Users' Mathboxes Mathbox for Brendan Leahy < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  poimirlem14 Structured version   Visualization version   GIF version

Theorem poimirlem14 34787
Description: Lemma for poimir 34806- 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 722 . . . . . . . 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 773 . . . . . . . 8 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → 𝑧𝑆)
51nngt0d 11674 . . . . . . . . . 10 (𝜑 → 0 < 𝑁)
6 breq2 5061 . . . . . . . . . . 11 ((2nd𝑧) = 𝑁 → (0 < (2nd𝑧) ↔ 0 < 𝑁))
76biimparc 480 . . . . . . . . . 10 ((0 < 𝑁 ∧ (2nd𝑧) = 𝑁) → 0 < (2nd𝑧))
85, 7sylan 580 . . . . . . . . 9 ((𝜑 ∧ (2nd𝑧) = 𝑁) → 0 < (2nd𝑧))
98ad2ant2r 743 . . . . . . . 8 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → 0 < (2nd𝑧))
102, 3, 4, 9poimirlem5 34778 . . . . . . 7 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → (𝐹‘0) = (1st ‘(1st𝑧)))
11 simplrr 774 . . . . . . . 8 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → 𝑘𝑆)
12 breq2 5061 . . . . . . . . . . 11 ((2nd𝑘) = 𝑁 → (0 < (2nd𝑘) ↔ 0 < 𝑁))
1312biimparc 480 . . . . . . . . . 10 ((0 < 𝑁 ∧ (2nd𝑘) = 𝑁) → 0 < (2nd𝑘))
145, 13sylan 580 . . . . . . . . 9 ((𝜑 ∧ (2nd𝑘) = 𝑁) → 0 < (2nd𝑘))
1514ad2ant2rl 745 . . . . . . . 8 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → 0 < (2nd𝑘))
162, 3, 11, 15poimirlem5 34778 . . . . . . 7 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → (𝐹‘0) = (1st ‘(1st𝑘)))
1710, 16eqtr3d 2855 . . . . . 6 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → (1st ‘(1st𝑧)) = (1st ‘(1st𝑘)))
18 elrabi 3672 . . . . . . . . . . . . 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 2928 . . . . . . . . . . . 12 (𝑧𝑆𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
20 xp1st 7710 . . . . . . . . . . . 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 7711 . . . . . . . . . . . 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 6676 . . . . . . . . . . . 12 (2nd ‘(1st𝑧)) ∈ V
24 f1oeq1 6597 . . . . . . . . . . . 12 (𝑓 = (2nd ‘(1st𝑧)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁)))
2523, 24elab 3664 . . . . . . . . . . 11 ((2nd ‘(1st𝑧)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
2622, 25sylib 219 . . . . . . . . . 10 (𝑧𝑆 → (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
27 f1ofn 6609 . . . . . . . . . 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 481 . . . . . . . 8 ((𝑧𝑆𝑘𝑆) → (2nd ‘(1st𝑧)) Fn (1...𝑁))
3029ad2antlr 723 . . . . . . 7 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → (2nd ‘(1st𝑧)) Fn (1...𝑁))
31 elrabi 3672 . . . . . . . . . . . . 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 2928 . . . . . . . . . . . 12 (𝑘𝑆𝑘 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
33 xp1st 7710 . . . . . . . . . . . 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 7711 . . . . . . . . . . . 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 6676 . . . . . . . . . . . 12 (2nd ‘(1st𝑘)) ∈ V
37 f1oeq1 6597 . . . . . . . . . . . 12 (𝑓 = (2nd ‘(1st𝑘)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑘)):(1...𝑁)–1-1-onto→(1...𝑁)))
3836, 37elab 3664 . . . . . . . . . . 11 ((2nd ‘(1st𝑘)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑘)):(1...𝑁)–1-1-onto→(1...𝑁))
3935, 38sylib 219 . . . . . . . . . 10 (𝑘𝑆 → (2nd ‘(1st𝑘)):(1...𝑁)–1-1-onto→(1...𝑁))
40 f1ofn 6609 . . . . . . . . . 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 482 . . . . . . . 8 ((𝑧𝑆𝑘𝑆) → (2nd ‘(1st𝑘)) Fn (1...𝑁))
4342ad2antlr 723 . . . . . . 7 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → (2nd ‘(1st𝑘)) Fn (1...𝑁))
44 simpllr 772 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (𝑧𝑆𝑘𝑆))
45 oveq2 7153 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
4645imaeq2d 5922 . . . . . . . . . . . . . . 15 (𝑛 = 𝑁 → ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑧)) “ (1...𝑁)))
