Mathbox for Brendan Leahy < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  poimirlem13 Structured version   Visualization version   GIF version

Theorem poimirlem13 33965
 Description: Lemma for poimir 33985- for at most one simplex associated with a shared face is the opposite vertex first 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
poimirlem13 (𝜑 → ∃*𝑧𝑆 (2nd𝑧) = 0)
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 poimirlem13
Dummy variables 𝑘 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 poimir.0 . . . . . . . . 9 (𝜑𝑁 ∈ ℕ)
21ad2antrr 717 . . . . . . . 8 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → 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 poimirlem22.1 . . . . . . . . 9 (𝜑𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)))
54ad2antrr 717 . . . . . . . 8 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)))
6 simplrl 795 . . . . . . . 8 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → 𝑧𝑆)
7 simprl 787 . . . . . . . 8 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (2nd𝑧) = 0)
82, 3, 5, 6, 7poimirlem10 33962 . . . . . . 7 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → ((𝐹‘(𝑁 − 1)) ∘𝑓 − ((1...𝑁) × {1})) = (1st ‘(1st𝑧)))
9 simplrr 796 . . . . . . . 8 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → 𝑘𝑆)
10 simprr 789 . . . . . . . 8 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (2nd𝑘) = 0)
112, 3, 5, 9, 10poimirlem10 33962 . . . . . . 7 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → ((𝐹‘(𝑁 − 1)) ∘𝑓 − ((1...𝑁) × {1})) = (1st ‘(1st𝑘)))
128, 11eqtr3d 2863 . . . . . 6 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (1st ‘(1st𝑧)) = (1st ‘(1st𝑘)))
13 elrabi 3580 . . . . . . . . . . . . . 14 (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)))
1413, 3eleq2s 2924 . . . . . . . . . . . . 13 (𝑧𝑆𝑧 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
15 xp1st 7465 . . . . . . . . . . . . 13 (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)}))
1614, 15syl 17 . . . . . . . . . . . 12 (𝑧𝑆 → (1st𝑧) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
17 xp2nd 7466 . . . . . . . . . . . 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)})
1816, 17syl 17 . . . . . . . . . . 11 (𝑧𝑆 → (2nd ‘(1st𝑧)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
19 fvex 6450 . . . . . . . . . . . 12 (2nd ‘(1st𝑧)) ∈ V
20 f1oeq1 6371 . . . . . . . . . . . 12 (𝑓 = (2nd ‘(1st𝑧)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁)))
2119, 20elab 3571 . . . . . . . . . . 11 ((2nd ‘(1st𝑧)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
2218, 21sylib 210 . . . . . . . . . 10 (𝑧𝑆 → (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
23 f1ofn 6383 . . . . . . . . . 10 ((2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑧)) Fn (1...𝑁))
2422, 23syl 17 . . . . . . . . 9 (𝑧𝑆 → (2nd ‘(1st𝑧)) Fn (1...𝑁))
2524adantr 474 . . . . . . . 8 ((𝑧𝑆𝑘𝑆) → (2nd ‘(1st𝑧)) Fn (1...𝑁))
2625ad2antlr 718 . . . . . . 7 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (2nd ‘(1st𝑧)) Fn (1...𝑁))
27 elrabi 3580 . . . . . . . . . . . . . 14 (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)))
2827, 3eleq2s 2924 . . . . . . . . . . . . 13 (𝑘𝑆𝑘 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
29 xp1st 7465 . . . . . . . . . . . . 13 (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)}))
