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Theorem poimirlem14 37621
Description: Lemma for poimir 37640- for at most one simplex associated with a shared face is the opposite vertex last on the walk. (Contributed by Brendan Leahy, 21-Aug-2020.)
Hypotheses
Ref Expression
poimir.0 (𝜑𝑁 ∈ ℕ)
poimirlem22.s 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
poimirlem22.1 (𝜑𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))
Assertion
Ref Expression
poimirlem14 (𝜑 → ∃*𝑧𝑆 (2nd𝑧) = 𝑁)
Distinct variable groups:   𝑓,𝑗,𝑡,𝑦,𝑧   𝜑,𝑗,𝑦   𝑗,𝐹,𝑦   𝑗,𝑁,𝑦   𝜑,𝑡   𝑓,𝐾,𝑗,𝑡   𝑓,𝑁,𝑡   𝜑,𝑧   𝑓,𝐹,𝑡,𝑧   𝑧,𝐾   𝑧,𝑁   𝑆,𝑗,𝑡,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑓)   𝑆(𝑓)   𝐾(𝑦)

Proof of Theorem poimirlem14
Dummy variables 𝑘 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 poimir.0 . . . . . . . . 9 (𝜑𝑁 ∈ ℕ)
21ad2antrr 726 . . . . . . . 8 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → 𝑁 ∈ ℕ)
3 poimirlem22.s . . . . . . . 8 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
4 simplrl 777 . . . . . . . 8 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → 𝑧𝑆)
51nngt0d 12313 . . . . . . . . . 10 (𝜑 → 0 < 𝑁)
6 breq2 5152 . . . . . . . . . . 11 ((2nd𝑧) = 𝑁 → (0 < (2nd𝑧) ↔ 0 < 𝑁))
76biimparc 479 . . . . . . . . . 10 ((0 < 𝑁 ∧ (2nd𝑧) = 𝑁) → 0 < (2nd𝑧))
85, 7sylan 580 . . . . . . . . 9 ((𝜑 ∧ (2nd𝑧) = 𝑁) → 0 < (2nd𝑧))
98ad2ant2r 747 . . . . . . . 8 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → 0 < (2nd𝑧))
102, 3, 4, 9poimirlem5 37612 . . . . . . 7 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → (𝐹‘0) = (1st ‘(1st𝑧)))
11 simplrr 778 . . . . . . . 8 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → 𝑘𝑆)
12 breq2 5152 . . . . . . . . . . 11 ((2nd𝑘) = 𝑁 → (0 < (2nd𝑘) ↔ 0 < 𝑁))
1312biimparc 479 . . . . . . . . . 10 ((0 < 𝑁 ∧ (2nd𝑘) = 𝑁) → 0 < (2nd𝑘))
145, 13sylan 580 . . . . . . . . 9 ((𝜑 ∧ (2nd𝑘) = 𝑁) → 0 < (2nd𝑘))
1514ad2ant2rl 749 . . . . . . . 8 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → 0 < (2nd𝑘))
162, 3, 11, 15poimirlem5 37612 . . . . . . 7 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → (𝐹‘0) = (1st ‘(1st𝑘)))
1710, 16eqtr3d 2777 . . . . . 6 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → (1st ‘(1st𝑧)) = (1st ‘(1st𝑘)))
18 elrabi 3690 . . . . . . . . . . . . 13 (𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
1918, 3eleq2s 2857 . . . . . . . . . . . 12 (𝑧𝑆𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
20 xp1st 8045 . . . . . . . . . . . 12 (𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st𝑧) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
21 xp2nd 8046 . . . . . . . . . . . 12 ((1st𝑧) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘(1st𝑧)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
2219, 20, 213syl 18 . . . . . . . . . . 11 (𝑧𝑆 → (2nd ‘(1st𝑧)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
23 fvex 6920 . . . . . . . . . . . 12 (2nd ‘(1st𝑧)) ∈ V
24 f1oeq1 6837 . . . . . . . . . . . 12 (𝑓 = (2nd ‘(1st𝑧)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁)))
2523, 24elab 3681 . . . . . . . . . . 11 ((2nd ‘(1st𝑧)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
2622, 25sylib 218 . . . . . . . . . 10 (𝑧𝑆 → (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
27 f1ofn 6850 . . . . . . . . . 10 ((2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑧)) Fn (1...𝑁))
2826, 27syl 17 . . . . . . . . 9 (𝑧𝑆 → (2nd ‘(1st𝑧)) Fn (1...𝑁))
2928adantr 480 . . . . . . . 8 ((𝑧𝑆𝑘𝑆) → (2nd ‘(1st𝑧)) Fn (1...𝑁))
3029ad2antlr 727 . . . . . . 7 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → (2nd ‘(1st𝑧)) Fn (1...𝑁))