47 f1ofo 6615 . . . . . . . . . . . . . . . 16 ((2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑧)):(1...𝑁)–onto→(1...𝑁))
48 foima 6588 . . . . . . . . . . . . . . . 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 2875 . . . . . . . . . . . . . 14 ((𝑧𝑆𝑛 = 𝑁) → ((2nd ‘(1st𝑧)) “ (1...𝑛)) = (1...𝑁))
5150adantlr 711 . . . . . . . . . . . . 13 (((𝑧𝑆𝑘𝑆) ∧ 𝑛 = 𝑁) → ((2nd ‘(1st𝑧)) “ (1...𝑛)) = (1...𝑁))
5245imaeq2d 5922 . . . . . . . . . . . . . . 15 (𝑛 = 𝑁 → ((2nd ‘(1st𝑘)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑁)))
53 f1ofo 6615 . . . . . . . . . . . . . . . 16 ((2nd ‘(1st𝑘)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑘)):(1...𝑁)–onto→(1...𝑁))
54 foima 6588 . . . . . . . . . . . . . . . 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 2875 . . . . . . . . . . . . . 14 ((𝑘𝑆𝑛 = 𝑁) → ((2nd ‘(1st𝑘)) “ (1...𝑛)) = (1...𝑁))
5756adantll 710 . . . . . . . . . . . . 13 (((𝑧𝑆𝑘𝑆) ∧ 𝑛 = 𝑁) → ((2nd ‘(1st𝑘)) “ (1...𝑛)) = (1...𝑁))
5851, 57eqtr4d 2856 . . . . . . . . . . . 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 763 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → 𝜑)
61 elnnuz 12270 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ‘1))
621, 61sylib 219 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑁 ∈ (ℤ‘1))
63 fzm1 12975 . . . . . . . . . . . . . . . . . . 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 629 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑛 ∈ (1...𝑁) ∧ 𝑛𝑁) ↔ ((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁) ∧ 𝑛𝑁)))
6665biimpa 477 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛𝑁)) → ((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁) ∧ 𝑛𝑁))
67 df-ne 3014 . . . . . . . . . . . . . . . . . 18 (𝑛𝑁 ↔ ¬ 𝑛 = 𝑁)
6867anbi2i 622 . . . . . . . . . . . . . . . . 17 (((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁) ∧ 𝑛𝑁) ↔ ((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁) ∧ ¬ 𝑛 = 𝑁))
69 pm5.61 994 . . . . . . . . . . . . . . . . 17 (((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁) ∧ ¬ 𝑛 = 𝑁) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∧ ¬ 𝑛 = 𝑁))
7068, 69bitri 276 . . . . . . . . . . . . . . . 16 (((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁) ∧ 𝑛𝑁) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∧ ¬ 𝑛 = 𝑁))
7166, 70sylib 219 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛𝑁)) → (𝑛 ∈ (1...(𝑁 − 1)) ∧ ¬ 𝑛 = 𝑁))
72 fz1ssfz0 12991 . . . . . . . . . . . . . . . . 17 (1...(𝑁 − 1)) ⊆ (0...(𝑁 − 1))
7372sseli 3960 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ (0...(𝑁 − 1)))
7473adantr 481 . . . . . . . . . . . . . . 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 2897 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑛 → (𝑚 ∈ (0...(𝑁 − 1)) ↔ 𝑛 ∈ (0...(𝑁 − 1))))
7877anbi2d 628 . . . . . . . . . . . . . . 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 7153 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
8079imaeq2d 5922 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑛 → ((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑧)) “ (1...𝑛)))
8179imaeq2d 5922 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑛 → ((2nd ‘(1st𝑘)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...𝑛)))
8280, 81eqeq12d 2834 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → (((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...𝑚)) ↔ ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑛))))
8378, 82imbi12d 346 . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))
87 simpl 483 . . . . . . . . . . . . . . . . 17 ((𝑧𝑆𝑘𝑆) → 𝑧𝑆)
8887ad3antlr 727 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → 𝑧𝑆)
89 simplrl 773 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → (2nd𝑧) = 𝑁)
