Theorem poimirlem18 32597

Description: Lemma for poimir 32612 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 32596	. 2 ⊢ (𝜑 → ∃𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)
8	6	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (2nd ‘𝑇) = 0)
9		0nnn 10929	. . . . . . . . . . . . 13 ⊢ ¬ 0 ∈ ℕ
10		elfznn 12241	. . . . . . . . . . . . 13 ⊢ (0 ∈ (1...(𝑁 − 1)) → 0 ∈ ℕ)
11	9, 10	mto 187	. . . . . . . . . . . 12 ⊢ ¬ 0 ∈ (1...(𝑁 − 1))
12		eleq1 2676	. . . . . . . . . . . 12 ⊢ ((2nd ‘𝑧) = 0 → ((2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ↔ 0 ∈ (1...(𝑁 − 1))))
13	11, 12	mtbiri 316	. . . . . . . . . . 11 ⊢ ((2nd ‘𝑧) = 0 → ¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)))
14	13	necon2ai 2811	. . . . . . . . . 10 ⊢ ((2nd ‘𝑧) ∈ (1...(𝑁 − 1)) → (2nd ‘𝑧) ≠ 0)
15	1	ad2antrr 758	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → 𝑁 ∈ ℕ)
16		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑧 → (2nd ‘𝑡) = (2nd ‘𝑧))
17	16	breq2d 4595	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = 𝑧 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑧)))
18	17	ifbid 4058	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑧 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑧), 𝑦, (𝑦 + 1)))
19	18	csbeq1d 3506	. . . . . . . . . . . . . . . . . . . 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		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑧 → (1st ‘𝑡) = (1st ‘𝑧))
21	20	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = 𝑧 → (1st ‘(1st ‘𝑡)) = (1st ‘(1st ‘𝑧)))
22	20	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑡 = 𝑧 → (2nd ‘(1st ‘𝑡)) = (2nd ‘(1st ‘𝑧)))
23	22	imaeq1d 5384	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑧 → ((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑧)) “ (1...𝑗)))
24	23	xpeq1d 5062	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑧 → (((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st ‘𝑧)) “ (1...𝑗)) × {1}))
25	22	imaeq1d 5384	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑧 → ((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑧)) “ ((𝑗 + 1)...𝑁)))
26	25	xpeq1d 5062	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑧 → (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))
27	24, 26	uneq12d 3730	. . . . . . . . . . . . . . . . . . . . . 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})))
28	21, 27	oveq12d 6567	. . . . . . . . . . . . . . . . . . . . 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}))))
29	28	csbeq2dv 3944	. . . . . . . . . . . . . . . . . . . 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}))))
30	19, 29	eqtrd 2644	. . . . . . . . . . . . . . . . . . 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}))))
31	30	mpteq2dv 4673	. . . . . . . . . . . . . . . . . 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})))))
32	31	eqeq2d 2620	. . . . . . . . . . . . . . . . 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}))))))
33	32, 2	elrab2 3333	. . . . . . . . . . . . . . . 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}))))))
34	33	simprbi 479	. . . . . . . . . . . . . . 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})))))
35	34	ad2antlr 759	. . . . . . . . . . . . . 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})))))
36		elrabi 3328	. . . . . . . . . . . . . . . . . . . 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)))
37	36, 2	eleq2s 2706	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
38		xp1st 7089	. . . . . . . . . . . . . . . . . . 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)}))
39	37, 38	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ 𝑆 → (1st ‘𝑧) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
40		xp1st 7089	. . . . . . . . . . . . . . . . . 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)))