31 elrabi 3690 . . . . . . . . . . . . 13 (𝑘 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 𝑘 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
3231, 3eleq2s 2857 . . . . . . . . . . . 12 (𝑘𝑆𝑘 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
33 xp1st 8045 . . . . . . . . . . . 12 (𝑘 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st𝑘) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
34 xp2nd 8046 . . . . . . . . . . . 12 ((1st𝑘) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘(1st𝑘)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
3532, 33, 343syl 18 . . . . . . . . . . 11 (𝑘𝑆 → (2nd ‘(1st𝑘)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
36 fvex 6920 . . . . . . . . . . . 12 (2nd ‘(1st𝑘)) ∈ V
37 f1oeq1 6837 . . . . . . . . . . . 12 (𝑓 = (2nd ‘(1st𝑘)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑘)):(1...𝑁)–1-1-onto→(1...𝑁)))
3836, 37elab 3681 . . . . . . . . . . 11 ((2nd ‘(1st𝑘)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑘)):(1...𝑁)–1-1-onto→(1...𝑁))
3935, 38sylib 218 . . . . . . . . . 10 (𝑘𝑆 → (2nd ‘(1st𝑘)):(1...𝑁)–1-1-onto→(1...𝑁))
40 f1ofn 6850 . . . . . . . . . 10 ((2nd ‘(1st𝑘)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑘)) Fn (1...𝑁))
4139, 40syl 17 . . . . . . . . 9 (𝑘𝑆 → (2nd ‘(1st𝑘)) Fn (1...𝑁))
4241adantl 481 . . . . . . . 8 ((𝑧𝑆𝑘𝑆) → (2nd ‘(1st𝑘)) Fn (1...𝑁))
4342ad2antlr 727 . . . . . . 7 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → (2nd ‘(1st𝑘)) Fn (1...𝑁))
44 simpllr 776 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (𝑧𝑆𝑘𝑆))
45 oveq2 7439 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
4645imaeq2d 6080 . . . . . . . . . . . . . . 15 (𝑛 = 𝑁 → ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑧)) “ (1...𝑁)))
47 f1ofo 6856 . . . . . . . . . . . . . . . 16 ((2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑧)):(1...𝑁)–onto→(1...𝑁))
48 foima 6826 . . . . . . . . . . . . . . . 16 ((2nd ‘(1st𝑧)):(1...𝑁)–onto→(1...𝑁) → ((2nd ‘(1st𝑧)) “ (1...𝑁)) = (1...𝑁))
4926, 47, 483syl 18 . . . . . . . . . . . . . . 15 (𝑧𝑆 → ((2nd ‘(1st𝑧)) “ (1...𝑁)) = (1...𝑁))
5046, 49sylan9eqr 2797 . . . . . . . . . . . . . 14 ((𝑧𝑆𝑛 = 𝑁) → ((2nd ‘(1st𝑧)) “ (1...𝑛)) = (1...𝑁))
5150adantlr 715 . . . . . . . . . . . . 13 (((𝑧𝑆𝑘𝑆) ∧ 𝑛 = 𝑁) → ((2nd ‘(1st𝑧)) “ (1...𝑛)) = (1...𝑁))
5245imaeq2d 6080 . . . . . . . . . . . . . . 15 (𝑛 = 𝑁 → ((2nd ‘(1st𝑘)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑁)))
53 f1ofo 6856 . . . . . . . . . . . . . . . 16 ((2nd ‘(1st𝑘)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑘)):(1...𝑁)–onto→(1...𝑁))
54 foima 6826 . . . . . . . . . . . . . . . 16 ((2nd ‘(1st𝑘)):(1...𝑁)–onto→(1...𝑁) → ((2nd ‘(1st𝑘)) “ (1...𝑁)) = (1...𝑁))
5539, 53, 543syl 18 . . . . . . . . . . . . . . 15 (𝑘𝑆 → ((2nd ‘(1st𝑘)) “ (1...𝑁)) = (1...𝑁))
5652, 55sylan9eqr 2797 . . . . . . . . . . . . . 14 ((𝑘𝑆𝑛 = 𝑁) → ((2nd ‘(1st𝑘)) “ (1...𝑛)) = (1...𝑁))
5756adantll 714 . . . . . . . . . . . . 13 (((𝑧𝑆𝑘𝑆) ∧ 𝑛 = 𝑁) → ((2nd ‘(1st𝑘)) “ (1...𝑛)) = (1...𝑁))
5851, 57eqtr4d 2778 . . . . . . . . . . . 12 (((𝑧𝑆𝑘𝑆) ∧ 𝑛 = 𝑁) → ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑛)))
5944, 58sylan 580 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑛 = 𝑁) → ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑛)))
60 simpll 767 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → 𝜑)
61 elnnuz 12920 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ‘1))
621, 61sylib 218 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑁 ∈ (ℤ‘1))
63 fzm1 13644 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ (ℤ‘1) → (𝑛 ∈ (1...𝑁) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁)))
