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Theorem poimirlem13 37640
Description: Lemma for poimir 37660- 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 726 . . . . . . . 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 726 . . . . . . . 8 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))
6 simplrl 777 . . . . . . . 8 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → 𝑧𝑆)
7 simprl 771 . . . . . . . 8 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (2nd𝑧) = 0)
82, 3, 5, 6, 7poimirlem10 37637 . . . . . . 7 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → ((𝐹‘(𝑁 − 1)) ∘f − ((1...𝑁) × {1})) = (1st ‘(1st𝑧)))
9 simplrr 778 . . . . . . . 8 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → 𝑘𝑆)
10 simprr 773 . . . . . . . 8 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (2nd𝑘) = 0)
112, 3, 5, 9, 10poimirlem10 37637 . . . . . . 7 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → ((𝐹‘(𝑁 − 1)) ∘f − ((1...𝑁) × {1})) = (1st ‘(1st𝑘)))
128, 11eqtr3d 2779 . . . . . 6 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (1st ‘(1st𝑧)) = (1st ‘(1st𝑘)))
13 elrabi 3687 . . . . . . . . . . . . . 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 2859 . . . . . . . . . . . . 13 (𝑧𝑆𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
15 xp1st 8046 . . . . . . . . . . . . 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 8047 . . . . . . . . . . . 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 6919 . . . . . . . . . . . 12 (2nd ‘(1st𝑧)) ∈ V
20 f1oeq1 6836 . . . . . . . . . . . 12 (𝑓 = (2nd ‘(1st𝑧)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁)))
2119, 20elab 3679 . . . . . . . . . . 11 ((2nd ‘(1st𝑧)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
2218, 21sylib 218 . . . . . . . . . 10 (𝑧𝑆 → (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
23 f1ofn 6849 . . . . . . . . . 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 727 . . . . . . 7 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (2nd ‘(1st𝑧)) Fn (1...𝑁))
27 elrabi 3687 . . . . . . . . . . . . . 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 2859 . . . . . . . . . . . . 13 (𝑘𝑆𝑘 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
29 xp1st 8046 . . . . . . . . . . . . 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 8047 . . . . . . . . . . . 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 6919 . . . . . . . . . . . 12 (2nd ‘(1st𝑘)) ∈ V
34 f1oeq1 6836 . . . . . . . . . . . 12 (𝑓 = (2nd ‘(1st𝑘)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑘)):(1...𝑁)–1-1-onto→(1...𝑁)))
3533, 34elab 3679 . . . . . . . . . . 11 ((2nd ‘(1st𝑘)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑘)):(1...𝑁)–1-1-onto→(1...𝑁))
3632, 35sylib 218 . . . . . . . . . 10 (𝑘𝑆 → (2nd ‘(1st𝑘)):(1...𝑁)–1-1-onto→(1...𝑁))
37 f1ofn 6849 . . . . . . . . . 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 727 . . . . . . 7 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (2nd ‘(1st𝑘)) Fn (1...𝑁))
41 eleq1 2829 . . . . . . . . . . . . 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 7439 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
4443imaeq2d 6078 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → ((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑧)) “ (1...𝑛)))
4543imaeq2d 6078 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → ((2nd ‘(1st𝑘)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...𝑛)))
4644, 45eqeq12d 2753 . . . . . . . . . . . 12 (𝑚 = 𝑛 → (((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...𝑚)) ↔ ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑛))))
4742, 46imbi12d 344 . . . . . . . . . . 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 730 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑁 ∈ ℕ)
494ad3antrrr 730 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))