3028, 29syl 17 . . . . . . . . . . . 12 (𝑘𝑆 → (1st𝑘) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
31 xp2nd 7466 . . . . . . . . . . . 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)})
3230, 31syl 17 . . . . . . . . . . 11 (𝑘𝑆 → (2nd ‘(1st𝑘)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
33 fvex 6450 . . . . . . . . . . . 12 (2nd ‘(1st𝑘)) ∈ V
34 f1oeq1 6371 . . . . . . . . . . . 12 (𝑓 = (2nd ‘(1st𝑘)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑘)):(1...𝑁)–1-1-onto→(1...𝑁)))
3533, 34elab 3571 . . . . . . . . . . 11 ((2nd ‘(1st𝑘)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑘)):(1...𝑁)–1-1-onto→(1...𝑁))
3632, 35sylib 210 . . . . . . . . . 10 (𝑘𝑆 → (2nd ‘(1st𝑘)):(1...𝑁)–1-1-onto→(1...𝑁))
37 f1ofn 6383 . . . . . . . . . 10 ((2nd ‘(1st𝑘)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑘)) Fn (1...𝑁))
3836, 37syl 17 . . . . . . . . 9 (𝑘𝑆 → (2nd ‘(1st𝑘)) Fn (1...𝑁))
3938adantl 475 . . . . . . . 8 ((𝑧𝑆𝑘𝑆) → (2nd ‘(1st𝑘)) Fn (1...𝑁))
4039ad2antlr 718 . . . . . . 7 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (2nd ‘(1st𝑘)) Fn (1...𝑁))
41 eleq1 2894 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → (𝑚 ∈ (1...𝑁) ↔ 𝑛 ∈ (1...𝑁)))
4241anbi2d 622 . . . . . . . . . . . 12 (𝑚 = 𝑛 → ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) ↔ (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁))))
43 oveq2 6918 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
4443imaeq2d 5711 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → ((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑧)) “ (1...𝑛)))
4543imaeq2d 5711 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → ((2nd ‘(1st𝑘)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...𝑛)))
4644, 45eqeq12d 2840 . . . . . . . . . . . 12 (𝑚 = 𝑛 → (((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...𝑚)) ↔ ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑛))))
4742, 46imbi12d 336 . . . . . . . . . . 11 (𝑚 = 𝑛 → (((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...𝑚))) ↔ ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑛)))))
481ad3antrrr 721 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑁 ∈ ℕ)
494ad3antrrr 721 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)))
50 simpl 476 . . . . . . . . . . . . . 14 ((𝑧𝑆𝑘𝑆) → 𝑧𝑆)
5150ad3antlr 722 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑧𝑆)
52 simplrl 795 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → (2nd𝑧) = 0)
53 simpr 479 . . . . . . . . . . . . . 14 ((𝑧𝑆𝑘𝑆) → 𝑘𝑆)
5453ad3antlr 722 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑘𝑆)
55 simplrr 796 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → (2nd𝑘) = 0)
56 simpr 479 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑚 ∈ (1...𝑁))
5748, 3, 49, 51, 52, 54, 55, 56poimirlem11 33963 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ (1...𝑚)) ⊆ ((2nd ‘(1st𝑘)) “ (1...𝑚)))
5848, 3, 49, 54, 55, 51, 52, 56poimirlem11 33963 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → ((2nd ‘(1st𝑘)) “ (1...𝑚)) ⊆ ((2nd ‘(1st𝑧)) “ (1...𝑚)))
5957, 58eqssd 3844 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...𝑚)))
6047, 59chvarv 2416 . . . . . . . . . 10 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑛)))
61 simpll 783 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → 𝜑)
