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Theorem poimirlem13 33552
Description: Lemma for poimir 33572- 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 762 . . . . . . . 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 762 . . . . . . . 8 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)))
6 simplrl 817 . . . . . . . 8 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → 𝑧𝑆)
7 simprl 809 . . . . . . . 8 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (2nd𝑧) = 0)
82, 3, 5, 6, 7poimirlem10 33549 . . . . . . 7 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → ((𝐹‘(𝑁 − 1)) ∘𝑓 − ((1...𝑁) × {1})) = (1st ‘(1st𝑧)))
9 simplrr 818 . . . . . . . 8 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → 𝑘𝑆)
10 simprr 811 . . . . . . . 8 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (2nd𝑘) = 0)
112, 3, 5, 9, 10poimirlem10 33549 . . . . . . 7 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → ((𝐹‘(𝑁 − 1)) ∘𝑓 − ((1...𝑁) × {1})) = (1st ‘(1st𝑘)))
128, 11eqtr3d 2687 . . . . . 6 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (1st ‘(1st𝑧)) = (1st ‘(1st𝑘)))
13 elrabi 3391 . . . . . . . . . . . . . 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 2748 . . . . . . . . . . . . 13 (𝑧𝑆𝑧 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
15 xp1st 7242 . . . . . . . . . . . . 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 7243 . . . . . . . . . . . 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 6239 . . . . . . . . . . . 12 (2nd ‘(1st𝑧)) ∈ V
20 f1oeq1 6165 . . . . . . . . . . . 12 (𝑓 = (2nd ‘(1st𝑧)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁)))
2119, 20elab 3382 . . . . . . . . . . 11 ((2nd ‘(1st𝑧)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
2218, 21sylib 208 . . . . . . . . . 10 (𝑧𝑆 → (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
23 f1ofn 6176 . . . . . . . . . 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 480 . . . . . . . 8 ((𝑧𝑆𝑘𝑆) → (2nd ‘(1st𝑧)) Fn (1...𝑁))
2625ad2antlr 763 . . . . . . 7 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (2nd ‘(1st𝑧)) Fn (1...𝑁))
27 elrabi 3391 . . . . . . . . . . . . . 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 2748 . . . . . . . . . . . . 13 (𝑘𝑆𝑘 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
29 xp1st 7242 . . . . . . . . . . . . 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 7243 . . . . . . . . . . . 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 6239 . . . . . . . . . . . 12 (2nd ‘(1st𝑘)) ∈ V
34 f1oeq1 6165 . . . . . . . . . . . 12 (𝑓 = (2nd ‘(1st𝑘)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑘)):(1...𝑁)–1-1-onto→(1...𝑁)))
3533, 34elab 3382 . . . . . . . . . . 11 ((2nd ‘(1st𝑘)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑘)):(1...𝑁)–1-1-onto→(1...𝑁))
3632, 35sylib 208 . . . . . . . . . 10 (𝑘𝑆 → (2nd ‘(1st𝑘)):(1...𝑁)–1-1-onto→(1...𝑁))
37 f1ofn 6176 . . . . . . . . . 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 481 . . . . . . . 8 ((𝑧𝑆𝑘𝑆) → (2nd ‘(1st𝑘)) Fn (1...𝑁))
4039ad2antlr 763 . . . . . . 7 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (2nd ‘(1st𝑘)) Fn (1...𝑁))
41 eleq1 2718 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → (𝑚 ∈ (1...𝑁) ↔ 𝑛 ∈ (1...𝑁)))
4241anbi2d 740 . . . . . . . . . . . 12 (𝑚 = 𝑛 → ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) ↔ (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁))))
43 oveq2 6698 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
4443imaeq2d 5501 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → ((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑧)) “ (1...𝑛)))
4543imaeq2d 5501 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → ((2nd ‘(1st𝑘)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...𝑛)))
4644, 45eqeq12d 2666 . . . . . . . . . . . 12 (𝑚 = 𝑛 → (((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...𝑚)) ↔ ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑛))))
