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Theorem poimirlem13 34899
Description: Lemma for poimir 34919- 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 724 . . . . . . . 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 724 . . . . . . . 8 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))
6 simplrl 775 . . . . . . . 8 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → 𝑧𝑆)
7 simprl 769 . . . . . . . 8 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (2nd𝑧) = 0)
82, 3, 5, 6, 7poimirlem10 34896 . . . . . . 7 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → ((𝐹‘(𝑁 − 1)) ∘f − ((1...𝑁) × {1})) = (1st ‘(1st𝑧)))
9 simplrr 776 . . . . . . . 8 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → 𝑘𝑆)
10 simprr 771 . . . . . . . 8 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (2nd𝑘) = 0)
112, 3, 5, 9, 10poimirlem10 34896 . . . . . . 7 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → ((𝐹‘(𝑁 − 1)) ∘f − ((1...𝑁) × {1})) = (1st ‘(1st𝑘)))
128, 11eqtr3d 2858 . . . . . 6 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (1st ‘(1st𝑧)) = (1st ‘(1st𝑘)))
13 elrabi 3674 . . . . . . . . . . . . . 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 2931 . . . . . . . . . . . . 13 (𝑧𝑆𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
15 xp1st 7715 . . . . . . . . . . . . 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 7716 . . . . . . . . . . . 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 6677 . . . . . . . . . . . 12 (2nd ‘(1st𝑧)) ∈ V
20 f1oeq1 6598 . . . . . . . . . . . 12 (𝑓 = (2nd ‘(1st𝑧)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁)))
2119, 20elab 3666 . . . . . . . . . . 11 ((2nd ‘(1st𝑧)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
2218, 21sylib 220 . . . . . . . . . 10 (𝑧𝑆 → (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
23 f1ofn 6610 . . . . . . . . . 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 483 . . . . . . . 8 ((𝑧𝑆𝑘𝑆) → (2nd ‘(1st𝑧)) Fn (1...𝑁))
2625ad2antlr 725 . . . . . . 7 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (2nd ‘(1st𝑧)) Fn (1...𝑁))
27 elrabi 3674 . . . . . . . . . . . . . 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 2931 . . . . . . . . . . . . 13 (𝑘𝑆𝑘 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
29 xp1st 7715 . . . . . . . . . . . . 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 7716 . . . . . . . . . . . 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 6677 . . . . . . . . . . . 12 (2nd ‘(1st𝑘)) ∈ V
34 f1oeq1 6598 . . . . . . . . . . . 12 (𝑓 = (2nd ‘(1st𝑘)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑘)):(1...𝑁)–1-1-onto→(1...𝑁)))
3533, 34elab 3666 . . . . . . . . . . 11 ((2nd ‘(1st𝑘)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑘)):(1...𝑁)–1-1-onto→(1...𝑁))
3632, 35sylib 220 . . . . . . . . . 10 (𝑘𝑆 → (2nd ‘(1st𝑘)):(1...𝑁)–1-1-onto→(1...𝑁))
37 f1ofn 6610 . . . . . . . . . 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 484 . . . . . . . 8 ((𝑧𝑆𝑘𝑆) → (2nd ‘(1st𝑘)) Fn (1...𝑁))
4039ad2antlr 725 . . . . . . 7 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (2nd ‘(1st𝑘)) Fn (1...𝑁))
41 eleq1 2900 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → (𝑚 ∈ (1...𝑁) ↔ 𝑛 ∈ (1...𝑁)))
4241anbi2d 630 . . . . . . . . . . . 12 (𝑚 = 𝑛 → ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) ↔ (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁))))
43 oveq2 7158 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
4443imaeq2d 5923 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → ((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑧)) “ (1...𝑛)))
4543imaeq2d 5923 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → ((2nd ‘(1st𝑘)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...𝑛)))
4644, 45eqeq12d 2837 . . . . . . . . . . . 12 (𝑚 = 𝑛 → (((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...𝑚)) ↔ ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑛))))