62 elfznn 12670 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℕ)
63 nnm1nn0 11668 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
6462, 63syl 17 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ∈ ℕ0)
6564adantr 474 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1) → (𝑛 − 1) ∈ ℕ0)
6662nncnd 11375 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℂ)
67 ax-1cn 10317 . . . . . . . . . . . . . . . . . . . 20 1 ∈ ℂ
68 subeq0 10635 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) = 0 ↔ 𝑛 = 1))
6966, 67, 68sylancl 580 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) = 0 ↔ 𝑛 = 1))
7069necon3abid 3035 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) ≠ 0 ↔ ¬ 𝑛 = 1))
7170biimpar 471 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1) → (𝑛 − 1) ≠ 0)
72 elnnne0 11641 . . . . . . . . . . . . . . . . 17 ((𝑛 − 1) ∈ ℕ ↔ ((𝑛 − 1) ∈ ℕ0 ∧ (𝑛 − 1) ≠ 0))
7365, 71, 72sylanbrc 578 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1) → (𝑛 − 1) ∈ ℕ)
7473adantl 475 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → (𝑛 − 1) ∈ ℕ)
7564nn0red 11686 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ∈ ℝ)
7675adantl 475 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ∈ ℝ)
7762nnred 11374 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℝ)
7877adantl 475 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℝ)
791nnred 11374 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ ℝ)
8079adantr 474 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (1...𝑁)) → 𝑁 ∈ ℝ)
8177lem1d 11294 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ≤ 𝑛)
8281adantl 475 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ≤ 𝑛)
83 elfzle2 12645 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...𝑁) → 𝑛𝑁)
8483adantl 475 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (1...𝑁)) → 𝑛𝑁)
8576, 78, 80, 82, 84letrd 10520 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ≤ 𝑁)
8685adantrr 708 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → (𝑛 − 1) ≤ 𝑁)
871nnzd 11816 . . . . . . . . . . . . . . . . 17 (𝜑𝑁 ∈ ℤ)
88 fznn 12709 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℤ → ((𝑛 − 1) ∈ (1...𝑁) ↔ ((𝑛 − 1) ∈ ℕ ∧ (𝑛 − 1) ≤ 𝑁)))
8987, 88syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑛 − 1) ∈ (1...𝑁) ↔ ((𝑛 − 1) ∈ ℕ ∧ (𝑛 − 1) ≤ 𝑁)))
9089adantr 474 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → ((𝑛 − 1) ∈ (1...𝑁) ↔ ((𝑛 − 1) ∈ ℕ ∧ (𝑛 − 1) ≤ 𝑁)))
9174, 86, 90mpbir2and 704 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → (𝑛 − 1) ∈ (1...𝑁))
9261, 91sylan 575 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → (𝑛 − 1) ∈ (1...𝑁))
93 ovex 6942 . . . . . . . . . . . . . 14 (𝑛 − 1) ∈ V
94 eleq1 2894 . . . . . . . . . . . . . . . 16 (𝑚 = (𝑛 − 1) → (𝑚 ∈ (1...𝑁) ↔ (𝑛 − 1) ∈ (1...𝑁)))
9594anbi2d 622 . . . . . . . . . . . . . . 15 (𝑚 = (𝑛 − 1) → ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) ↔ (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ (𝑛 − 1) ∈ (1...𝑁))))
96 oveq2 6918 . . . . . . . . . . . . . . . . 17 (𝑚 = (𝑛 − 1) → (1...𝑚) = (1...(𝑛 − 1)))
9796imaeq2d 5711 . . . . . . . . . . . . . . . 16 (𝑚 = (𝑛 − 1) → ((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))))
9896imaeq2d 5711 . . . . . . . . . . . . . . . 16 (𝑚 = (𝑛 − 1) → ((2nd ‘(1st𝑘)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))
