nd ‘𝑧) = 0 ↔ ∀𝑧 ∈ 𝑆 ∀𝑠 ∈ 𝑆 (((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑠) = 0) → 𝑧 = 𝑠))
181	178, 180	sylib 210	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 ∀𝑠 ∈ 𝑆 (((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑠) = 0) → 𝑧 = 𝑠))
182	181	r19.21bi 3152	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ∀𝑠 ∈ 𝑆 (((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑠) = 0) → 𝑧 = 𝑠))
183	4	adantr 473	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝑇 ∈ 𝑆)
184		fveqeq2 6505	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑇 → ((2nd ‘𝑠) = 0 ↔ (2nd ‘𝑇) = 0))
185	184	anbi2d 619	. . . . . . . . . 10 ⊢ (𝑠 = 𝑇 → (((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑠) = 0) ↔ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑇) = 0)))
186		eqeq2 2783	. . . . . . . . . 10 ⊢ (𝑠 = 𝑇 → (𝑧 = 𝑠 ↔ 𝑧 = 𝑇))
187	185, 186	imbi12d 337	. . . . . . . . 9 ⊢ (𝑠 = 𝑇 → ((((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑠) = 0) → 𝑧 = 𝑠) ↔ (((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑇) = 0) → 𝑧 = 𝑇)))
188	187	rspccv 3526	. . . . . . . 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 681	. . . . . 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 2978	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝑧 ≠ 𝑇 → (2nd ‘𝑧) = 𝑁))
193	192	ralrimiva 3126	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 (𝑧 ≠ 𝑇 → (2nd ‘𝑧) = 𝑁))
194	1, 2, 3	poimirlem14 34376	. . 3 ⊢ (𝜑 → ∃*𝑧 ∈ 𝑆 (2nd ‘𝑧) = 𝑁)
195		rmoim 3637	. . 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 3364	. 2 ⊢ (∃!𝑧 ∈ 𝑆 𝑧 ≠ 𝑇 ↔ (∃𝑧 ∈ 𝑆 𝑧 ≠ 𝑇 ∧ ∃*𝑧 ∈ 𝑆 𝑧 ≠ 𝑇))
198	7, 196, 197	sylanbrc 575	1 ⊢ (𝜑 → ∃!𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 387   ∨ wo 833   = wceq 1507   ∈ wcel 2050  {cab 2752   ≠ wne 2961  ∀wral 3082  ∃wrex 3083  ∃!wreu 3084  ∃*wrmo 3085  {crab 3086  ⦋csb 3780   ∖ cdif 3820   ∪ cun 3821   ∩ cin 3822   ⊆ wss 3823  ∅c0 4172  ifcif 4344  {csn 4435   class class class wbr 4925   ↦ cmpt 5004   × cxp 5401  ran crn 5404   “ cima 5406  ⟶wf 6181  –1-1-onto→wf1o 6184  ‘cfv 6185  (class class class)co 6974   ∘_𝑓 cof 7223  1^st c1st 7497  2^nd c2nd 7498   ↑_𝑚 cmap 8204  ℂcc 10331  0cc0 10333  1c1 10334   + caddc 10336   < clt 10472   − cmin 10668  ℕcn 11437  ℕ₀cn0 11705  ℤcz 11791  ℤ_≥cuz 12056  ...cfz 12706  ..^cfzo 12847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277  ax-cnex 10389  ax-resscn 10390  ax-1cn 10391  ax-icn 10392  ax-addcl 10393  ax-addrcl 10394  ax-mulcl 10395  ax-mulrcl 10396  ax-mulcom 10397  ax-addass 10398  ax-mulass 10399  ax-distr 10400  ax-i2m1 10401  ax-1ne0 10402  ax-1rid 10403  ax-rnegex 10404  ax-rrecex 10405  ax-cnre 10406  ax-pre-lttri 10407  ax-pre-lttrn 10408  ax-pre-ltadd 10409  ax-pre-mulgt0 10410

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-pss 3839  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-pred 5983  df-ord 6029  df-on 6030  df-lim 6031  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-riota 6935  df-ov 6977  df-oprab 6978  df-mpo 6979  df-of 7225  df-om 7395  df-1st 7499  df-2nd 7500  df-wrecs 7748  df-recs 7810  df-rdg 7848  df-1o 7903  df-er 8087  df-map 8206  df-en 8305  df-dom 8306  df-sdom 8307  df-fin 8308  df-pnf 10474  df-mnf 10475  df-xr 10476  df-ltxr 10477  df-le 10478  df-sub 10670  df-neg 10671  df-nn 11438  df-n0 11706  df-z 11792  df-uz 12057  df-fz 12707  df-fzo 12848

This theorem is referenced by:  poimirlem22  34384