4742, 46imbi12d 333 . . . . . . . . . . 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 766 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑁 ∈ ℕ)
494ad3antrrr 766 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)))
50 simpl 472 . . . . . . . . . . . . . 14 ((𝑧𝑆𝑘𝑆) → 𝑧𝑆)
5150ad3antlr 767 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑧𝑆)
52 simplrl 817 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → (2nd𝑧) = 0)
53 simpr 476 . . . . . . . . . . . . . 14 ((𝑧𝑆𝑘𝑆) → 𝑘𝑆)
5453ad3antlr 767 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑘𝑆)
55 simplrr 818 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → (2nd𝑘) = 0)
56 simpr 476 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑚 ∈ (1...𝑁))
5748, 3, 49, 51, 52, 54, 55, 56poimirlem11 33550 . . . . . . . . . . . 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 33550 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → ((2nd ‘(1st𝑘)) “ (1...𝑚)) ⊆ ((2nd ‘(1st𝑧)) “ (1...𝑚)))
5957, 58eqssd 3653 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...𝑚)))
6047, 59chvarv 2299 . . . . . . . . . 10 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑛)))
61 simpll 805 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → 𝜑)
62 elfznn 12408 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℕ)
63 nnm1nn0 11372 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
6462, 63syl 17 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ∈ ℕ0)
6564adantr 480 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1) → (𝑛 − 1) ∈ ℕ0)
6662nncnd 11074 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℂ)
67 ax-1cn 10032 . . . . . . . . . . . . . . . . . . . 20 1 ∈ ℂ
68 subeq0 10345 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) = 0 ↔ 𝑛 = 1))
6966, 67, 68sylancl 695 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) = 0 ↔ 𝑛 = 1))
7069necon3abid 2859 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) ≠ 0 ↔ ¬ 𝑛 = 1))
7170biimpar 501 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1) → (𝑛 − 1) ≠ 0)
72 elnnne0 11344 . . . . . . . . . . . . . . . . 17 ((𝑛 − 1) ∈ ℕ ↔ ((𝑛 − 1) ∈ ℕ0 ∧ (𝑛 − 1) ≠ 0))
7365, 71, 72sylanbrc 699 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1) → (𝑛 − 1) ∈ ℕ)
7473adantl 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → (𝑛 − 1) ∈ ℕ)
7564nn0red 11390 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ∈ ℝ)
7675adantl 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ∈ ℝ)
7762nnred 11073 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℝ)
7877adantl 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℝ)
791nnred 11073 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ ℝ)
8079adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (1...𝑁)) → 𝑁 ∈ ℝ)
8177lem1d 10995 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ≤ 𝑛)
8281adantl 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ≤ 𝑛)
83 elfzle2 12383 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...𝑁) → 𝑛𝑁)
8483adantl 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (1...𝑁)) → 𝑛𝑁)
8576, 78, 80, 82, 84letrd 10232 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ≤ 𝑁)
8685adantrr 753 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → (𝑛 − 1) ≤ 𝑁)
871nnzd 11519 . . . . . . . . . . . . . . . . 17 (𝜑𝑁 ∈ ℤ)
88 fznn 12446 . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → ((𝑛 − 1) ∈ (1...𝑁) ↔ ((𝑛 − 1) ∈ ℕ ∧ (𝑛 − 1) ≤ 𝑁)))
9174, 86, 90mpbir2and 977 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → (𝑛 − 1) ∈ (1...𝑁))
9261, 91sylan 487 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → (𝑛 − 1) ∈ (1...𝑁))
93 ovex 6718 . . . . . . . . . . . . . 14 (𝑛 − 1) ∈ V
94 eleq1 2718 . . . . . . . . . . . . . . . 16 (𝑚 = (𝑛 − 1) → (𝑚 ∈ (1...𝑁) ↔ (𝑛 − 1) ∈ (1...𝑁)))
9594anbi2d 740 . . . . . . . . . . . . . . 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 6698 . . . . . . . . . . . . . . . . 17 (𝑚 = (𝑛 − 1) → (1...𝑚) = (1...(𝑛 − 1)))
9796imaeq2d 5501 . . . . . . . . . . . . . . . 16 (𝑚 = (𝑛 − 1) → ((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))))