9997, 98eqeq12d 2840 . . . . . . . . . . . . . . 15 (𝑚 = (𝑛 − 1) → (((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...𝑚)) ↔ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1)))))
10095, 99imbi12d 336 . . . . . . . . . . . . . 14 (𝑚 = (𝑛 − 1) → (((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...𝑚))) ↔ ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ (𝑛 − 1) ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))))
10193, 100, 59vtocl 3475 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ (𝑛 − 1) ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))
10292, 101syldan 585 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))
103102expr 450 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → (¬ 𝑛 = 1 → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1)))))
104 ima0 5726 . . . . . . . . . . . . 13 ((2nd ‘(1st𝑧)) “ ∅) = ∅
105 ima0 5726 . . . . . . . . . . . . 13 ((2nd ‘(1st𝑘)) “ ∅) = ∅
106104, 105eqtr4i 2852 . . . . . . . . . . . 12 ((2nd ‘(1st𝑧)) “ ∅) = ((2nd ‘(1st𝑘)) “ ∅)
107 oveq1 6917 . . . . . . . . . . . . . . . 16 (𝑛 = 1 → (𝑛 − 1) = (1 − 1))
108 1m1e0 11430 . . . . . . . . . . . . . . . 16 (1 − 1) = 0
109107, 108syl6eq 2877 . . . . . . . . . . . . . . 15 (𝑛 = 1 → (𝑛 − 1) = 0)
110109oveq2d 6926 . . . . . . . . . . . . . 14 (𝑛 = 1 → (1...(𝑛 − 1)) = (1...0))
111 fz10 12662 . . . . . . . . . . . . . 14 (1...0) = ∅
112110, 111syl6eq 2877 . . . . . . . . . . . . 13 (𝑛 = 1 → (1...(𝑛 − 1)) = ∅)
113112imaeq2d 5711 . . . . . . . . . . . 12 (𝑛 = 1 → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑧)) “ ∅))
114112imaeq2d 5711 . . . . . . . . . . . 12 (𝑛 = 1 → ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ ∅))
115106, 113, 1143eqtr4a 2887 . . . . . . . . . . 11 (𝑛 = 1 → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))
116103, 115pm2.61d2 174 . . . . . . . . . 10 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))
11760, 116difeq12d 3958 . . . . . . . . 9 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → (((2nd ‘(1st𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1)))) = (((2nd ‘(1st𝑘)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1)))))
118 fnsnfv 6509 . . . . . . . . . . . 12 (((2nd ‘(1st𝑧)) Fn (1...𝑁) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑧))‘𝑛)} = ((2nd ‘(1st𝑧)) “ {𝑛}))
11924, 118sylan 575 . . . . . . . . . . 11 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑧))‘𝑛)} = ((2nd ‘(1st𝑧)) “ {𝑛}))
12062adantl 475 . . . . . . . . . . . . 13 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℕ)
121 uncom 3986 . . . . . . . . . . . . . . . 16 ((1...(𝑛 − 1)) ∪ {𝑛}) = ({𝑛} ∪ (1...(𝑛 − 1)))
122121difeq1i 3953 . . . . . . . . . . . . . . 15 (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))) = (({𝑛} ∪ (1...(𝑛 − 1))) ∖ (1...(𝑛 − 1)))
123 difun2 4273 . . . . . . . . . . . . . . 15 (({𝑛} ∪ (1...(𝑛 − 1))) ∖ (1...(𝑛 − 1))) = ({𝑛} ∖ (1...(𝑛 − 1)))
124122, 123eqtri 2849 . . . . . . . . . . . . . 14 (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))) = ({𝑛} ∖ (1...(𝑛 − 1)))
125 nncn 11366 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
126 npcan1 10786 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℂ → ((𝑛 − 1) + 1) = 𝑛)
127125, 126syl 17 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) = 𝑛)
128 elnnuz 12013 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ‘1))