4742, 46imbi12d 347 . . . . . . . . . . 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 728 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑁 ∈ ℕ)
494ad3antrrr 728 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))
50 simpl 485 . . . . . . . . . . . . . 14 ((𝑧𝑆𝑘𝑆) → 𝑧𝑆)
5150ad3antlr 729 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑧𝑆)
52 simplrl 775 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → (2nd𝑧) = 0)
53 simpr 487 . . . . . . . . . . . . . 14 ((𝑧𝑆𝑘𝑆) → 𝑘𝑆)
5453ad3antlr 729 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑘𝑆)
55 simplrr 776 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → (2nd𝑘) = 0)
56 simpr 487 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑚 ∈ (1...𝑁))
5748, 3, 49, 51, 52, 54, 55, 56poimirlem11 34897 . . . . . . . . . . . 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 34897 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → ((2nd ‘(1st𝑘)) “ (1...𝑚)) ⊆ ((2nd ‘(1st𝑧)) “ (1...𝑚)))
5957, 58eqssd 3983 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...𝑚)))
6047, 59chvarvv 2001 . . . . . . . . . 10 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑛)))
61 simpll 765 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → 𝜑)
62 elfznn 12930 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℕ)
63 nnm1nn0 11932 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
6462, 63syl 17 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ∈ ℕ0)
6564adantr 483 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1) → (𝑛 − 1) ∈ ℕ0)
6662nncnd 11648 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℂ)
67 ax-1cn 10589 . . . . . . . . . . . . . . . . . . . 20 1 ∈ ℂ
68 subeq0 10906 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) = 0 ↔ 𝑛 = 1))
6966, 67, 68sylancl 588 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) = 0 ↔ 𝑛 = 1))
7069necon3abid 3052 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) ≠ 0 ↔ ¬ 𝑛 = 1))
7170biimpar 480 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1) → (𝑛 − 1) ≠ 0)
72 elnnne0 11905 . . . . . . . . . . . . . . . . 17 ((𝑛 − 1) ∈ ℕ ↔ ((𝑛 − 1) ∈ ℕ0 ∧ (𝑛 − 1) ≠ 0))
7365, 71, 72sylanbrc 585 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1) → (𝑛 − 1) ∈ ℕ)
7473adantl 484 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → (𝑛 − 1) ∈ ℕ)
7564nn0red 11950 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ∈ ℝ)
7675adantl 484 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ∈ ℝ)
7762nnred 11647 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℝ)
7877adantl 484 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℝ)
791nnred 11647 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ ℝ)
8079adantr 483 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (1...𝑁)) → 𝑁 ∈ ℝ)
8177lem1d 11567 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ≤ 𝑛)
8281adantl 484 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ≤ 𝑛)
83 elfzle2 12905 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...𝑁) → 𝑛𝑁)
8483adantl 484 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (1...𝑁)) → 𝑛𝑁)
8576, 78, 80, 82, 84letrd 10791 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ≤ 𝑁)
8685adantrr 715 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → (𝑛 − 1) ≤ 𝑁)
871nnzd 12080 . . . . . . . . . . . . . . . . 17 (𝜑𝑁 ∈ ℤ)
88 fznn 12969 . . . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → ((𝑛 − 1) ∈ (1...𝑁) ↔ ((𝑛 − 1) ∈ ℕ ∧ (𝑛 − 1) ≤ 𝑁)))
9174, 86, 90mpbir2and 711 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → (𝑛 − 1) ∈ (1...𝑁))
9261, 91sylan 582 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → (𝑛 − 1) ∈ (1...𝑁))
93 ovex 7183 . . . . . . . . . . . . . 14 (𝑛 − 1) ∈ V
94 eleq1 2900 . . . . . . . . . . . . . . . 16 (𝑚 = (𝑛 − 1) → (𝑚 ∈ (1...𝑁) ↔ (𝑛 − 1) ∈ (1...𝑁)))
9594anbi2d 630 . . . . . . . . . . . . . . 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 7158 . . . . . . . . . . . . . . . . 17 (𝑚 = (𝑛 − 1) → (1...𝑚) = (1...(𝑛 − 1)))