6462, 63syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑛 ∈ (1...𝑁) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁)))
6564anbi1d 631 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑛 ∈ (1...𝑁) ∧ 𝑛𝑁) ↔ ((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁) ∧ 𝑛𝑁)))
6665biimpa 476 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛𝑁)) → ((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁) ∧ 𝑛𝑁))
67 df-ne 2939 . . . . . . . . . . . . . . . . . 18 (𝑛𝑁 ↔ ¬ 𝑛 = 𝑁)
6867anbi2i 623 . . . . . . . . . . . . . . . . 17 (((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁) ∧ 𝑛𝑁) ↔ ((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁) ∧ ¬ 𝑛 = 𝑁))
69 pm5.61 1002 . . . . . . . . . . . . . . . . 17 (((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁) ∧ ¬ 𝑛 = 𝑁) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∧ ¬ 𝑛 = 𝑁))
7068, 69bitri 275 . . . . . . . . . . . . . . . 16 (((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁) ∧ 𝑛𝑁) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∧ ¬ 𝑛 = 𝑁))
7166, 70sylib 218 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛𝑁)) → (𝑛 ∈ (1...(𝑁 − 1)) ∧ ¬ 𝑛 = 𝑁))
72 fz1ssfz0 13660 . . . . . . . . . . . . . . . . 17 (1...(𝑁 − 1)) ⊆ (0...(𝑁 − 1))
7372sseli 3991 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ (0...(𝑁 − 1)))
7473adantr 480 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (1...(𝑁 − 1)) ∧ ¬ 𝑛 = 𝑁) → 𝑛 ∈ (0...(𝑁 − 1)))
7571, 74syl 17 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛𝑁)) → 𝑛 ∈ (0...(𝑁 − 1)))
7660, 75sylan 580 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛𝑁)) → 𝑛 ∈ (0...(𝑁 − 1)))
77 eleq1 2827 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑛 → (𝑚 ∈ (0...(𝑁 − 1)) ↔ 𝑛 ∈ (0...(𝑁 − 1))))
7877anbi2d 630 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) ↔ (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 1)))))
79 oveq2 7439 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
8079imaeq2d 6080 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑛 → ((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑧)) “ (1...𝑛)))
8179imaeq2d 6080 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑛 → ((2nd ‘(1st𝑘)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...𝑛)))
8280, 81eqeq12d 2751 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → (((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...𝑚)) ↔ ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑛))))
8378, 82imbi12d 344 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → (((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...𝑚))) ↔ ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑛)))))
841ad3antrrr 730 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℕ)
85 poimirlem22.1 . . . . . . . . . . . . . . . . 17 (𝜑𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))
8685ad3antrrr 730 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))
87 simpl 482 . . . . . . . . . . . . . . . . 17 ((𝑧𝑆𝑘𝑆) → 𝑧𝑆)
8887ad3antlr 731 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → 𝑧𝑆)
89 simplrl 777 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → (2nd𝑧) = 𝑁)
90 simpr 484 . . . . . . . . . . . . . . . . 17 ((𝑧𝑆𝑘𝑆) → 𝑘𝑆)
9190ad3antlr 731 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → 𝑘𝑆)
92 simplrr 778 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → (2nd𝑘) = 𝑁)
93 simpr 484 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → 𝑚 ∈ (0...(𝑁 − 1)))
9484, 3, 86, 88, 89, 91, 92, 93poimirlem12 37619 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑧)) “ (1...𝑚)) ⊆ ((2nd ‘(1st𝑘)) “ (1...𝑚)))