50 simpl 482 . . . . . . . . . . . . . 14 ((𝑧𝑆𝑘𝑆) → 𝑧𝑆)
5150ad3antlr 731 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑧𝑆)
52 simplrl 777 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → (2nd𝑧) = 0)
53 simpr 484 . . . . . . . . . . . . . 14 ((𝑧𝑆𝑘𝑆) → 𝑘𝑆)
5453ad3antlr 731 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑘𝑆)
55 simplrr 778 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → (2nd𝑘) = 0)
56 simpr 484 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → 𝑚 ∈ (1...𝑁))
5748, 3, 49, 51, 52, 54, 55, 56poimirlem11 37638 . . . . . . . . . . . 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 37638 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → ((2nd ‘(1st𝑘)) “ (1...𝑚)) ⊆ ((2nd ‘(1st𝑧)) “ (1...𝑚)))
5957, 58eqssd 4001 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑚 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...𝑚)))
6047, 59chvarvv 1998 . . . . . . . . . 10 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑛)))
61 simpll 767 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → 𝜑)
62 elfznn 13593 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℕ)
63 nnm1nn0 12567 . . . . . . . . . . . . . . . . . . 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 12282 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℂ)
67 ax-1cn 11213 . . . . . . . . . . . . . . . . . . . 20 1 ∈ ℂ
68 subeq0 11535 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) = 0 ↔ 𝑛 = 1))
6966, 67, 68sylancl 586 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) = 0 ↔ 𝑛 = 1))
7069necon3abid 2977 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...𝑁) → ((𝑛 − 1) ≠ 0 ↔ ¬ 𝑛 = 1))
7170biimpar 477 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1) → (𝑛 − 1) ≠ 0)
72 elnnne0 12540 . . . . . . . . . . . . . . . . 17 ((𝑛 − 1) ∈ ℕ ↔ ((𝑛 − 1) ∈ ℕ0 ∧ (𝑛 − 1) ≠ 0))
7365, 71, 72sylanbrc 583 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1) → (𝑛 − 1) ∈ ℕ)
7473adantl 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → (𝑛 − 1) ∈ ℕ)
7564nn0red 12588 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ∈ ℝ)
7675adantl 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ∈ ℝ)
7762nnred 12281 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℝ)
7877adantl 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℝ)
791nnred 12281 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ ℝ)
8079adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (1...𝑁)) → 𝑁 ∈ ℝ)
8177lem1d 12201 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...𝑁) → (𝑛 − 1) ≤ 𝑛)
8281adantl 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ≤ 𝑛)
83 elfzle2 13568 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...𝑁) → 𝑛𝑁)
8483adantl 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (1...𝑁)) → 𝑛𝑁)
8576, 78, 80, 82, 84letrd 11418 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ≤ 𝑁)
8685adantrr 717 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → (𝑛 − 1) ≤ 𝑁)
871nnzd 12640 . . . . . . . . . . . . . . . . 17 (𝜑𝑁 ∈ ℤ)
88 fznn 13632 . . . . . . . . . . . . . . . . 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 713 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → (𝑛 − 1) ∈ (1...𝑁))
9261, 91sylan 580 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → (𝑛 − 1) ∈ (1...𝑁))
93 ovex 7464 . . . . . . . . . . . . . 14 (𝑛 − 1) ∈ V
94 eleq1 2829 . . . . . . . . . . . . . . . 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 7439 . . . . . . . . . . . . . . . . 17 (𝑚 = (𝑛 − 1) → (1...𝑚) = (1...(𝑛 − 1)))
9796imaeq2d 6078 . . . . . . . . . . . . . . . 16 (𝑚 = (𝑛 − 1) → ((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))))
9896imaeq2d 6078 . . . . . . . . . . . . . . . 16 (𝑚 = (𝑛 − 1) → ((2nd ‘(1st𝑘)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))