9896imaeq2d 5501 . . . . . . . . . . . . . . . 16 (𝑚 = (𝑛 − 1) → ((2nd ‘(1st𝑘)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))
9997, 98eqeq12d 2666 . . . . . . . . . . . . . . 15 (𝑚 = (𝑛 − 1) → (((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...𝑚)) ↔ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1)))))
10095, 99imbi12d 333 . . . . . . . . . . . . . 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 3290 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ (𝑛 − 1) ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))
10292, 101syldan 486 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))
103102expr 642 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → (¬ 𝑛 = 1 → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1)))))
104 ima0 5516 . . . . . . . . . . . . 13 ((2nd ‘(1st𝑧)) “ ∅) = ∅
105 ima0 5516 . . . . . . . . . . . . 13 ((2nd ‘(1st𝑘)) “ ∅) = ∅
106104, 105eqtr4i 2676 . . . . . . . . . . . 12 ((2nd ‘(1st𝑧)) “ ∅) = ((2nd ‘(1st𝑘)) “ ∅)
107 oveq1 6697 . . . . . . . . . . . . . . . 16 (𝑛 = 1 → (𝑛 − 1) = (1 − 1))
108 1m1e0 11127 . . . . . . . . . . . . . . . 16 (1 − 1) = 0
109107, 108syl6eq 2701 . . . . . . . . . . . . . . 15 (𝑛 = 1 → (𝑛 − 1) = 0)
110109oveq2d 6706 . . . . . . . . . . . . . 14 (𝑛 = 1 → (1...(𝑛 − 1)) = (1...0))
111 fz10 12400 . . . . . . . . . . . . . 14 (1...0) = ∅
112110, 111syl6eq 2701 . . . . . . . . . . . . 13 (𝑛 = 1 → (1...(𝑛 − 1)) = ∅)
113112imaeq2d 5501 . . . . . . . . . . . 12 (𝑛 = 1 → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑧)) “ ∅))
114112imaeq2d 5501 . . . . . . . . . . . 12 (𝑛 = 1 → ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ ∅))
115106, 113, 1143eqtr4a 2711 . . . . . . . . . . 11 (𝑛 = 1 → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))
116103, 115pm2.61d2 172 . . . . . . . . . 10 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))
11760, 116difeq12d 3762 . . . . . . . . 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 6297 . . . . . . . . . . . 12 (((2nd ‘(1st𝑧)) Fn (1...𝑁) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑧))‘𝑛)} = ((2nd ‘(1st𝑧)) “ {𝑛}))
11924, 118sylan 487 . . . . . . . . . . 11 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑧))‘𝑛)} = ((2nd ‘(1st𝑧)) “ {𝑛}))
12062adantl 481 . . . . . . . . . . . . 13 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℕ)
121 uncom 3790 . . . . . . . . . . . . . . . 16 ((1...(𝑛 − 1)) ∪ {𝑛}) = ({𝑛} ∪ (1...(𝑛 − 1)))
122121difeq1i 3757 . . . . . . . . . . . . . . 15 (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))) = (({𝑛} ∪ (1...(𝑛 − 1))) ∖ (1...(𝑛 − 1)))
123 difun2 4081 . . . . . . . . . . . . . . 15 (({𝑛} ∪ (1...(𝑛 − 1))) ∖ (1...(𝑛 − 1))) = ({𝑛} ∖ (1...(𝑛 − 1)))
124122, 123eqtri 2673 . . . . . . . . . . . . . 14 (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))) = ({𝑛} ∖ (1...(𝑛 − 1)))
125 nncn 11066 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
126 npcan1 10493 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℂ → ((𝑛 − 1) + 1) = 𝑛)
127125, 126syl 17 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) = 𝑛)
128 elnnuz 11762 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ‘1))
129128biimpi 206 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ‘1))
130127, 129eqeltrd 2730 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) ∈ (ℤ‘1))
13163nn0zd 11518 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℤ)
132 uzid 11740 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 − 1) ∈ ℤ → (𝑛 − 1) ∈ (ℤ‘(𝑛 − 1)))
133131, 132syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ (ℤ‘(𝑛 − 1)))
134 peano2uz 11779 . . . . . . . . . . . . . . . . . . 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 2731 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ‘(𝑛 − 1)))
137 fzsplit2 12404 . . . . . . . . . . . . . . . . 17 ((((𝑛 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑛 ∈ (ℤ‘(𝑛 − 1))) → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)))
138130, 136, 137syl2anc 694 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)))