90 simpr 485 . . . . . . . . . . . . . . . . 17 ((𝑧𝑆𝑘𝑆) → 𝑘𝑆)
9190ad3antlr 727 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → 𝑘𝑆)
92 simplrr 774 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → (2nd𝑘) = 𝑁)
93 simpr 485 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → 𝑚 ∈ (0...(𝑁 − 1)))
9484, 3, 86, 88, 89, 91, 92, 93poimirlem12 34785 . . . . . . . . . . . . . . 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 34785 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑘)) “ (1...𝑚)) ⊆ ((2nd ‘(1st𝑧)) “ (1...𝑚)))
9694, 95eqssd 3981 . . . . . . . . . . . . . 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 468 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑛𝑁) → ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑛)))
10059, 99pm2.61dane 3101 . . . . . . . . . 10 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑛)))
101 simpr 485 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...𝑁)) → 𝑛 ∈ (1...𝑁))
102 elfzelz 12896 . . . . . . . . . . . . . 14 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℤ)
1031nnzd 12074 . . . . . . . . . . . . . 14 (𝜑𝑁 ∈ ℤ)
104 elfzm1b 12973 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑛 ∈ (1...𝑁) ↔ (𝑛 − 1) ∈ (0...(𝑁 − 1))))
105102, 103, 104syl2anr 596 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑛 ∈ (1...𝑁) ↔ (𝑛 − 1) ∈ (0...(𝑁 − 1))))
106101, 105mpbid 233 . . . . . . . . . . . 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 7178 . . . . . . . . . . . 12 (𝑛 − 1) ∈ V
109 eleq1 2897 . . . . . . . . . . . . . 14 (𝑚 = (𝑛 − 1) → (𝑚 ∈ (0...(𝑁 − 1)) ↔ (𝑛 − 1) ∈ (0...(𝑁 − 1))))
110109anbi2d 628 . . . . . . . . . . . . 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 7153 . . . . . . . . . . . . . . 15 (𝑚 = (𝑛 − 1) → (1...𝑚) = (1...(𝑛 − 1)))
112111imaeq2d 5922 . . . . . . . . . . . . . 14 (𝑚 = (𝑛 − 1) → ((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))))
113111imaeq2d 5922 . . . . . . . . . . . . . 14 (𝑚 = (𝑛 − 1) → ((2nd ‘(1st𝑘)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))
114112, 113eqeq12d 2834 . . . . . . . . . . . . 13 (𝑚 = (𝑛 − 1) → (((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...𝑚)) ↔ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1)))))
115110, 114imbi12d 346 . . . . . . . . . . . 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 3557 . . . . . . . . . . 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 4097 . . . . . . . . 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 6736 . . . . . . . . . . . 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 12924 . . . . . . . . . . . . . 14 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℕ)
122 uncom 4126 . . . . . . . . . . . . . . . . 17 ((1...(𝑛 − 1)) ∪ {𝑛}) = ({𝑛} ∪ (1...(𝑛 − 1)))
123122difeq1i 4092 . . . . . . . . . . . . . . . 16 (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))) = (({𝑛} ∪ (1...(𝑛 − 1))) ∖ (1...(𝑛 − 1)))
124 difun2 4425 . . . . . . . . . . . . . . . 16 (({𝑛} ∪ (1...(𝑛 − 1))) ∖ (1...(𝑛 − 1))) = ({𝑛} ∖ (1...(𝑛 − 1)))
125123, 124eqtri 2841 . . . . . . . . . . . . . . 15 (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))) = ({𝑛} ∖ (1...(𝑛 − 1)))
126 nncn 11634 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
127 npcan1 11053 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℂ → ((𝑛 − 1) + 1) = 𝑛)
128126, 127syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) = 𝑛)
129 elnnuz 12270 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ‘1))
130129biimpi 217 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ‘1))
131128, 130eqeltrd 2910 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) ∈ (ℤ‘1))
132 nnm1nn0 11926 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
133132nn0zd 12073 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℤ)
134 uzid 12246 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 − 1) ∈ ℤ → (𝑛 − 1) ∈ (ℤ‘(𝑛 − 1)))