9997, 98eqeq12d 2753 . . . . . . . . . . . . . . 15 (𝑚 = (𝑛 − 1) → (((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...𝑚)) ↔ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1)))))
10095, 99imbi12d 344 . . . . . . . . . . . . . 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 3558 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ (𝑛 − 1) ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))
10292, 101syldan 591 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = 1)) → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))
103102expr 456 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → (¬ 𝑛 = 1 → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1)))))
104 ima0 6095 . . . . . . . . . . . . 13 ((2nd ‘(1st𝑧)) “ ∅) = ∅
105 ima0 6095 . . . . . . . . . . . . 13 ((2nd ‘(1st𝑘)) “ ∅) = ∅
106104, 105eqtr4i 2768 . . . . . . . . . . . 12 ((2nd ‘(1st𝑧)) “ ∅) = ((2nd ‘(1st𝑘)) “ ∅)
107 oveq1 7438 . . . . . . . . . . . . . . . 16 (𝑛 = 1 → (𝑛 − 1) = (1 − 1))
108 1m1e0 12338 . . . . . . . . . . . . . . . 16 (1 − 1) = 0
109107, 108eqtrdi 2793 . . . . . . . . . . . . . . 15 (𝑛 = 1 → (𝑛 − 1) = 0)
110109oveq2d 7447 . . . . . . . . . . . . . 14 (𝑛 = 1 → (1...(𝑛 − 1)) = (1...0))
111 fz10 13585 . . . . . . . . . . . . . 14 (1...0) = ∅
112110, 111eqtrdi 2793 . . . . . . . . . . . . 13 (𝑛 = 1 → (1...(𝑛 − 1)) = ∅)
113112imaeq2d 6078 . . . . . . . . . . . 12 (𝑛 = 1 → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑧)) “ ∅))
114112imaeq2d 6078 . . . . . . . . . . . 12 (𝑛 = 1 → ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ ∅))
115106, 113, 1143eqtr4a 2803 . . . . . . . . . . 11 (𝑛 = 1 → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))
116103, 115pm2.61d2 181 . . . . . . . . . 10 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))
11760, 116difeq12d 4127 . . . . . . . . 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 6988 . . . . . . . . . . . 12 (((2nd ‘(1st𝑧)) Fn (1...𝑁) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑧))‘𝑛)} = ((2nd ‘(1st𝑧)) “ {𝑛}))
11924, 118sylan 580 . . . . . . . . . . 11 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑧))‘𝑛)} = ((2nd ‘(1st𝑧)) “ {𝑛}))
12062adantl 481 . . . . . . . . . . . . 13 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → 𝑛 ∈ ℕ)
121 uncom 4158 . . . . . . . . . . . . . . . 16 ((1...(𝑛 − 1)) ∪ {𝑛}) = ({𝑛} ∪ (1...(𝑛 − 1)))
122121difeq1i 4122 . . . . . . . . . . . . . . 15 (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))) = (({𝑛} ∪ (1...(𝑛 − 1))) ∖ (1...(𝑛 − 1)))
123 difun2 4481 . . . . . . . . . . . . . . 15 (({𝑛} ∪ (1...(𝑛 − 1))) ∖ (1...(𝑛 − 1))) = ({𝑛} ∖ (1...(𝑛 − 1)))
124122, 123eqtri 2765 . . . . . . . . . . . . . 14 (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))) = ({𝑛} ∖ (1...(𝑛 − 1)))
125 nncn 12274 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
126 npcan1 11688 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℂ → ((𝑛 − 1) + 1) = 𝑛)
127125, 126syl 17 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) = 𝑛)
128 elnnuz 12922 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ‘1))
129128biimpi 216 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ‘1))
130127, 129eqeltrd 2841 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) ∈ (ℤ‘1))
13163nn0zd 12639 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℤ)
132 uzid 12893 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 − 1) ∈ ℤ → (𝑛 − 1) ∈ (ℤ‘(𝑛 − 1)))
133131, 132syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ (ℤ‘(𝑛 − 1)))
134 peano2uz 12943 . . . . . . . . . . . . . . . . . . 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 2842 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ‘(𝑛 − 1)))
137 fzsplit2 13589 . . . . . . . . . . . . . . . . 17 ((((𝑛 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑛 ∈ (ℤ‘(𝑛 − 1))) → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)))
138130, 136, 137syl2anc 584 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)))