9584, 3, 86, 91, 92, 88, 89, 93poimirlem12 37619 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑘)) “ (1...𝑚)) ⊆ ((2nd ‘(1st𝑧)) “ (1...𝑚)))
9694, 95eqssd 4013 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...𝑚)))
9783, 96chvarvv 1996 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑛)))
9876, 97syldan 591 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛𝑁)) → ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑛)))
9998anassrs 467 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑛𝑁) → ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑛)))
10059, 99pm2.61dane 3027 . . . . . . . . . 10 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑛)))
101 simpr 484 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...𝑁)) → 𝑛 ∈ (1...𝑁))
102 elfzelz 13561 . . . . . . . . . . . . . 14 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℤ)
1031nnzd 12638 . . . . . . . . . . . . . 14 (𝜑𝑁 ∈ ℤ)
104 elfzm1b 13639 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑛 ∈ (1...𝑁) ↔ (𝑛 − 1) ∈ (0...(𝑁 − 1))))
105102, 103, 104syl2anr 597 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑛 ∈ (1...𝑁) ↔ (𝑛 − 1) ∈ (0...(𝑁 − 1))))
106101, 105mpbid 232 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ∈ (0...(𝑁 − 1)))
10760, 106sylan 580 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ∈ (0...(𝑁 − 1)))
108 ovex 7464 . . . . . . . . . . . 12 (𝑛 − 1) ∈ V
109 eleq1 2827 . . . . . . . . . . . . . 14 (𝑚 = (𝑛 − 1) → (𝑚 ∈ (0...(𝑁 − 1)) ↔ (𝑛 − 1) ∈ (0...(𝑁 − 1))))
110109anbi2d 630 . . . . . . . . . . . . 13 (𝑚 = (𝑛 − 1) → ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) ↔ (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ (𝑛 − 1) ∈ (0...(𝑁 − 1)))))
111 oveq2 7439 . . . . . . . . . . . . . . 15 (𝑚 = (𝑛 − 1) → (1...𝑚) = (1...(𝑛 − 1)))
112111imaeq2d 6080 . . . . . . . . . . . . . 14 (𝑚 = (𝑛 − 1) → ((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))))
113111imaeq2d 6080 . . . . . . . . . . . . . 14 (𝑚 = (𝑛 − 1) → ((2nd ‘(1st𝑘)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))
114112, 113eqeq12d 2751 . . . . . . . . . . . . 13 (𝑚 = (𝑛 − 1) → (((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...𝑚)) ↔ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1)))))
115110, 114imbi12d 344 . . . . . . . . . . . 12 (𝑚 = (𝑛 − 1) → (((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...𝑚))) ↔ ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ (𝑛 − 1) ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))))
116108, 115, 96vtocl 3558 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ (𝑛 − 1) ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))
117107, 116syldan 591 . . . . . . . . . 10 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))
118100, 117difeq12d 4137 . . . . . . . . 9 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (((2nd ‘(1st𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1)))) = (((2nd ‘(1st𝑘)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1)))))
119 fnsnfv 6988 . . . . . . . . . . . 12 (((2nd ‘(1st𝑧)) Fn (1...𝑁) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑧))‘𝑛)} = ((2nd ‘(1st𝑧)) “ {𝑛}))
12028, 119sylan 580 . . . . . . . . . . 11 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑧))‘𝑛)} = ((2nd ‘(1st𝑧)) “ {𝑛}))
121 elfznn 13590 . . . . . . . . . . . . . 14 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℕ)
122 uncom 4168 . . . . . . . . . . . . . . . . 17 ((1...(𝑛 − 1)) ∪ {𝑛}) = ({𝑛} ∪ (1...(𝑛 − 1)))
123122difeq1i 4132 . . . . . . . . . . . . . . . 16 (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))) = (({𝑛} ∪ (1...(𝑛 − 1))) ∖ (1...(𝑛 − 1)))
124 difun2 4487 . . . . . . . . . . . . . . . 16 (({𝑛} ∪ (1...(𝑛 − 1))) ∖ (1...(𝑛 − 1))) = ({𝑛} ∖ (1...(𝑛 − 1)))