135 peano2uz 12289 . . . . . . . . . . . . . . . . . . . 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 2911 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ‘(𝑛 − 1)))
138 fzsplit2 12920 . . . . . . . . . . . . . . . . . 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 7160 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → (((𝑛 − 1) + 1)...𝑛) = (𝑛...𝑛))
141 nnz 11992 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
142 fzsn 12937 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℤ → (𝑛...𝑛) = {𝑛})
143141, 142syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → (𝑛...𝑛) = {𝑛})
144140, 143eqtrd 2853 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → (((𝑛 − 1) + 1)...𝑛) = {𝑛})
145144uneq2d 4136 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ → ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)) = ((1...(𝑛 − 1)) ∪ {𝑛}))
146139, 145eqtrd 2853 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → (1...𝑛) = ((1...(𝑛 − 1)) ∪ {𝑛}))
147146difeq1d 4095 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → ((1...𝑛) ∖ (1...(𝑛 − 1))) = (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))))
148 nnre 11633 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
149 ltm1 11470 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℝ → (𝑛 − 1) < 𝑛)
150 peano2rem 10941 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℝ → (𝑛 − 1) ∈ ℝ)
151 ltnle 10708 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 − 1) ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((𝑛 − 1) < 𝑛 ↔ ¬ 𝑛 ≤ (𝑛 − 1)))
152150, 151mpancom 684 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℝ → ((𝑛 − 1) < 𝑛 ↔ ¬ 𝑛 ≤ (𝑛 − 1)))
153149, 152mpbid 233 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℝ → ¬ 𝑛 ≤ (𝑛 − 1))
154 elfzle2 12899 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...(𝑛 − 1)) → 𝑛 ≤ (𝑛 − 1))
155153, 154nsyl 142 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℝ → ¬ 𝑛 ∈ (1...(𝑛 − 1)))
156148, 155syl 17 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → ¬ 𝑛 ∈ (1...(𝑛 − 1)))
157 incom 4175 . . . . . . . . . . . . . . . . . 18 ((1...(𝑛 − 1)) ∩ {𝑛}) = ({𝑛} ∩ (1...(𝑛 − 1)))
158157eqeq1i 2823 . . . . . . . . . . . . . . . . 17 (((1...(𝑛 − 1)) ∩ {𝑛}) = ∅ ↔ ({𝑛} ∩ (1...(𝑛 − 1))) = ∅)
159 disjsn 4639 . . . . . . . . . . . . . . . . 17 (((1...(𝑛 − 1)) ∩ {𝑛}) = ∅ ↔ ¬ 𝑛 ∈ (1...(𝑛 − 1)))
160 disj3 4399 . . . . . . . . . . . . . . . . 17 (({𝑛} ∩ (1...(𝑛 − 1))) = ∅ ↔ {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1))))
161158, 159, 1603bitr3i 302 . . . . . . . . . . . . . . . 16 𝑛 ∈ (1...(𝑛 − 1)) ↔ {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1))))
162156, 161sylib 219 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1))))
163125, 147, 1623eqtr4a 2879 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → ((1...𝑛) ∖ (1...(𝑛 − 1))) = {𝑛})
164121, 163syl 17 . . . . . . . . . . . . 13 (𝑛 ∈ (1...𝑁) → ((1...𝑛) ∖ (1...(𝑛 − 1))) = {𝑛})
165164imaeq2d 5922 . . . . . . . . . . . 12 (𝑛 ∈ (1...𝑁) → ((2nd ‘(1st𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = ((2nd ‘(1st𝑧)) “ {𝑛}))
166165adantl 482 . . . . . . . . . . 11 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = ((2nd ‘(1st𝑧)) “ {𝑛}))
167 dff1o3 6614 . . . . . . . . . . . . . 14 ((2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd ‘(1st𝑧)):(1...𝑁)–onto→(1...𝑁) ∧ Fun (2nd ‘(1st𝑧))))
168167simprbi 497 . . . . . . . . . . . . 13 ((2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun (2nd ‘(1st𝑧)))
169 imadif 6431 . . . . . . . . . . . . 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 481 . . . . . . . . . . 11 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = (((2nd ‘(1st𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1)))))