41	39, 40	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ 𝑆 → (1st ‘(1st ‘𝑧)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
42		elmapi 7765	. . . . . . . . . . . . . . . . 17 ⊢ ((1st ‘(1st ‘𝑧)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st ‘(1st ‘𝑧)):(1...𝑁)⟶(0..^𝐾))
43	41, 42	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑆 → (1st ‘(1st ‘𝑧)):(1...𝑁)⟶(0..^𝐾))
44		elfzoelz 12339	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (0..^𝐾) → 𝑛 ∈ ℤ)
45	44	ssriv 3572	. . . . . . . . . . . . . . . 16 ⊢ (0..^𝐾) ⊆ ℤ
46		fss 5969	. . . . . . . . . . . . . . . 16 ⊢ (((1st ‘(1st ‘𝑧)):(1...𝑁)⟶(0..^𝐾) ∧ (0..^𝐾) ⊆ ℤ) → (1st ‘(1st ‘𝑧)):(1...𝑁)⟶ℤ)
47	43, 45, 46	sylancl 693	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑆 → (1st ‘(1st ‘𝑧)):(1...𝑁)⟶ℤ)
48	47	ad2antlr 759	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → (1st ‘(1st ‘𝑧)):(1...𝑁)⟶ℤ)
49		xp2nd 7090	. . . . . . . . . . . . . . . . 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)})
50	39, 49	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑆 → (2nd ‘(1st ‘𝑧)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
51		fvex 6113	. . . . . . . . . . . . . . . . 17 ⊢ (2nd ‘(1st ‘𝑧)) ∈ V
52		f1oeq1 6040	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (2nd ‘(1st ‘𝑧)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁)))
53	51, 52	elab 3319	. . . . . . . . . . . . . . . 16 ⊢ ((2nd ‘(1st ‘𝑧)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
54	50, 53	sylib 207	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑆 → (2nd ‘(1st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
55	54	ad2antlr 759	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → (2nd ‘(1st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
56		simpr 476	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → (2nd ‘𝑧) ∈ (1...(𝑁 − 1)))
57	15, 35, 48, 55, 56	poimirlem1 32580	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd ‘𝑧) − 1))‘𝑛) ≠ ((𝐹‘(2nd ‘𝑧))‘𝑛))
58	1	ad2antrr 758	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)) → 𝑁 ∈ ℕ)
59		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 = 𝑇 → (2nd ‘𝑡) = (2nd ‘𝑇))
60	59	breq2d 4595	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 = 𝑇 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑇)))
61	60	ifbid 4058	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑡 = 𝑇 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)))
62	61	csbeq1d 3506	. . . . . . . . . . . . . . . . . . . . . . . 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}))))
63		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 = 𝑇 → (1st ‘𝑡) = (1st ‘𝑇))
64	63	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 = 𝑇 → (1st ‘(1st ‘𝑡)) = (1st ‘(1st ‘𝑇)))
65	63	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑡 = 𝑇 → (2nd ‘(1st ‘𝑡)) = (2nd ‘(1st ‘𝑇)))
66	65	imaeq1d 5384	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑡 = 𝑇 → ((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑇)) “ (1...𝑗)))
67	66	xpeq1d 5062	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 = 𝑇 → (((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}))
68	65	imaeq1d 5384	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑡 = 𝑇 → ((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)))
69	68	xpeq1d 5062	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 = 𝑇 → (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
70	67, 69	uneq12d 3730	. . . . . . . . . . . . . . . . . . . . . . . . . 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})))
71	64, 70	oveq12d 6567	. . . . . . . . . . . . . . . . . . . . . . . . 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}))))
72	71	csbeq2dv 3944	. . . . . . . . . . . . . . . . . . . . . . . 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}))))