125123, 124eqtri 2763 . . . . . . . . . . . . . . 15 (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))) = ({𝑛} ∖ (1...(𝑛 − 1)))
126 nncn 12272 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
127 npcan1 11686 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℂ → ((𝑛 − 1) + 1) = 𝑛)
128126, 127syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) = 𝑛)
129 elnnuz 12920 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ‘1))
130129biimpi 216 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ‘1))
131128, 130eqeltrd 2839 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) ∈ (ℤ‘1))
132 nnm1nn0 12565 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
133132nn0zd 12637 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℤ)
134 uzid 12891 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 − 1) ∈ ℤ → (𝑛 − 1) ∈ (ℤ‘(𝑛 − 1)))
135 peano2uz 12941 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 − 1) ∈ (ℤ‘(𝑛 − 1)) → ((𝑛 − 1) + 1) ∈ (ℤ‘(𝑛 − 1)))
136133, 134, 1353syl 18 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) ∈ (ℤ‘(𝑛 − 1)))
137128, 136eqeltrrd 2840 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ‘(𝑛 − 1)))
138 fzsplit2 13586 . . . . . . . . . . . . . . . . . 18 ((((𝑛 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑛 ∈ (ℤ‘(𝑛 − 1))) → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)))
139131, 137, 138syl2anc 584 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)))
140128oveq1d 7446 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → (((𝑛 − 1) + 1)...𝑛) = (𝑛...𝑛))
141 nnz 12632 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
142 fzsn 13603 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℤ → (𝑛...𝑛) = {𝑛})
143141, 142syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → (𝑛...𝑛) = {𝑛})
144140, 143eqtrd 2775 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → (((𝑛 − 1) + 1)...𝑛) = {𝑛})
145144uneq2d 4178 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ → ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)) = ((1...(𝑛 − 1)) ∪ {𝑛}))
146139, 145eqtrd 2775 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → (1...𝑛) = ((1...(𝑛 − 1)) ∪ {𝑛}))
147146difeq1d 4135 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → ((1...𝑛) ∖ (1...(𝑛 − 1))) = (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))))
148 nnre 12271 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
149 ltm1 12107 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℝ → (𝑛 − 1) < 𝑛)
150 peano2rem 11574 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℝ → (𝑛 − 1) ∈ ℝ)
151 ltnle 11338 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 − 1) ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((𝑛 − 1) < 𝑛 ↔ ¬ 𝑛 ≤ (𝑛 − 1)))
152150, 151mpancom 688 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℝ → ((𝑛 − 1) < 𝑛 ↔ ¬ 𝑛 ≤ (𝑛 − 1)))
153149, 152mpbid 232 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℝ → ¬ 𝑛 ≤ (𝑛 − 1))
154 elfzle2 13565 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...(𝑛 − 1)) → 𝑛 ≤ (𝑛 − 1))
155153, 154nsyl 140 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℝ → ¬ 𝑛 ∈ (1...(𝑛 − 1)))
156148, 155syl 17 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → ¬ 𝑛 ∈ (1...(𝑛 − 1)))
157 incom 4217 . . . . . . . . . . . . . . . . . 18 ((1...(𝑛 − 1)) ∩ {𝑛}) = ({𝑛} ∩ (1...(𝑛 − 1)))
158157eqeq1i 2740 . . . . . . . . . . . . . . . . 17 (((1...(𝑛 − 1)) ∩ {𝑛}) = ∅ ↔ ({𝑛} ∩ (1...(𝑛 − 1))) = ∅)
159 disjsn 4716 . . . . . . . . . . . . . . . . 17 (((1...(𝑛 − 1)) ∩ {𝑛}) = ∅ ↔ ¬ 𝑛 ∈ (1...(𝑛 − 1)))