9796imaeq2d 5923 . . . . . . . . . . . . . . . 16 (𝑚 = (𝑛 − 1) → ((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))))
9896imaeq2d 5923 . . . . . . . . . . . . . . . 16 (𝑚 = (𝑛 − 1) → ((2nd ‘(1st𝑘)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))
9997, 98eqeq12d 2837 . . . . . . . . . . . . . . 15 (𝑚 = (𝑛 − 1) → (((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...𝑚)) ↔ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1)))))
10095, 99imbi12d 347 . . . . . . . . . . . . . 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 3559 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ (𝑛 − 1) ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))
10292, 101syldan 593 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))
103102expr 459 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → (¬ 𝑛 = 1 → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1)))))
104 ima0 5939 . . . . . . . . . . . . 13 ((2nd ‘(1st𝑧)) “ ∅) = ∅
105 ima0 5939 . . . . . . . . . . . . 13 ((2nd ‘(1st𝑘)) “ ∅) = ∅
106104, 105eqtr4i 2847 . . . . . . . . . . . 12 ((2nd ‘(1st𝑧)) “ ∅) = ((2nd ‘(1st𝑘)) “ ∅)
107 oveq1 7157 . . . . . . . . . . . . . . . 16 (𝑛 = 1 → (𝑛 − 1) = (1 − 1))
108 1m1e0 11703 . . . . . . . . . . . . . . . 16 (1 − 1) = 0
109107, 108syl6eq 2872 . . . . . . . . . . . . . . 15 (𝑛 = 1 → (𝑛 − 1) = 0)
110109oveq2d 7166 . . . . . . . . . . . . . 14 (𝑛 = 1 → (1...(𝑛 − 1)) = (1...0))
111 fz10 12922 . . . . . . . . . . . . . 14 (1...0) = ∅
112110, 111syl6eq 2872 . . . . . . . . . . . . 13 (𝑛 = 1 → (1...(𝑛 − 1)) = ∅)
113112imaeq2d 5923 . . . . . . . . . . . 12 (𝑛 = 1 → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑧)) “ ∅))
114112imaeq2d 5923 . . . . . . . . . . . 12 (𝑛 = 1 → ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ ∅))
115106, 113, 1143eqtr4a 2882 . . . . . . . . . . 11 (𝑛 = 1 → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))
116103, 115pm2.61d2 183 . . . . . . . . . 10 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))
11760, 116difeq12d 4099 . . . . . . . . 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 6737 . . . . . . . . . . . 12 (((2nd ‘(1st𝑧)) Fn (1...𝑁) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑧))‘𝑛)} = ((2nd ‘(1st𝑧)) “ {𝑛}))
11924, 118sylan 582 . . . . . . . . . . 11 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑧))‘𝑛)} = ((2nd ‘(1st𝑧)) “ {𝑛}))
12062adantl 484 . . . . . . . . . . . . 13 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℕ)
121 uncom 4128 . . . . . . . . . . . . . . . 16 ((1...(𝑛 − 1)) ∪ {𝑛}) = ({𝑛} ∪ (1...(𝑛 − 1)))
122121difeq1i 4094 . . . . . . . . . . . . . . 15 (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))) = (({𝑛} ∪ (1...(𝑛 − 1))) ∖ (1...(𝑛 − 1)))
123 difun2 4428 . . . . . . . . . . . . . . 15 (({𝑛} ∪ (1...(𝑛 − 1))) ∖ (1...(𝑛 − 1))) = ({𝑛} ∖ (1...(𝑛 − 1)))
124122, 123eqtri 2844 . . . . . . . . . . . . . 14 (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))) = ({𝑛} ∖ (1...(𝑛 − 1)))
125 nncn 11640 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
126 npcan1 11059 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℂ → ((𝑛 − 1) + 1) = 𝑛)
127125, 126syl 17 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) = 𝑛)
128 elnnuz 12276 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ‘1))
129128biimpi 218 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ‘1))
130127, 129eqeltrd 2913 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) ∈ (ℤ‘1))
13163nn0zd 12079 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℤ)
132 uzid 12252 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 − 1) ∈ ℤ → (𝑛 − 1) ∈ (ℤ‘(𝑛 − 1)))
133131, 132syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ (ℤ‘(𝑛 − 1)))
134 peano2uz 12295 . . . . . . . . . . . . . . . . . . 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 2914 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ‘(𝑛 − 1)))
137 fzsplit2 12926 . . . . . . . . . . . . . . . . 17 ((((𝑛 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑛 ∈ (ℤ‘(𝑛 − 1))) → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)))