73	62, 72	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . 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}))))
74	73	mpteq2dv 4673	. . . . . . . . . . . . . . . . . . . . . 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})))))
75	74	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . 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}))))))
76	75, 2	elrab2 3333	. . . . . . . . . . . . . . . . . . . 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}))))))
77	76	simprbi 479	. . . . . . . . . . . . . . . . . . 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})))))
78	4, 77	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})))))
79	78	ad2antrr 758	. . . . . . . . . . . . . . . . 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})))))
80		elrabi 3328	. . . . . . . . . . . . . . . . . . . . . . . 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)))
81	80, 2	eleq2s 2706	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑇 ∈ 𝑆 → 𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
82	4, 81	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
83		xp1st 7089	. . . . . . . . . . . . . . . . . . . . . 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)}))
84	82, 83	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (1st ‘𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
85		xp1st 7089	. . . . . . . . . . . . . . . . . . . . 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)))
86	84, 85	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
87		elmapi 7765	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
88	86, 87	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
89		fss 5969	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1st ‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾) ∧ (0..^𝐾) ⊆ ℤ) → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶ℤ)
90	88, 45, 89	sylancl 693	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶ℤ)
91	90	ad2antrr 758	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)) → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶ℤ)
92		xp2nd 7090	. . . . . . . . . . . . . . . . . . . 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)})
93	84, 92	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (2nd ‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
94		fvex 6113	. . . . . . . . . . . . . . . . . . . 20 ⊢ (2nd ‘(1st ‘𝑇)) ∈ V
95		f1oeq1 6040	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = (2nd ‘(1st ‘𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
96	94, 95	elab 3319	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2nd ‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
97	93, 96	sylib 207	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
98	97	ad2antrr 758	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)) → (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
99		simplr 788	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)) → (2nd ‘𝑧) ∈ (1...(𝑁 − 1)))
100		xp2nd 7090	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd ‘𝑇) ∈ (0...𝑁))
101	82, 100	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (2nd ‘𝑇) ∈ (0...𝑁))
102	101	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → (2nd ‘𝑇) ∈ (0...𝑁))
103		eldifsn 4260	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2nd ‘𝑇) ∈ ((0...𝑁) ∖ {(2nd ‘𝑧)}) ↔ ((2nd ‘𝑇) ∈ (0...𝑁) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)))
104	103	biimpri 217	. . . . . . . . . . . . . . . . . 18 ⊢ (((2nd ‘𝑇) ∈ (0...𝑁) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)) → (2nd ‘𝑇) ∈ ((0...𝑁) ∖ {(2nd ‘𝑧)}))
105	102, 104	sylan 487	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)) → (2nd ‘𝑇) ∈ ((0...𝑁) ∖ {(2nd ‘𝑧)}))