160 disj3 4460 . . . . . . . . . . . . . . . . 17 (({𝑛} ∩ (1...(𝑛 − 1))) = ∅ ↔ {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1))))
161158, 159, 1603bitr3i 301 . . . . . . . . . . . . . . . 16 𝑛 ∈ (1...(𝑛 − 1)) ↔ {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1))))
162156, 161sylib 218 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1))))
163125, 147, 1623eqtr4a 2801 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → ((1...𝑛) ∖ (1...(𝑛 − 1))) = {𝑛})
164121, 163syl 17 . . . . . . . . . . . . 13 (𝑛 ∈ (1...𝑁) → ((1...𝑛) ∖ (1...(𝑛 − 1))) = {𝑛})
165164imaeq2d 6080 . . . . . . . . . . . 12 (𝑛 ∈ (1...𝑁) → ((2nd ‘(1st𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = ((2nd ‘(1st𝑧)) “ {𝑛}))
166165adantl 481 . . . . . . . . . . 11 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = ((2nd ‘(1st𝑧)) “ {𝑛}))
167 dff1o3 6855 . . . . . . . . . . . . . 14 ((2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd ‘(1st𝑧)):(1...𝑁)–onto→(1...𝑁) ∧ Fun (2nd ‘(1st𝑧))))
168167simprbi 496 . . . . . . . . . . . . 13 ((2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun (2nd ‘(1st𝑧)))
169 imadif 6652 . . . . . . . . . . . . 13 (Fun (2nd ‘(1st𝑧)) → ((2nd ‘(1st𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = (((2nd ‘(1st𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1)))))
17026, 168, 1693syl 18 . . . . . . . . . . . 12 (𝑧𝑆 → ((2nd ‘(1st𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = (((2nd ‘(1st𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1)))))
171170adantr 480 . . . . . . . . . . 11 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = (((2nd ‘(1st𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1)))))
172120, 166, 1713eqtr2d 2781 . . . . . . . . . 10 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑧))‘𝑛)} = (((2nd ‘(1st𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1)))))
1734, 172sylan 580 . . . . . . . . 9 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑧))‘𝑛)} = (((2nd ‘(1st𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1)))))
174 eleq1 2827 . . . . . . . . . . . . 13 (𝑧 = 𝑘 → (𝑧𝑆𝑘𝑆))
175174anbi1d 631 . . . . . . . . . . . 12 (𝑧 = 𝑘 → ((𝑧𝑆𝑛 ∈ (1...𝑁)) ↔ (𝑘𝑆𝑛 ∈ (1...𝑁))))
176 2fveq3 6912 . . . . . . . . . . . . . . 15 (𝑧 = 𝑘 → (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑘)))
177176fveq1d 6909 . . . . . . . . . . . . . 14 (𝑧 = 𝑘 → ((2nd ‘(1st𝑧))‘𝑛) = ((2nd ‘(1st𝑘))‘𝑛))
178177sneqd 4643 . . . . . . . . . . . . 13 (𝑧 = 𝑘 → {((2nd ‘(1st𝑧))‘𝑛)} = {((2nd ‘(1st𝑘))‘𝑛)})
179176imaeq1d 6079 . . . . . . . . . . . . . 14 (𝑧 = 𝑘 → ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑛)))
180176imaeq1d 6079 . . . . . . . . . . . . . 14 (𝑧 = 𝑘 → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))
181179, 180difeq12d 4137 . . . . . . . . . . . . 13 (𝑧 = 𝑘 → (((2nd ‘(1st𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1)))) = (((2nd ‘(1st𝑘)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1)))))
182178, 181eqeq12d 2751 . . . . . . . . . . . 12 (𝑧 = 𝑘 → ({((2nd ‘(1st𝑧))‘𝑛)} = (((2nd ‘(1st𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1)))) ↔ {((2nd ‘(1st𝑘))‘𝑛)} = (((2nd ‘(1st𝑘)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))))
183175, 182imbi12d 344 . . . . . . . . . . 11 (𝑧 = 𝑘 → (((𝑧𝑆𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑧))‘𝑛)} = (((2nd ‘(1st𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))))) ↔ ((𝑘𝑆𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑘))‘𝑛)} = (((2nd ‘(1st𝑘)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1)))))))
184183, 172chvarvv 1996 . . . . . . . . . 10 ((𝑘𝑆𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑘))‘𝑛)} = (((2nd ‘(1st𝑘)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1)))))