129128biimpi 208 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ‘1))
130127, 129eqeltrd 2906 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) ∈ (ℤ‘1))
13163nn0zd 11815 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℤ)
132 uzid 11990 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 − 1) ∈ ℤ → (𝑛 − 1) ∈ (ℤ‘(𝑛 − 1)))
133131, 132syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ (ℤ‘(𝑛 − 1)))
134 peano2uz 12030 . . . . . . . . . . . . . . . . . . 19 ((𝑛 − 1) ∈ (ℤ‘(𝑛 − 1)) → ((𝑛 − 1) + 1) ∈ (ℤ‘(𝑛 − 1)))
135133, 134syl 17 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) ∈ (ℤ‘(𝑛 − 1)))
136127, 135eqeltrrd 2907 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ‘(𝑛 − 1)))
137 fzsplit2 12666 . . . . . . . . . . . . . . . . 17 ((((𝑛 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑛 ∈ (ℤ‘(𝑛 − 1))) → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)))
138130, 136, 137syl2anc 579 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)))
139127oveq1d 6925 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → (((𝑛 − 1) + 1)...𝑛) = (𝑛...𝑛))
140 nnz 11734 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
141 fzsn 12683 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℤ → (𝑛...𝑛) = {𝑛})
142140, 141syl 17 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → (𝑛...𝑛) = {𝑛})
143139, 142eqtrd 2861 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ → (((𝑛 − 1) + 1)...𝑛) = {𝑛})
144143uneq2d 3996 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)) = ((1...(𝑛 − 1)) ∪ {𝑛}))
145138, 144eqtrd 2861 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → (1...𝑛) = ((1...(𝑛 − 1)) ∪ {𝑛}))
146145difeq1d 3956 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → ((1...𝑛) ∖ (1...(𝑛 − 1))) = (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))))
147 nnre 11365 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
148 ltm1 11200 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℝ → (𝑛 − 1) < 𝑛)
149 peano2rem 10676 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℝ → (𝑛 − 1) ∈ ℝ)
150 ltnle 10443 . . . . . . . . . . . . . . . . . . 19 (((𝑛 − 1) ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((𝑛 − 1) < 𝑛 ↔ ¬ 𝑛 ≤ (𝑛 − 1)))
151149, 150mpancom 679 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℝ → ((𝑛 − 1) < 𝑛 ↔ ¬ 𝑛 ≤ (𝑛 − 1)))
152148, 151mpbid 224 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℝ → ¬ 𝑛 ≤ (𝑛 − 1))
153 elfzle2 12645 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (1...(𝑛 − 1)) → 𝑛 ≤ (𝑛 − 1))
154152, 153nsyl 138 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℝ → ¬ 𝑛 ∈ (1...(𝑛 − 1)))
155147, 154syl 17 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → ¬ 𝑛 ∈ (1...(𝑛 − 1)))
156 incom 4034 . . . . . . . . . . . . . . . . 17 ((1...(𝑛 − 1)) ∩ {𝑛}) = ({𝑛} ∩ (1...(𝑛 − 1)))
157156eqeq1i 2830 . . . . . . . . . . . . . . . 16 (((1...(𝑛 − 1)) ∩ {𝑛}) = ∅ ↔ ({𝑛} ∩ (1...(𝑛 − 1))) = ∅)
158 disjsn 4467 . . . . . . . . . . . . . . . 16 (((1...(𝑛 − 1)) ∩ {𝑛}) = ∅ ↔ ¬ 𝑛 ∈ (1...(𝑛 − 1)))
159 disj3 4247 . . . . . . . . . . . . . . . 16 (({𝑛} ∩ (1...(𝑛 − 1))) = ∅ ↔ {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1))))
160157, 158, 1593bitr3i 293 . . . . . . . . . . . . . . 15 𝑛 ∈ (1...(𝑛 − 1)) ↔ {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1))))
161155, 160sylib 210 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1))))