139127oveq1d 7446 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → (((𝑛 − 1) + 1)...𝑛) = (𝑛...𝑛))
140 nnz 12634 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
141 fzsn 13606 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℤ → (𝑛...𝑛) = {𝑛})
142140, 141syl 17 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → (𝑛...𝑛) = {𝑛})
143139, 142eqtrd 2777 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ → (((𝑛 − 1) + 1)...𝑛) = {𝑛})
144143uneq2d 4168 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)) = ((1...(𝑛 − 1)) ∪ {𝑛}))
145138, 144eqtrd 2777 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → (1...𝑛) = ((1...(𝑛 − 1)) ∪ {𝑛}))
146145difeq1d 4125 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → ((1...𝑛) ∖ (1...(𝑛 − 1))) = (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))))
147 nnre 12273 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
148 ltm1 12109 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℝ → (𝑛 − 1) < 𝑛)
149 peano2rem 11576 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℝ → (𝑛 − 1) ∈ ℝ)
150 ltnle 11340 . . . . . . . . . . . . . . . . . . 19 (((𝑛 − 1) ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((𝑛 − 1) < 𝑛 ↔ ¬ 𝑛 ≤ (𝑛 − 1)))
151149, 150mpancom 688 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℝ → ((𝑛 − 1) < 𝑛 ↔ ¬ 𝑛 ≤ (𝑛 − 1)))
152148, 151mpbid 232 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℝ → ¬ 𝑛 ≤ (𝑛 − 1))
153 elfzle2 13568 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (1...(𝑛 − 1)) → 𝑛 ≤ (𝑛 − 1))
154152, 153nsyl 140 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℝ → ¬ 𝑛 ∈ (1...(𝑛 − 1)))
155147, 154syl 17 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → ¬ 𝑛 ∈ (1...(𝑛 − 1)))
156 incom 4209 . . . . . . . . . . . . . . . . 17 ((1...(𝑛 − 1)) ∩ {𝑛}) = ({𝑛} ∩ (1...(𝑛 − 1)))
157156eqeq1i 2742 . . . . . . . . . . . . . . . 16 (((1...(𝑛 − 1)) ∩ {𝑛}) = ∅ ↔ ({𝑛} ∩ (1...(𝑛 − 1))) = ∅)
158 disjsn 4711 . . . . . . . . . . . . . . . 16 (((1...(𝑛 − 1)) ∩ {𝑛}) = ∅ ↔ ¬ 𝑛 ∈ (1...(𝑛 − 1)))
159 disj3 4454 . . . . . . . . . . . . . . . 16 (({𝑛} ∩ (1...(𝑛 − 1))) = ∅ ↔ {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1))))
160157, 158, 1593bitr3i 301 . . . . . . . . . . . . . . 15 𝑛 ∈ (1...(𝑛 − 1)) ↔ {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1))))
161155, 160sylib 218 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1))))
162124, 146, 1613eqtr4a 2803 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → ((1...𝑛) ∖ (1...(𝑛 − 1))) = {𝑛})
163120, 162syl 17 . . . . . . . . . . . 12 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → ((1...𝑛) ∖ (1...(𝑛 − 1))) = {𝑛})
164163imaeq2d 6078 . . . . . . . . . . 11 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = ((2nd ‘(1st𝑧)) “ {𝑛}))
165 dff1o3 6854 . . . . . . . . . . . . . . 15 ((2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd ‘(1st𝑧)):(1...𝑁)–onto→(1...𝑁) ∧ Fun (2nd ‘(1st𝑧))))
166165simprbi 496 . . . . . . . . . . . . . 14 ((2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun (2nd ‘(1st𝑧)))
16722, 166syl 17 . . . . . . . . . . . . 13 (𝑧𝑆 → Fun (2nd ‘(1st𝑧)))
168 imadif 6650 . . . . . . . . . . . . 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 2783 . . . . . . . . . 10 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑧))‘𝑛)} = (((2nd ‘(1st𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1)))))
1726, 171sylan 580 . . . . . . . . 9 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑧))‘𝑛)} = (((2nd ‘(1st𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1)))))
173 eleq1 2829 . . . . . . . . . . . . 13 (𝑧 = 𝑘 → (𝑧𝑆𝑘𝑆))
174173anbi1d 631 . . . . . . . . . . . 12 (𝑧 = 𝑘 → ((𝑧𝑆𝑛 ∈ (1...𝑁)) ↔ (𝑘𝑆𝑛 ∈ (1...𝑁))))
175 2fveq3 6911 . . . . . . . . . . . . . . 15 (𝑧 = 𝑘 → (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑘)))
