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Theorem poimirlem14 33553
 Description: Lemma for poimir 33572- for at most one simplex associated with a shared face is the opposite vertex last on the walk. (Contributed by Brendan Leahy, 21-Aug-2020.)
Hypotheses
Ref Expression
poimir.0 (𝜑𝑁 ∈ ℕ)
poimirlem22.s 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
poimirlem22.1 (𝜑𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)))
Assertion
Ref Expression
poimirlem14 (𝜑 → ∃*𝑧𝑆 (2nd𝑧) = 𝑁)
Distinct variable groups:   𝑓,𝑗,𝑡,𝑦,𝑧   𝜑,𝑗,𝑦   𝑗,𝐹,𝑦   𝑗,𝑁,𝑦   𝜑,𝑡   𝑓,𝐾,𝑗,𝑡   𝑓,𝑁,𝑡   𝜑,𝑧   𝑓,𝐹,𝑡,𝑧   𝑧,𝐾   𝑧,𝑁   𝑆,𝑗,𝑡,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑓)   𝑆(𝑓)   𝐾(𝑦)

Proof of Theorem poimirlem14
Dummy variables 𝑘 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 poimir.0 . . . . . . . . 9 (𝜑𝑁 ∈ ℕ)
21ad2antrr 762 . . . . . . . 8 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → 𝑁 ∈ ℕ)
3 poimirlem22.s . . . . . . . 8 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
4 simplrl 817 . . . . . . . 8 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → 𝑧𝑆)
51nngt0d 11102 . . . . . . . . . 10 (𝜑 → 0 < 𝑁)
6 breq2 4689 . . . . . . . . . . 11 ((2nd𝑧) = 𝑁 → (0 < (2nd𝑧) ↔ 0 < 𝑁))
76biimparc 503 . . . . . . . . . 10 ((0 < 𝑁 ∧ (2nd𝑧) = 𝑁) → 0 < (2nd𝑧))
85, 7sylan 487 . . . . . . . . 9 ((𝜑 ∧ (2nd𝑧) = 𝑁) → 0 < (2nd𝑧))
98ad2ant2r 798 . . . . . . . 8 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → 0 < (2nd𝑧))
102, 3, 4, 9poimirlem5 33544 . . . . . . 7 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → (𝐹‘0) = (1st ‘(1st𝑧)))
11 simplrr 818 . . . . . . . 8 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → 𝑘𝑆)
12 breq2 4689 . . . . . . . . . . 11 ((2nd𝑘) = 𝑁 → (0 < (2nd𝑘) ↔ 0 < 𝑁))
1312biimparc 503 . . . . . . . . . 10 ((0 < 𝑁 ∧ (2nd𝑘) = 𝑁) → 0 < (2nd𝑘))
145, 13sylan 487 . . . . . . . . 9 ((𝜑 ∧ (2nd𝑘) = 𝑁) → 0 < (2nd𝑘))
1514ad2ant2rl 800 . . . . . . . 8 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → 0 < (2nd𝑘))
162, 3, 11, 15poimirlem5 33544 . . . . . . 7 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → (𝐹‘0) = (1st ‘(1st𝑘)))
1710, 16eqtr3d 2687 . . . . . 6 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → (1st ‘(1st𝑧)) = (1st ‘(1st𝑘)))
18 elrabi 3391 . . . . . . . . . . . . 13 (𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 𝑧 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
1918, 3eleq2s 2748 . . . . . . . . . . . 12 (𝑧𝑆𝑧 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
20 xp1st 7242 . . . . . . . . . . . 12 (𝑧 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st𝑧) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
21 xp2nd 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)})
2219, 20, 213syl 18 . . . . . . . . . . 11 (𝑧𝑆 → (2nd ‘(1st𝑧)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
23 fvex 6239 . . . . . . . . . . . 12 (2nd ‘(1st𝑧)) ∈ V
24 f1oeq1 6165 . . . . . . . . . . . 12 (𝑓 = (2nd ‘(1st𝑧)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁)))
2523, 24elab 3382 . . . . . . . . . . 11 ((2nd ‘(1st𝑧)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
2622, 25sylib 208 . . . . . . . . . 10 (𝑧𝑆 → (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
27 f1ofn 6176 . . . . . . . . . 10 ((2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑧)) Fn (1...𝑁))
2826, 27syl 17 . . . . . . . . 9 (𝑧𝑆 → (2nd ‘(1st𝑧)) Fn (1...𝑁))
2928adantr 480 . . . . . . . 8 ((𝑧𝑆𝑘𝑆) → (2nd ‘(1st𝑧)) Fn (1...𝑁))
3029ad2antlr 763 . . . . . . 7 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → (2nd ‘(1st𝑧)) Fn (1...𝑁))