162124, 146, 1613eqtr4a 2887 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → ((1...𝑛) ∖ (1...(𝑛 − 1))) = {𝑛})
163120, 162syl 17 . . . . . . . . . . . 12 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → ((1...𝑛) ∖ (1...(𝑛 − 1))) = {𝑛})
164163imaeq2d 5711 . . . . . . . . . . 11 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = ((2nd ‘(1st𝑧)) “ {𝑛}))
165 dff1o3 6388 . . . . . . . . . . . . . . 15 ((2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd ‘(1st𝑧)):(1...𝑁)–onto→(1...𝑁) ∧ Fun (2nd ‘(1st𝑧))))
166165simprbi 492 . . . . . . . . . . . . . 14 ((2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun (2nd ‘(1st𝑧)))
16722, 166syl 17 . . . . . . . . . . . . 13 (𝑧𝑆 → Fun (2nd ‘(1st𝑧)))
168 imadif 6210 . . . . . . . . . . . . 13 (Fun (2nd ‘(1st𝑧)) → ((2nd ‘(1st𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = (((2nd ‘(1st𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1)))))
169167, 168syl 17 . . . . . . . . . . . 12 (𝑧𝑆 → ((2nd ‘(1st𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = (((2nd ‘(1st𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1)))))
170169adantr 474 . . . . . . . . . . 11 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = (((2nd ‘(1st𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1)))))
171119, 164, 1703eqtr2d 2867 . . . . . . . . . 10 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑧))‘𝑛)} = (((2nd ‘(1st𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1)))))
1726, 171sylan 575 . . . . . . . . 9 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑧))‘𝑛)} = (((2nd ‘(1st𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1)))))
173 eleq1 2894 . . . . . . . . . . . . 13 (𝑧 = 𝑘 → (𝑧𝑆𝑘𝑆))
174173anbi1d 623 . . . . . . . . . . . 12 (𝑧 = 𝑘 → ((𝑧𝑆𝑛 ∈ (1...𝑁)) ↔ (𝑘𝑆𝑛 ∈ (1...𝑁))))
175 2fveq3 6442 . . . . . . . . . . . . . . 15 (𝑧 = 𝑘 → (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑘)))
176175fveq1d 6439 . . . . . . . . . . . . . 14 (𝑧 = 𝑘 → ((2nd ‘(1st𝑧))‘𝑛) = ((2nd ‘(1st𝑘))‘𝑛))
177176sneqd 4411 . . . . . . . . . . . . 13 (𝑧 = 𝑘 → {((2nd ‘(1st𝑧))‘𝑛)} = {((2nd ‘(1st𝑘))‘𝑛)})
178175imaeq1d 5710 . . . . . . . . . . . . . 14 (𝑧 = 𝑘 → ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑛)))
179175imaeq1d 5710 . . . . . . . . . . . . . 14 (𝑧 = 𝑘 → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))
180178, 179difeq12d 3958 . . . . . . . . . . . . 13 (𝑧 = 𝑘 → (((2nd ‘(1st𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1)))) = (((2nd ‘(1st𝑘)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1)))))
181177, 180eqeq12d 2840 . . . . . . . . . . . 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))))))
182174, 181imbi12d 336 . . . . . . . . . . 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)))))))
183182, 171chvarv 2416 . . . . . . . . . 10 ((𝑘𝑆𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑘))‘𝑛)} = (((2nd ‘(1st𝑘)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1)))))
1849, 183sylan 575 . . . . . . . . 9 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑘))‘𝑛)} = (((2nd ‘(1st𝑘)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1)))))
185117, 172, 1843eqtr4d 2871 . . . . . . . 8 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑧))‘𝑛)} = {((2nd ‘(1st𝑘))‘𝑛)})
186 fvex 6450 . . . . . . . . 9 ((2nd ‘(1st𝑧))‘𝑛) ∈ V