106	58, 79, 91, 98, 99, 105	poimirlem2 32581	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)) → ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd ‘𝑧) − 1))‘𝑛) ≠ ((𝐹‘(2nd ‘𝑧))‘𝑛))
107	106	ex 449	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → ((2nd ‘𝑇) ≠ (2nd ‘𝑧) → ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd ‘𝑧) − 1))‘𝑛) ≠ ((𝐹‘(2nd ‘𝑧))‘𝑛)))
108	107	necon1bd 2800	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → (¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd ‘𝑧) − 1))‘𝑛) ≠ ((𝐹‘(2nd ‘𝑧))‘𝑛) → (2nd ‘𝑇) = (2nd ‘𝑧)))
109	108	adantlr 747	. . . . . . . . . . . . 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)))
110	57, 109	mpd 15	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → (2nd ‘𝑇) = (2nd ‘𝑧))
111	110	neeq1d 2841	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → ((2nd ‘𝑇) ≠ 0 ↔ (2nd ‘𝑧) ≠ 0))
112	111	exbiri 650	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ((2nd ‘𝑧) ∈ (1...(𝑁 − 1)) → ((2nd ‘𝑧) ≠ 0 → (2nd ‘𝑇) ≠ 0)))
113	14, 112	mpdi 44	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ((2nd ‘𝑧) ∈ (1...(𝑁 − 1)) → (2nd ‘𝑇) ≠ 0))
114	113	necon2bd 2798	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ((2nd ‘𝑇) = 0 → ¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))))
115	8, 114	mpd 15	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)))
116		xp2nd 7090	. . . . . . . . 9 ⊢ (𝑧 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd ‘𝑧) ∈ (0...𝑁))
117	37, 116	syl 17	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑆 → (2nd ‘𝑧) ∈ (0...𝑁))
118	1	nncnd 10913	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℂ)
119		npcan1 10334	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
120	118, 119	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
121		nnuz 11599	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ = (ℤ≥‘1)
122	1, 121	syl6eleq 2698	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘1))
123	120, 122	eqeltrd 2688	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ≥‘1))
124	1	nnzd 11357	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑁 ∈ ℤ)
125		peano2zm 11297	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
126	124, 125	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑁 − 1) ∈ ℤ)
127		uzid 11578	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ≥‘(𝑁 − 1)))
128		peano2uz 11617	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 − 1) ∈ (ℤ≥‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ≥‘(𝑁 − 1)))
129	126, 127, 128	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ≥‘(𝑁 − 1)))
130	120, 129	eqeltrrd 2689	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑁 − 1)))
131		fzsplit2 12237	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 − 1) + 1) ∈ (ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘(𝑁 − 1))) → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
132	123, 130, 131	syl2anc 691	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
133	120	oveq1d 6564	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = (𝑁...𝑁))
134		fzsn 12254	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
135	124, 134	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑁...𝑁) = {𝑁})
136	133, 135	eqtrd 2644	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = {𝑁})
137	136	uneq2d 3729	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = ((1...(𝑁 − 1)) ∪ {𝑁}))
138	132, 137	eqtrd 2644	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁}))
139	138	eleq2d 2673	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2nd ‘𝑧) ∈ (1...𝑁) ↔ (2nd ‘𝑧) ∈ ((1...(𝑁 − 1)) ∪ {𝑁})))
140	139	notbid 307	. . . . . . . . . . . . 13 ⊢ (𝜑 → (¬ (2nd ‘𝑧) ∈ (1...𝑁) ↔ ¬ (2nd ‘𝑧) ∈ ((1...(𝑁 − 1)) ∪ {𝑁})))
141		ioran 510	. . . . . . . . . . . . . 14 ⊢ (¬ ((2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∨ (2nd ‘𝑧) = 𝑁) ↔ (¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd ‘𝑧) = 𝑁))