139127oveq1d 6705 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → (((𝑛 − 1) + 1)...𝑛) = (𝑛...𝑛))
140 nnz 11437 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
141 fzsn 12421 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℤ → (𝑛...𝑛) = {𝑛})
142140, 141syl 17 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → (𝑛...𝑛) = {𝑛})
143139, 142eqtrd 2685 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ → (((𝑛 − 1) + 1)...𝑛) = {𝑛})
144143uneq2d 3800 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)) = ((1...(𝑛 − 1)) ∪ {𝑛}))
145138, 144eqtrd 2685 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → (1...𝑛) = ((1...(𝑛 − 1)) ∪ {𝑛}))
146145difeq1d 3760 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → ((1...𝑛) ∖ (1...(𝑛 − 1))) = (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))))
147 nnre 11065 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
148 ltm1 10901 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℝ → (𝑛 − 1) < 𝑛)
149 peano2rem 10386 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℝ → (𝑛 − 1) ∈ ℝ)
150 ltnle 10155 . . . . . . . . . . . . . . . . . . 19 (((𝑛 − 1) ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((𝑛 − 1) < 𝑛 ↔ ¬ 𝑛 ≤ (𝑛 − 1)))
151149, 150mpancom 704 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℝ → ((𝑛 − 1) < 𝑛 ↔ ¬ 𝑛 ≤ (𝑛 − 1)))
152148, 151mpbid 222 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℝ → ¬ 𝑛 ≤ (𝑛 − 1))
153 elfzle2 12383 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (1...(𝑛 − 1)) → 𝑛 ≤ (𝑛 − 1))
154152, 153nsyl 135 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℝ → ¬ 𝑛 ∈ (1...(𝑛 − 1)))
155147, 154syl 17 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → ¬ 𝑛 ∈ (1...(𝑛 − 1)))
156 incom 3838 . . . . . . . . . . . . . . . . 17 ((1...(𝑛 − 1)) ∩ {𝑛}) = ({𝑛} ∩ (1...(𝑛 − 1)))
157156eqeq1i 2656 . . . . . . . . . . . . . . . 16 (((1...(𝑛 − 1)) ∩ {𝑛}) = ∅ ↔ ({𝑛} ∩ (1...(𝑛 − 1))) = ∅)
158 disjsn 4278 . . . . . . . . . . . . . . . 16 (((1...(𝑛 − 1)) ∩ {𝑛}) = ∅ ↔ ¬ 𝑛 ∈ (1...(𝑛 − 1)))
159 disj3 4054 . . . . . . . . . . . . . . . 16 (({𝑛} ∩ (1...(𝑛 − 1))) = ∅ ↔ {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1))))
160157, 158, 1593bitr3i 290 . . . . . . . . . . . . . . 15 𝑛 ∈ (1...(𝑛 − 1)) ↔ {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1))))
161155, 160sylib 208 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1))))
162124, 146, 1613eqtr4a 2711 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → ((1...𝑛) ∖ (1...(𝑛 − 1))) = {𝑛})
163120, 162syl 17 . . . . . . . . . . . 12 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → ((1...𝑛) ∖ (1...(𝑛 − 1))) = {𝑛})
164163imaeq2d 5501 . . . . . . . . . . 11 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = ((2nd ‘(1st𝑧)) “ {𝑛}))
165 dff1o3 6181 . . . . . . . . . . . . . . 15 ((2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd ‘(1st𝑧)):(1...𝑁)–onto→(1...𝑁) ∧ Fun (2nd ‘(1st𝑧))))
166165simprbi 479 . . . . . . . . . . . . . 14 ((2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun (2nd ‘(1st𝑧)))
16722, 166syl 17 . . . . . . . . . . . . 13 (𝑧𝑆 → Fun (2nd ‘(1st𝑧)))
168 imadif 6011 . . . . . . . . . . . . 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 480 . . . . . . . . . . 11 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = (((2nd ‘(1st𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1)))))
171119, 164, 1703eqtr2d 2691 . . . . . . . . . 10 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑧))‘𝑛)} = (((2nd ‘(1st𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1)))))
1726, 171sylan 487 . . . . . . . . 9 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑧))‘𝑛)} = (((2nd ‘(1st𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1)))))
173 eleq1 2718 . . . . . . . . . . . . 13 (𝑧 = 𝑘 → (𝑧𝑆𝑘𝑆))
174173anbi1d 741 . . . . . . . . . . . 12 (𝑧 = 𝑘 → ((𝑧𝑆𝑛 ∈ (1...𝑁)) ↔ (𝑘𝑆𝑛 ∈ (1...𝑁))))
175 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑘 → (1st𝑧) = (1st𝑘))
176175fveq2d 6233 . . . . . . . . . . . . . . 15 (𝑧 = 𝑘 → (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑘)))