172120, 166, 1713eqtr2d 2859 . . . . . . . . . 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 2897 . . . . . . . . . . . . 13 (𝑧 = 𝑘 → (𝑧𝑆𝑘𝑆))
175174anbi1d 629 . . . . . . . . . . . 12 (𝑧 = 𝑘 → ((𝑧𝑆𝑛 ∈ (1...𝑁)) ↔ (𝑘𝑆𝑛 ∈ (1...𝑁))))
176 2fveq3 6668 . . . . . . . . . . . . . . 15 (𝑧 = 𝑘 → (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑘)))
177176fveq1d 6665 . . . . . . . . . . . . . 14 (𝑧 = 𝑘 → ((2nd ‘(1st𝑧))‘𝑛) = ((2nd ‘(1st𝑘))‘𝑛))
178177sneqd 4569 . . . . . . . . . . . . 13 (𝑧 = 𝑘 → {((2nd ‘(1st𝑧))‘𝑛)} = {((2nd ‘(1st𝑘))‘𝑛)})
179176imaeq1d 5921 . . . . . . . . . . . . . 14 (𝑧 = 𝑘 → ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑛)))
180176imaeq1d 5921 . . . . . . . . . . . . . 14 (𝑧 = 𝑘 → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))
181179, 180difeq12d 4097 . . . . . . . . . . . . 13 (𝑧 = 𝑘 → (((2nd ‘(1st𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1)))) = (((2nd ‘(1st𝑘)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1)))))
182178, 181eqeq12d 2834 . . . . . . . . . . . 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 346 . . . . . . . . . . 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 2863 . . . . . . . 8 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑧))‘𝑛)} = {((2nd ‘(1st𝑘))‘𝑛)})
187 fvex 6676 . . . . . . . . 9 ((2nd ‘(1st𝑧))‘𝑛) ∈ V
188187sneqr 4763 . . . . . . . 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 6797 . . . . . 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 7719 . . . . . . . 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 595 . . . . . . 7 ((𝑧𝑆𝑘𝑆) → (((1st ‘(1st𝑧)) = (1st ‘(1st𝑘)) ∧ (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑘))) ↔ (1st𝑧) = (1st𝑘)))
195194ad2antlr 723 . . . . . 6 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → (((1st ‘(1st𝑧)) = (1st ‘(1st𝑘)) ∧ (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑘))) ↔ (1st𝑧) = (1st𝑘)))
19617, 190, 195mpbi2and 708 . . . . 5 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → (1st𝑧) = (1st𝑘))
197 eqtr3 2840 . . . . . 6 (((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁) → (2nd𝑧) = (2nd𝑘))
198197adantl 482 . . . . 5 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → (2nd𝑧) = (2nd𝑘))
199 xpopth 7719 . . . . . . 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 595 . . . . . 6 ((𝑧𝑆𝑘𝑆) → (((1st𝑧) = (1st𝑘) ∧ (2nd𝑧) = (2nd𝑘)) ↔ 𝑧 = 𝑘))
201200ad2antlr 723 . . . . 5 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → (((1st𝑧) = (1st𝑘) ∧ (2nd𝑧) = (2nd𝑘)) ↔ 𝑧 = 𝑘))
202196, 198, 201mpbi2and 708 . . . 4 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → 𝑧 = 𝑘)
203202ex 413 . . 3 ((𝜑 ∧ (𝑧𝑆𝑘𝑆)) → (((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁) → 𝑧 = 𝑘))
204203ralrimivva 3188 . 2 (𝜑 → ∀𝑧𝑆𝑘𝑆 (((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁) → 𝑧 = 𝑘))
205 fveqeq2 6672 . . 3 (𝑧 = 𝑘 → ((2nd𝑧) = 𝑁 ↔ (2nd𝑘) = 𝑁))
206205rmo4 3718 . 2 (∃*𝑧𝑆 (2nd𝑧) = 𝑁 ↔ ∀𝑧𝑆𝑘𝑆 (((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁) → 𝑧 = 𝑘))
207204, 206sylibr 235 1 (𝜑 → ∃*𝑧𝑆 (2nd𝑧) = 𝑁)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841   = wceq 1528  wcel 2105  {cab 2796  wne 3013  wral 3135  ∃*wrmo 3138  {crab 3139  csb 3880  cdif 3930  cun 3931  cin 3932  c0 4288  ifcif 4463  {csn 4557   class class class wbr 5057  cmpt 5137   × cxp 5546  ccnv 5547  cima 5551  Fun wfun 6342   Fn wfn 6343  wf 6344  ontowfo 6346  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145  f cof 7396  1st c1st 7676  2nd c2nd 7677  m cmap 8395  cc 10523  cr 10524  0cc0 10525  1c1 10526   + caddc 10528   < clt 10663  cle 10664  cmin 10858  cn 11626  cz 11969  cuz 12231  ...cfz 12880  ..^cfzo 13021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12881  df-fzo 13022
This theorem is referenced by:  poimirlem18  34791  poimirlem21  34794
  Copyright terms: Public domain W3C validator