142		elun 3715	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘𝑧) ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ ((2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∨ (2nd ‘𝑧) ∈ {𝑁}))
143		fvex 6113	. . . . . . . . . . . . . . . . 17 ⊢ (2nd ‘𝑧) ∈ V
144	143	elsn 4140	. . . . . . . . . . . . . . . 16 ⊢ ((2nd ‘𝑧) ∈ {𝑁} ↔ (2nd ‘𝑧) = 𝑁)
145	144	orbi2i 540	. . . . . . . . . . . . . . 15 ⊢ (((2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∨ (2nd ‘𝑧) ∈ {𝑁}) ↔ ((2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∨ (2nd ‘𝑧) = 𝑁))
146	142, 145	bitri 263	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝑧) ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ ((2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∨ (2nd ‘𝑧) = 𝑁))
147	141, 146	xchnxbir 322	. . . . . . . . . . . . 13 ⊢ (¬ (2nd ‘𝑧) ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ (¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd ‘𝑧) = 𝑁))
148	140, 147	syl6bb 275	. . . . . . . . . . . 12 ⊢ (𝜑 → (¬ (2nd ‘𝑧) ∈ (1...𝑁) ↔ (¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd ‘𝑧) = 𝑁)))
149	148	anbi2d 736	. . . . . . . . . . 11 ⊢ (𝜑 → (((2nd ‘𝑧) ∈ (0...𝑁) ∧ ¬ (2nd ‘𝑧) ∈ (1...𝑁)) ↔ ((2nd ‘𝑧) ∈ (0...𝑁) ∧ (¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd ‘𝑧) = 𝑁))))
150	1	nnnn0d 11228	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
151		nn0uz 11598	. . . . . . . . . . . . . . . . 17 ⊢ ℕ0 = (ℤ≥‘0)
152	150, 151	syl6eleq 2698	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0))
153		fzpred 12259	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ≥‘0) → (0...𝑁) = ({0} ∪ ((0 + 1)...𝑁)))
154	152, 153	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0...𝑁) = ({0} ∪ ((0 + 1)...𝑁)))
155	154	difeq1d 3689	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((0...𝑁) ∖ (1...𝑁)) = (({0} ∪ ((0 + 1)...𝑁)) ∖ (1...𝑁)))
156		difun2 4000	. . . . . . . . . . . . . . 15 ⊢ (({0} ∪ (1...𝑁)) ∖ (1...𝑁)) = ({0} ∖ (1...𝑁))
157		0p1e1 11009	. . . . . . . . . . . . . . . . . 18 ⊢ (0 + 1) = 1
158	157	oveq1i 6559	. . . . . . . . . . . . . . . . 17 ⊢ ((0 + 1)...𝑁) = (1...𝑁)
159	158	uneq2i 3726	. . . . . . . . . . . . . . . 16 ⊢ ({0} ∪ ((0 + 1)...𝑁)) = ({0} ∪ (1...𝑁))
160	159	difeq1i 3686	. . . . . . . . . . . . . . 15 ⊢ (({0} ∪ ((0 + 1)...𝑁)) ∖ (1...𝑁)) = (({0} ∪ (1...𝑁)) ∖ (1...𝑁))
161		incom 3767	. . . . . . . . . . . . . . . . 17 ⊢ ({0} ∩ (1...𝑁)) = ((1...𝑁) ∩ {0})
162		elfznn 12241	. . . . . . . . . . . . . . . . . . 19 ⊢ (0 ∈ (1...𝑁) → 0 ∈ ℕ)
163	9, 162	mto 187	. . . . . . . . . . . . . . . . . 18 ⊢ ¬ 0 ∈ (1...𝑁)
164		disjsn 4192	. . . . . . . . . . . . . . . . . 18 ⊢ (((1...𝑁) ∩ {0}) = ∅ ↔ ¬ 0 ∈ (1...𝑁))
165	163, 164	mpbir 220	. . . . . . . . . . . . . . . . 17 ⊢ ((1...𝑁) ∩ {0}) = ∅
166	161, 165	eqtri 2632	. . . . . . . . . . . . . . . 16 ⊢ ({0} ∩ (1...𝑁)) = ∅
167		disj3 3973	. . . . . . . . . . . . . . . 16 ⊢ (({0} ∩ (1...𝑁)) = ∅ ↔ {0} = ({0} ∖ (1...𝑁)))
168	166, 167	mpbi 219	. . . . . . . . . . . . . . 15 ⊢ {0} = ({0} ∖ (1...𝑁))
169	156, 160, 168	3eqtr4i 2642	. . . . . . . . . . . . . 14 ⊢ (({0} ∪ ((0 + 1)...𝑁)) ∖ (1...𝑁)) = {0}
170	155, 169	syl6eq 2660	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((0...𝑁) ∖ (1...𝑁)) = {0})
171	170	eleq2d 2673	. . . . . . . . . . . 12 ⊢ (𝜑 → ((2nd ‘𝑧) ∈ ((0...𝑁) ∖ (1...𝑁)) ↔ (2nd ‘𝑧) ∈ {0}))
172		eldif 3550	. . . . . . . . . . . 12 ⊢ ((2nd ‘𝑧) ∈ ((0...𝑁) ∖ (1...𝑁)) ↔ ((2nd ‘𝑧) ∈ (0...𝑁) ∧ ¬ (2nd ‘𝑧) ∈ (1...𝑁)))
173	143	elsn 4140	. . . . . . . . . . . 12 ⊢ ((2nd ‘𝑧) ∈ {0} ↔ (2nd ‘𝑧) = 0)
174	171, 172, 173	3bitr3g 301	. . . . . . . . . . 11 ⊢ (𝜑 → (((2nd ‘𝑧) ∈ (0...𝑁) ∧ ¬ (2nd ‘𝑧) ∈ (1...𝑁)) ↔ (2nd ‘𝑧) = 0))
175	149, 174	bitr3d 269	. . . . . . . . . 10 ⊢ (𝜑 → (((2nd ‘𝑧) ∈ (0...𝑁) ∧ (¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd ‘𝑧) = 𝑁)) ↔ (2nd ‘𝑧) = 0))