177176fveq1d 6231 . . . . . . . . . . . . . 14 (𝑧 = 𝑘 → ((2nd ‘(1st𝑧))‘𝑛) = ((2nd ‘(1st𝑘))‘𝑛))
178177sneqd 4222 . . . . . . . . . . . . 13 (𝑧 = 𝑘 → {((2nd ‘(1st𝑧))‘𝑛)} = {((2nd ‘(1st𝑘))‘𝑛)})
179176imaeq1d 5500 . . . . . . . . . . . . . 14 (𝑧 = 𝑘 → ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑛)))
180176imaeq1d 5500 . . . . . . . . . . . . . 14 (𝑧 = 𝑘 → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))
181179, 180difeq12d 3762 . . . . . . . . . . . . 13 (𝑧 = 𝑘 → (((2nd ‘(1st𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1)))) = (((2nd ‘(1st𝑘)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1)))))
182178, 181eqeq12d 2666 . . . . . . . . . . . 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))))))
183174, 182imbi12d 333 . . . . . . . . . . 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, 171chvarv 2299 . . . . . . . . . 10 ((𝑘𝑆𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑘))‘𝑛)} = (((2nd ‘(1st𝑘)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1)))))
1859, 184sylan 487 . . . . . . . . 9 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑘))‘𝑛)} = (((2nd ‘(1st𝑘)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1)))))
186117, 172, 1853eqtr4d 2695 . . . . . . . 8 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑧))‘𝑛)} = {((2nd ‘(1st𝑘))‘𝑛)})
187 fvex 6239 . . . . . . . . 9 ((2nd ‘(1st𝑧))‘𝑛) ∈ V
188187sneqr 4403 . . . . . . . 8 ({((2nd ‘(1st𝑧))‘𝑛)} = {((2nd ‘(1st𝑘))‘𝑛)} → ((2nd ‘(1st𝑧))‘𝑛) = ((2nd ‘(1st𝑘))‘𝑛))
189186, 188syl 17 . . . . . . 7 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧))‘𝑛) = ((2nd ‘(1st𝑘))‘𝑛))
19026, 40, 189eqfnfvd 6354 . . . . . 6 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑘)))
191 xpopth 7251 . . . . . . . 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)))
19216, 30, 191syl2an 493 . . . . . . 7 ((𝑧𝑆𝑘𝑆) → (((1st ‘(1st𝑧)) = (1st ‘(1st𝑘)) ∧ (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑘))) ↔ (1st𝑧) = (1st𝑘)))
193192ad2antlr 763 . . . . . 6 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (((1st ‘(1st𝑧)) = (1st ‘(1st𝑘)) ∧ (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑘))) ↔ (1st𝑧) = (1st𝑘)))
19412, 190, 193mpbi2and 976 . . . . 5 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (1st𝑧) = (1st𝑘))
195 eqtr3 2672 . . . . . 6 (((2nd𝑧) = 0 ∧ (2nd𝑘) = 0) → (2nd𝑧) = (2nd𝑘))
196195adantl 481 . . . . 5 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (2nd𝑧) = (2nd𝑘))
197 xpopth 7251 . . . . . . 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))
19814, 28, 197syl2an 493 . . . . . 6 ((𝑧𝑆𝑘𝑆) → (((1st𝑧) = (1st𝑘) ∧ (2nd𝑧) = (2nd𝑘)) ↔ 𝑧 = 𝑘))
199198ad2antlr 763 . . . . 5 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (((1st𝑧) = (1st𝑘) ∧ (2nd𝑧) = (2nd𝑘)) ↔ 𝑧 = 𝑘))
200194, 196, 199mpbi2and 976 . . . 4 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → 𝑧 = 𝑘)
201200ex 449 . . 3 ((𝜑 ∧ (𝑧𝑆𝑘𝑆)) → (((2nd𝑧) = 0 ∧ (2nd𝑘) = 0) → 𝑧 = 𝑘))
202201ralrimivva 3000 . 2 (𝜑 → ∀𝑧𝑆𝑘𝑆 (((2nd𝑧) = 0 ∧ (2nd𝑘) = 0) → 𝑧 = 𝑘))
203 fveq2 6229 . . . 4 (𝑧 = 𝑘 → (2nd𝑧) = (2nd𝑘))
204203eqeq1d 2653 . . 3 (𝑧 = 𝑘 → ((2nd𝑧) = 0 ↔ (2nd𝑘) = 0))
205204rmo4 3432 . 2 (∃*𝑧𝑆 (2nd𝑧) = 0 ↔ ∀𝑧𝑆𝑘𝑆 (((2nd𝑧) = 0 ∧ (2nd𝑘) = 0) → 𝑧 = 𝑘))
206202, 205sylibr 224 1 (𝜑 → ∃*𝑧𝑆 (2nd𝑧) = 0)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  {cab 2637  wne 2823  wral 2941  ∃*wrmo 2944  {crab 2945  csb 3566  cdif 3604  cun 3605  cin 3606  c0 3948  ifcif 4119  {csn 4210   class class class wbr 4685  cmpt 4762   × cxp 5141  ccnv 5142  cima 5146  Fun wfun 5920   Fn wfn 5921  wf 5922  ontowfo 5924  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  𝑓 cof 6937  1st c1st 7208  2nd c2nd 7209  𝑚 cmap 7899  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   < clt 10112  cle 10113  cmin 10304  cn 11058  0cn0 11330  cz 11415  cuz 11725  ...cfz 12364  ..^cfzo 12504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505
This theorem is referenced by:  poimirlem18  33557  poimirlem21  33560
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