18511, 184sylan 580 . . . . . . . . 9 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑘))‘𝑛)} = (((2nd ‘(1st𝑘)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1)))))
186118, 173, 1853eqtr4d 2785 . . . . . . . 8 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑧))‘𝑛)} = {((2nd ‘(1st𝑘))‘𝑛)})
187 fvex 6920 . . . . . . . . 9 ((2nd ‘(1st𝑧))‘𝑛) ∈ V
188187sneqr 4845 . . . . . . . 8 ({((2nd ‘(1st𝑧))‘𝑛)} = {((2nd ‘(1st𝑘))‘𝑛)} → ((2nd ‘(1st𝑧))‘𝑛) = ((2nd ‘(1st𝑘))‘𝑛))
189186, 188syl 17 . . . . . . 7 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧))‘𝑛) = ((2nd ‘(1st𝑘))‘𝑛))
19030, 43, 189eqfnfvd 7054 . . . . . 6 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑘)))
19119, 20syl 17 . . . . . . . 8 (𝑧𝑆 → (1st𝑧) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
19232, 33syl 17 . . . . . . . 8 (𝑘𝑆 → (1st𝑘) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
193 xpopth 8054 . . . . . . . 8 (((1st𝑧) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (1st𝑘) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) → (((1st ‘(1st𝑧)) = (1st ‘(1st𝑘)) ∧ (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑘))) ↔ (1st𝑧) = (1st𝑘)))
194191, 192, 193syl2an 596 . . . . . . 7 ((𝑧𝑆𝑘𝑆) → (((1st ‘(1st𝑧)) = (1st ‘(1st𝑘)) ∧ (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑘))) ↔ (1st𝑧) = (1st𝑘)))
195194ad2antlr 727 . . . . . 6 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → (((1st ‘(1st𝑧)) = (1st ‘(1st𝑘)) ∧ (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑘))) ↔ (1st𝑧) = (1st𝑘)))
19617, 190, 195mpbi2and 712 . . . . 5 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → (1st𝑧) = (1st𝑘))
197 eqtr3 2761 . . . . . 6 (((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁) → (2nd𝑧) = (2nd𝑘))
198197adantl 481 . . . . 5 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → (2nd𝑧) = (2nd𝑘))
199 xpopth 8054 . . . . . . 7 ((𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑘 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (((1st𝑧) = (1st𝑘) ∧ (2nd𝑧) = (2nd𝑘)) ↔ 𝑧 = 𝑘))
20019, 32, 199syl2an 596 . . . . . 6 ((𝑧𝑆𝑘𝑆) → (((1st𝑧) = (1st𝑘) ∧ (2nd𝑧) = (2nd𝑘)) ↔ 𝑧 = 𝑘))
201200ad2antlr 727 . . . . 5 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → (((1st𝑧) = (1st𝑘) ∧ (2nd𝑧) = (2nd𝑘)) ↔ 𝑧 = 𝑘))
202196, 198, 201mpbi2and 712 . . . 4 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → 𝑧 = 𝑘)
203202ex 412 . . 3 ((𝜑 ∧ (𝑧𝑆𝑘𝑆)) → (((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁) → 𝑧 = 𝑘))
204203ralrimivva 3200 . 2 (𝜑 → ∀𝑧𝑆𝑘𝑆 (((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁) → 𝑧 = 𝑘))
205 fveqeq2 6916 . . 3 (𝑧 = 𝑘 → ((2nd𝑧) = 𝑁 ↔ (2nd𝑘) = 𝑁))
206205rmo4 3739 . 2 (∃*𝑧𝑆 (2nd𝑧) = 𝑁 ↔ ∀𝑧𝑆𝑘𝑆 (((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁) → 𝑧 = 𝑘))
207204, 206sylibr 234 1 (𝜑 → ∃*𝑧𝑆 (2nd𝑧) = 𝑁)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847   = wceq 1537  wcel 2106  {cab 2712  wne 2938  wral 3059  ∃*wrmo 3377  {crab 3433  csb 3908  cdif 3960  cun 3961  cin 3962  c0 4339  ifcif 4531  {csn 4631   class class class wbr 5148  cmpt 5231   × cxp 5687  ccnv 5688  cima 5692  Fun wfun 6557   Fn wfn 6558  wf 6559  ontowfo 6561  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  f cof 7695  1st c1st 8011  2nd c2nd 8012  m cmap 8865  cc 11151  cr 11152  0cc0 11153  1c1 11154   + caddc 11156   < clt 11293  cle 11294  cmin 11490  cn 12264  cz 12611  cuz 12876  ...cfz 13544  ..^cfzo 13691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692
This theorem is referenced by:  poimirlem18  37625  poimirlem21  37628
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