176	175	biimpd 218	. . . . . . . . 9 ⊢ (𝜑 → (((2nd ‘𝑧) ∈ (0...𝑁) ∧ (¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd ‘𝑧) = 𝑁)) → (2nd ‘𝑧) = 0))
177	176	expdimp 452	. . . . . . . 8 ⊢ ((𝜑 ∧ (2nd ‘𝑧) ∈ (0...𝑁)) → ((¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd ‘𝑧) = 𝑁) → (2nd ‘𝑧) = 0))
178	117, 177	sylan2 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ((¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd ‘𝑧) = 𝑁) → (2nd ‘𝑧) = 0))
179	115, 178	mpand 707	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (¬ (2nd ‘𝑧) = 𝑁 → (2nd ‘𝑧) = 0))
180	1, 2, 3	poimirlem13 32592	. . . . . . . . . 10 ⊢ (𝜑 → ∃*𝑧 ∈ 𝑆 (2nd ‘𝑧) = 0)
181		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑠 → (2nd ‘𝑧) = (2nd ‘𝑠))
182	181	eqeq1d 2612	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑠 → ((2nd ‘𝑧) = 0 ↔ (2nd ‘𝑠) = 0))
183	182	rmo4 3366	. . . . . . . . . 10 ⊢ (∃*𝑧 ∈ 𝑆 (2nd ‘𝑧) = 0 ↔ ∀𝑧 ∈ 𝑆 ∀𝑠 ∈ 𝑆 (((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑠) = 0) → 𝑧 = 𝑠))
184	180, 183	sylib 207	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 ∀𝑠 ∈ 𝑆 (((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑠) = 0) → 𝑧 = 𝑠))
185	184	r19.21bi 2916	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ∀𝑠 ∈ 𝑆 (((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑠) = 0) → 𝑧 = 𝑠))
186	4	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝑇 ∈ 𝑆)
187		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑇 → (2nd ‘𝑠) = (2nd ‘𝑇))
188	187	eqeq1d 2612	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑇 → ((2nd ‘𝑠) = 0 ↔ (2nd ‘𝑇) = 0))
189	188	anbi2d 736	. . . . . . . . . 10 ⊢ (𝑠 = 𝑇 → (((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑠) = 0) ↔ ((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑇) = 0)))
190		eqeq2 2621	. . . . . . . . . 10 ⊢ (𝑠 = 𝑇 → (𝑧 = 𝑠 ↔ 𝑧 = 𝑇))
191	189, 190	imbi12d 333	. . . . . . . . 9 ⊢ (𝑠 = 𝑇 → ((((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑠) = 0) → 𝑧 = 𝑠) ↔ (((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑇) = 0) → 𝑧 = 𝑇)))
192	191	rspccv 3279	. . . . . . . 8 ⊢ (∀𝑠 ∈ 𝑆 (((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑠) = 0) → 𝑧 = 𝑠) → (𝑇 ∈ 𝑆 → (((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑇) = 0) → 𝑧 = 𝑇)))
193	185, 186, 192	sylc 63	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (((2nd ‘𝑧) = 0 ∧ (2nd ‘𝑇) = 0) → 𝑧 = 𝑇))
194	8, 193	mpan2d 706	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ((2nd ‘𝑧) = 0 → 𝑧 = 𝑇))
195	179, 194	syld 46	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (¬ (2nd ‘𝑧) = 𝑁 → 𝑧 = 𝑇))
196	195	necon1ad 2799	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝑧 ≠ 𝑇 → (2nd ‘𝑧) = 𝑁))
197	196	ralrimiva 2949	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 (𝑧 ≠ 𝑇 → (2nd ‘𝑧) = 𝑁))
198	1, 2, 3	poimirlem14 32593	. . 3 ⊢ (𝜑 → ∃*𝑧 ∈ 𝑆 (2nd ‘𝑧) = 𝑁)
199		rmoim 3374	. . 3 ⊢ (∀𝑧 ∈ 𝑆 (𝑧 ≠ 𝑇 → (2nd ‘𝑧) = 𝑁) → (∃*𝑧 ∈ 𝑆 (2nd ‘𝑧) = 𝑁 → ∃*𝑧 ∈ 𝑆 𝑧 ≠ 𝑇))
200	197, 198, 199	sylc 63	. 2 ⊢ (𝜑 → ∃*𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)
201		reu5 3136	. 2 ⊢ (∃!𝑧 ∈ 𝑆 𝑧 ≠ 𝑇 ↔ (∃𝑧 ∈ 𝑆 𝑧 ≠ 𝑇 ∧ ∃*𝑧 ∈ 𝑆 𝑧 ≠ 𝑇))
202	7, 200, 201	sylanbrc 695	1 ⊢ (𝜑 → ∃!𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898  ∃*wrmo 2899  {crab 2900  ⦋csb 3499   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  ifcif 4036  {csn 4125   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  ran crn 5039   “ cima 5041  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ∘_𝑓 cof 6793  1^st c1st 7057  2^nd c2nd 7058   ↑_𝑚 cmap 7744  ℂcc 9813  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953   − cmin 10145  ℕcn 10897  ℕ₀cn0 11169  ℤcz 11254  ℤ_≥cuz 11563  ...cfz 12197  ..^cfzo 12334

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335

This theorem is referenced by:  poimirlem22  32601