187186sneqr 4589 . . . . . . . 8 ({((2nd ‘(1st𝑧))‘𝑛)} = {((2nd ‘(1st𝑘))‘𝑛)} → ((2nd ‘(1st𝑧))‘𝑛) = ((2nd ‘(1st𝑘))‘𝑛))
188185, 187syl 17 . . . . . . 7 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧))‘𝑛) = ((2nd ‘(1st𝑘))‘𝑛))
18926, 40, 188eqfnfvd 6568 . . . . . 6 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑘)))
190 xpopth 7474 . . . . . . . 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)))
19116, 30, 190syl2an 589 . . . . . . 7 ((𝑧𝑆𝑘𝑆) → (((1st ‘(1st𝑧)) = (1st ‘(1st𝑘)) ∧ (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑘))) ↔ (1st𝑧) = (1st𝑘)))
192191ad2antlr 718 . . . . . 6 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (((1st ‘(1st𝑧)) = (1st ‘(1st𝑘)) ∧ (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑘))) ↔ (1st𝑧) = (1st𝑘)))
19312, 189, 192mpbi2and 703 . . . . 5 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (1st𝑧) = (1st𝑘))
194 eqtr3 2848 . . . . . 6 (((2nd𝑧) = 0 ∧ (2nd𝑘) = 0) → (2nd𝑧) = (2nd𝑘))
195194adantl 475 . . . . 5 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (2nd𝑧) = (2nd𝑘))
196 xpopth 7474 . . . . . . 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))
19714, 28, 196syl2an 589 . . . . . 6 ((𝑧𝑆𝑘𝑆) → (((1st𝑧) = (1st𝑘) ∧ (2nd𝑧) = (2nd𝑘)) ↔ 𝑧 = 𝑘))
198197ad2antlr 718 . . . . 5 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (((1st𝑧) = (1st𝑘) ∧ (2nd𝑧) = (2nd𝑘)) ↔ 𝑧 = 𝑘))
199193, 195, 198mpbi2and 703 . . . 4 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → 𝑧 = 𝑘)
200199ex 403 . . 3 ((𝜑 ∧ (𝑧𝑆𝑘𝑆)) → (((2nd𝑧) = 0 ∧ (2nd𝑘) = 0) → 𝑧 = 𝑘))
201200ralrimivva 3180 . 2 (𝜑 → ∀𝑧𝑆𝑘𝑆 (((2nd𝑧) = 0 ∧ (2nd𝑘) = 0) → 𝑧 = 𝑘))
202 fveqeq2 6446 . . 3 (𝑧 = 𝑘 → ((2nd𝑧) = 0 ↔ (2nd𝑘) = 0))
203202rmo4 3624 . 2 (∃*𝑧𝑆 (2nd𝑧) = 0 ↔ ∀𝑧𝑆𝑘𝑆 (((2nd𝑧) = 0 ∧ (2nd𝑘) = 0) → 𝑧 = 𝑘))
204201, 203sylibr 226 1 (𝜑 → ∃*𝑧𝑆 (2nd𝑧) = 0)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1656   ∈ wcel 2164  {cab 2811   ≠ wne 2999  ∀wral 3117  ∃*wrmo 3120  {crab 3121  ⦋csb 3757   ∖ cdif 3795   ∪ cun 3796   ∩ cin 3797  ∅c0 4146  ifcif 4308  {csn 4399   class class class wbr 4875   ↦ cmpt 4954   × cxp 5344  ◡ccnv 5345   “ cima 5349  Fun wfun 6121   Fn wfn 6122  ⟶wf 6123  –onto→wfo 6125  –1-1-onto→wf1o 6126  ‘cfv 6127  (class class class)co 6910   ∘𝑓 cof 7160  1st c1st 7431  2nd c2nd 7432   ↑𝑚 cmap 8127  ℂcc 10257  ℝcr 10258  0cc0 10259  1c1 10260   + caddc 10262   < clt 10398   ≤ cle 10399   − cmin 10592  ℕcn 11357  ℕ0cn0 11625  ℤcz 11711  ℤ≥cuz 11975  ...cfz 12626  ..^cfzo 12767 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-cnex 10315  ax-resscn 10316  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-addrcl 10320  ax-mulcl 10321  ax-mulrcl 10322  ax-mulcom 10323  ax-addass 10324  ax-mulass 10325  ax-distr 10326  ax-i2m1 10327  ax-1ne0 10328  ax-1rid 10329  ax-rnegex 10330  ax-rrecex 10331  ax-cnre 10332  ax-pre-lttri 10333  ax-pre-lttrn 10334  ax-pre-ltadd 10335  ax-pre-mulgt0 10336 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-of 7162  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-er 8014  df-map 8129  df-en 8229  df-dom 8230  df-sdom 8231  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-sub 10594  df-neg 10595  df-nn 11358  df-n0 11626  df-z 11712  df-uz 11976  df-fz 12627  df-fzo 12768 This theorem is referenced by:  poimirlem18  33970  poimirlem21  33973
 Copyright terms: Public domain W3C validator