176175fveq1d 6908 . . . . . . . . . . . . . 14 (𝑧 = 𝑘 → ((2nd ‘(1st𝑧))‘𝑛) = ((2nd ‘(1st𝑘))‘𝑛))
177176sneqd 4638 . . . . . . . . . . . . 13 (𝑧 = 𝑘 → {((2nd ‘(1st𝑧))‘𝑛)} = {((2nd ‘(1st𝑘))‘𝑛)})
178175imaeq1d 6077 . . . . . . . . . . . . . 14 (𝑧 = 𝑘 → ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑛)))
179175imaeq1d 6077 . . . . . . . . . . . . . 14 (𝑧 = 𝑘 → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))
180178, 179difeq12d 4127 . . . . . . . . . . . . 13 (𝑧 = 𝑘 → (((2nd ‘(1st𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1)))) = (((2nd ‘(1st𝑘)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1)))))
181177, 180eqeq12d 2753 . . . . . . . . . . . 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 344 . . . . . . . . . . 11 (𝑧 = 𝑘 → (((𝑧𝑆𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑧))‘𝑛)} = (((2nd ‘(1st𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))))) ↔ ((𝑘𝑆𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑘))‘𝑛)} = (((2nd ‘(1st𝑘)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1)))))))
183182, 171chvarvv 1998 . . . . . . . . . 10 ((𝑘𝑆𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑘))‘𝑛)} = (((2nd ‘(1st𝑘)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1)))))
1849, 183sylan 580 . . . . . . . . 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 2787 . . . . . . . 8 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑧))‘𝑛)} = {((2nd ‘(1st𝑘))‘𝑛)})
186 fvex 6919 . . . . . . . . 9 ((2nd ‘(1st𝑧))‘𝑛) ∈ V
187186sneqr 4840 . . . . . . . 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 7054 . . . . . 6 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑘)))
190 xpopth 8055 . . . . . . . 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 596 . . . . . . 7 ((𝑧𝑆𝑘𝑆) → (((1st ‘(1st𝑧)) = (1st ‘(1st𝑘)) ∧ (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑘))) ↔ (1st𝑧) = (1st𝑘)))
192191ad2antlr 727 . . . . . 6 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (((1st ‘(1st𝑧)) = (1st ‘(1st𝑘)) ∧ (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑘))) ↔ (1st𝑧) = (1st𝑘)))
19312, 189, 192mpbi2and 712 . . . . 5 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (1st𝑧) = (1st𝑘))
194 eqtr3 2763 . . . . . 6 (((2nd𝑧) = 0 ∧ (2nd𝑘) = 0) → (2nd𝑧) = (2nd𝑘))
195194adantl 481 . . . . 5 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (2nd𝑧) = (2nd𝑘))
196 xpopth 8055 . . . . . . 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 596 . . . . . 6 ((𝑧𝑆𝑘𝑆) → (((1st𝑧) = (1st𝑘) ∧ (2nd𝑧) = (2nd𝑘)) ↔ 𝑧 = 𝑘))
198197ad2antlr 727 . . . . 5 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → (((1st𝑧) = (1st𝑘) ∧ (2nd𝑧) = (2nd𝑘)) ↔ 𝑧 = 𝑘))
199193, 195, 198mpbi2and 712 . . . 4 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 0 ∧ (2nd𝑘) = 0)) → 𝑧 = 𝑘)
200199ex 412 . . 3 ((𝜑 ∧ (𝑧𝑆𝑘𝑆)) → (((2nd𝑧) = 0 ∧ (2nd𝑘) = 0) → 𝑧 = 𝑘))
201200ralrimivva 3202 . 2 (𝜑 → ∀𝑧𝑆𝑘𝑆 (((2nd𝑧) = 0 ∧ (2nd𝑘) = 0) → 𝑧 = 𝑘))
202 fveqeq2 6915 . . 3 (𝑧 = 𝑘 → ((2nd𝑧) = 0 ↔ (2nd𝑘) = 0))
203202rmo4 3736 . 2 (∃*𝑧𝑆 (2nd𝑧) = 0 ↔ ∀𝑧𝑆𝑘𝑆 (((2nd𝑧) = 0 ∧ (2nd𝑘) = 0) → 𝑧 = 𝑘))
204201, 203sylibr 234 1 (𝜑 → ∃*𝑧𝑆 (2nd𝑧) = 0)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  {cab 2714  wne 2940  wral 3061  ∃*wrmo 3379  {crab 3436  csb 3899  cdif 3948  cun 3949  cin 3950  c0 4333  ifcif 4525  {csn 4626   class class class wbr 5143  cmpt 5225   × cxp 5683  ccnv 5684  cima 5688  Fun wfun 6555   Fn wfn 6556  wf 6557  ontowfo 6559  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  f cof 7695  1st c1st 8012  2nd c2nd 8013  m cmap 8866  cc 11153  cr 11154  0cc0 11155  1c1 11156   + caddc 11158   < clt 11295  cle 11296  cmin 11492  cn 12266  0cn0 12526  cz 12613  cuz 12878  ...cfz 13547  ..^cfzo 13694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695
This theorem is referenced by:  poimirlem18  37645  poimirlem21  37648
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