138130, 136, 137syl2anc 586 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)))
139127oveq1d 7165 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → (((𝑛 − 1) + 1)...𝑛) = (𝑛...𝑛))
140 nnz 11998 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
141 fzsn 12943 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℤ → (𝑛...𝑛) = {𝑛})
142140, 141syl 17 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → (𝑛...𝑛) = {𝑛})
143139, 142eqtrd 2856 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ → (((𝑛 − 1) + 1)...𝑛) = {𝑛})
144143uneq2d 4138 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)) = ((1...(𝑛 − 1)) ∪ {𝑛}))
145138, 144eqtrd 2856 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → (1...𝑛) = ((1...(𝑛 − 1)) ∪ {𝑛}))
146145difeq1d 4097 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → ((1...𝑛) ∖ (1...(𝑛 − 1))) = (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))))
147 nnre 11639 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
148 ltm1 11476 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℝ → (𝑛 − 1) < 𝑛)
149 peano2rem 10947 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℝ → (𝑛 − 1) ∈ ℝ)
150 ltnle 10714 . . . . . . . . . . . . . . . . . . 19 (((𝑛 − 1) ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((𝑛 − 1) < 𝑛 ↔ ¬ 𝑛 ≤ (𝑛 − 1)))
151149, 150mpancom 686 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℝ → ((𝑛 − 1) < 𝑛 ↔ ¬ 𝑛 ≤ (𝑛 − 1)))
152148, 151mpbid 234 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℝ → ¬ 𝑛 ≤ (𝑛 − 1))
153 elfzle2 12905 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (1...(𝑛 − 1)) → 𝑛 ≤ (𝑛 − 1))
154152, 153nsyl 142 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℝ → ¬ 𝑛 ∈ (1...(𝑛 − 1)))
155147, 154syl 17 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → ¬ 𝑛 ∈ (1...(𝑛 − 1)))
156 incom 4177 . . . . . . . . . . . . . . . . 17 ((1...(𝑛 − 1)) ∩ {𝑛}) = ({𝑛} ∩ (1...(𝑛 − 1)))
157156eqeq1i 2826 . . . . . . . . . . . . . . . 16 (((1...(𝑛 − 1)) ∩ {𝑛}) = ∅ ↔ ({𝑛} ∩ (1...(𝑛 − 1))) = ∅)
158 disjsn 4640 . . . . . . . . . . . . . . . 16 (((1...(𝑛 − 1)) ∩ {𝑛}) = ∅ ↔ ¬ 𝑛 ∈ (1...(𝑛 − 1)))
159 disj3 4402 . . . . . . . . . . . . . . . 16 (({𝑛} ∩ (1...(𝑛 − 1))) = ∅ ↔ {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1))))
160157, 158, 1593bitr3i 303 . . . . . . . . . . . . . . 15 𝑛 ∈ (1...(𝑛 − 1)) ↔ {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1))))
161155, 160sylib 220 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1))))
162124, 146, 1613eqtr4a 2882 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → ((1...𝑛) ∖ (1...(𝑛 − 1))) = {𝑛})
163120, 162syl 17 . . . . . . . . . . . 12 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → ((1...𝑛) ∖ (1...(𝑛 − 1))) = {𝑛})
164163imaeq2d 5923 . . . . . . . . . . 11 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = ((2nd ‘(1st𝑧)) “ {𝑛}))
165 dff1o3 6615 . . . . . . . . . . . . . . 15 ((2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd ‘(1st𝑧)):(1...𝑁)–onto→(1...𝑁) ∧ Fun (2nd ‘(1st𝑧))))
166165simprbi 499 . . . . . . . . . . . . . 14 ((2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun (2nd ‘(1st𝑧)))
16722, 166syl 17 . . . . . . . . . . . . 13 (𝑧𝑆 → Fun (2nd ‘(1st𝑧)))
168 imadif 6432 . . . . . . . . . . . . 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 483 . . . . . . . . . . 11 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = (((2nd ‘(1st𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1)))))
171119, 164, 1703eqtr2d 2862 . . . . . . . . . 10 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑧))‘𝑛)} = (((2nd ‘(1st𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1)))))
1726, 171sylan 582 . . . . . . . . 9 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑧))‘𝑛)} = (((2nd ‘(1st𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1)))))
173 eleq1 2900 . . . . . . . . . . . . 13 (𝑧 = 𝑘 → (𝑧𝑆𝑘𝑆))
174173anbi1d 631 . . . . . . . . . . . 12 (𝑧 = 𝑘 → ((𝑧𝑆𝑛 ∈ (1...𝑁)) ↔ (𝑘𝑆𝑛 ∈ (1...𝑁))))
175 2fveq3 6669 . . . . . . . . . . . . . . 15 (𝑧 = 𝑘 → (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑘)))
176175fveq1d 6666 . . . . . . . . . . . . . 14 (𝑧 = 𝑘 → ((2nd ‘(1st𝑧))‘𝑛) = ((2nd ‘(1st𝑘))‘𝑛))
177176sneqd 4572 . . . . . . . . . . . . 13 (𝑧 = 𝑘 → {((2nd ‘(1st𝑧))‘𝑛)} = {((2nd ‘(1st𝑘))‘𝑛)})
178175imaeq1d 5922 . . . . . . . . . . . . . 14 (𝑧 = 𝑘 → ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑛)))
179175imaeq1d 5922 . . . . . . . . . . . . . 14 (𝑧 = 𝑘 → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))
180178, 179difeq12d 4099 . . . . . . . . . . . . 13 (𝑧 = 𝑘 → (((2nd ‘(1st𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1)))) = (((2nd ‘(1st𝑘)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1)))))
181177, 180eqeq12d 2837 . . . . . . . . . . . 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 347 . . . . . . . . . . 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, 171chvarvv 2001 . . . . . . . . . 10 ((𝑘𝑆𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑘))‘𝑛)} = (((2nd ‘(1st𝑘)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1)))))
1849, 183sylan 582 . . . . . . . . 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 2866 . . . . . . . 8 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑧))‘𝑛)} = {((2nd ‘(1st𝑘))‘𝑛)})
186 fvex 6677 . . . . . . . . 9 ((2nd ‘(1st𝑧))‘𝑛) ∈ V
187186sneqr 4764 . . . . . . . 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 6799 . . . . . 6 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑘)))
190 xpopth 7724 . . . . . . . 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 597 . . . . . . 7 ((𝑧𝑆𝑘𝑆) → (((1st ‘(1st𝑧)) = (1st ‘(1st𝑘)) ∧ (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑘))) ↔ (1st𝑧) = (1st𝑘)))
192191ad2antlr 725 . . . . . 6 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (((1st ‘(1st𝑧)) = (1st ‘(1st𝑘)) ∧ (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑘))) ↔ (1st𝑧) = (1st𝑘)))
19312, 189, 192mpbi2and 710 . . . . 5 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (1st𝑧) = (1st𝑘))
194 eqtr3 2843 . . . . . 6 (((2nd𝑧) = 0 ∧ (2nd𝑘) = 0) → (2nd𝑧) = (2nd𝑘))
195194adantl 484 . . . . 5 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (2nd𝑧) = (2nd𝑘))
196 xpopth 7724 . . . . . . 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 597 . . . . . 6 ((𝑧𝑆𝑘𝑆) → (((1st𝑧) = (1st𝑘) ∧ (2nd𝑧) = (2nd𝑘)) ↔ 𝑧 = 𝑘))
198197ad2antlr 725 . . . . 5 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (((1st𝑧) = (1st𝑘) ∧ (2nd𝑧) = (2nd𝑘)) ↔ 𝑧 = 𝑘))
199193, 195, 198mpbi2and 710 . . . 4 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → 𝑧 = 𝑘)
200199ex 415 . . 3 ((𝜑 ∧ (𝑧𝑆𝑘𝑆)) → (((2nd𝑧) = 0 ∧ (2nd𝑘) = 0) → 𝑧 = 𝑘))
201200ralrimivva 3191 . 2 (𝜑 → ∀𝑧𝑆𝑘𝑆 (((2nd𝑧) = 0 ∧ (2nd𝑘) = 0) → 𝑧 = 𝑘))
202 fveqeq2 6673 . . 3 (𝑧 = 𝑘 → ((2nd𝑧) = 0 ↔ (2nd𝑘) = 0))
203202rmo4 3720 . 2 (∃*𝑧𝑆 (2nd𝑧) = 0 ↔ ∀𝑧𝑆𝑘𝑆 (((2nd𝑧) = 0 ∧ (2nd𝑘) = 0) → 𝑧 = 𝑘))
204201, 203sylibr 236 1 (𝜑 → ∃*𝑧𝑆 (2nd𝑧) = 0)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398   = wceq 1533  wcel 2110  {cab 2799  wne 3016  wral 3138  ∃*wrmo 3141  {crab 3142  csb 3882  cdif 3932  cun 3933  cin 3934  c0 4290  ifcif 4466  {csn 4560   class class class wbr 5058  cmpt 5138   × cxp 5547  ccnv 5548  cima 5552  Fun wfun 6343   Fn wfn 6344  wf 6345  ontowfo 6347  1-1-ontowf1o 6348  cfv 6349  (class class class)co 7150  f cof 7401  1st c1st 7681  2nd c2nd 7682  m cmap 8400  cc 10529  cr 10530  0cc0 10531  1c1 10532   + caddc 10534   < clt 10669  cle 10670  cmin 10864  cn 11632  0cn0 11891  cz 11975  cuz 12237  ...cfz 12886  ..^cfzo 13027
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-of 7403  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8283  df-map 8402  df-en 8504  df-dom 8505  df-sdom 8506  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-n0 11892  df-z 11976  df-uz 12238  df-fz 12887  df-fzo 13028
This theorem is referenced by:  poimirlem18  34904  poimirlem21  34907
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