31 elrabi 3391 . . . . . . . . . . . . 13 (𝑘 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 𝑘 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
3231, 3eleq2s 2748 . . . . . . . . . . . 12 (𝑘𝑆𝑘 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
33 xp1st 7242 . . . . . . . . . . . 12 (𝑘 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st𝑘) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
34 xp2nd 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)})
3532, 33, 343syl 18 . . . . . . . . . . 11 (𝑘𝑆 → (2nd ‘(1st𝑘)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
36 fvex 6239 . . . . . . . . . . . 12 (2nd ‘(1st𝑘)) ∈ V
37 f1oeq1 6165 . . . . . . . . . . . 12 (𝑓 = (2nd ‘(1st𝑘)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑘)):(1...𝑁)–1-1-onto→(1...𝑁)))
3836, 37elab 3382 . . . . . . . . . . 11 ((2nd ‘(1st𝑘)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑘)):(1...𝑁)–1-1-onto→(1...𝑁))
3935, 38sylib 208 . . . . . . . . . 10 (𝑘𝑆 → (2nd ‘(1st𝑘)):(1...𝑁)–1-1-onto→(1...𝑁))
40 f1ofn 6176 . . . . . . . . . 10 ((2nd ‘(1st𝑘)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑘)) Fn (1...𝑁))
4139, 40syl 17 . . . . . . . . 9 (𝑘𝑆 → (2nd ‘(1st𝑘)) Fn (1...𝑁))
4241adantl 481 . . . . . . . 8 ((𝑧𝑆𝑘𝑆) → (2nd ‘(1st𝑘)) Fn (1...𝑁))
4342ad2antlr 763 . . . . . . 7 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → (2nd ‘(1st𝑘)) Fn (1...𝑁))
44 simpllr 815 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (𝑧𝑆𝑘𝑆))
45 oveq2 6698 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
4645imaeq2d 5501 . . . . . . . . . . . . . . 15 (𝑛 = 𝑁 → ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑧)) “ (1...𝑁)))
47 f1ofo 6182 . . . . . . . . . . . . . . . 16 ((2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑧)):(1...𝑁)–onto→(1...𝑁))
48 foima 6158 . . . . . . . . . . . . . . . 16 ((2nd ‘(1st𝑧)):(1...𝑁)–onto→(1...𝑁) → ((2nd ‘(1st𝑧)) “ (1...𝑁)) = (1...𝑁))
4926, 47, 483syl 18 . . . . . . . . . . . . . . 15 (𝑧𝑆 → ((2nd ‘(1st𝑧)) “ (1...𝑁)) = (1...𝑁))
5046, 49sylan9eqr 2707 . . . . . . . . . . . . . 14 ((𝑧𝑆𝑛 = 𝑁) → ((2nd ‘(1st𝑧)) “ (1...𝑛)) = (1...𝑁))
5150adantlr 751 . . . . . . . . . . . . 13 (((𝑧𝑆𝑘𝑆) ∧ 𝑛 = 𝑁) → ((2nd ‘(1st𝑧)) “ (1...𝑛)) = (1...𝑁))
5245imaeq2d 5501 . . . . . . . . . . . . . . 15 (𝑛 = 𝑁 → ((2nd ‘(1st𝑘)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑁)))
53 f1ofo 6182 . . . . . . . . . . . . . . . 16 ((2nd ‘(1st𝑘)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑘)):(1...𝑁)–onto→(1...𝑁))
54 foima 6158 . . . . . . . . . . . . . . . 16 ((2nd ‘(1st𝑘)):(1...𝑁)–onto→(1...𝑁) → ((2nd ‘(1st𝑘)) “ (1...𝑁)) = (1...𝑁))
5539, 53, 543syl 18 . . . . . . . . . . . . . . 15 (𝑘𝑆 → ((2nd ‘(1st𝑘)) “ (1...𝑁)) = (1...𝑁))
5652, 55sylan9eqr 2707 . . . . . . . . . . . . . 14 ((𝑘𝑆𝑛 = 𝑁) → ((2nd ‘(1st𝑘)) “ (1...𝑛)) = (1...𝑁))
5756adantll 750 . . . . . . . . . . . . 13 (((𝑧𝑆𝑘𝑆) ∧ 𝑛 = 𝑁) → ((2nd ‘(1st𝑘)) “ (1...𝑛)) = (1...𝑁))
5851, 57eqtr4d 2688 . . . . . . . . . . . 12 (((𝑧𝑆𝑘𝑆) ∧ 𝑛 = 𝑁) → ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑛)))
5944, 58sylan 487 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑛 = 𝑁) → ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑛)))
60 simpll 805 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → 𝜑)
61 elnnuz 11762 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ‘1))
621, 61sylib 208 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑁 ∈ (ℤ‘1))
63 fzm1 12458 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ (ℤ‘1) → (𝑛 ∈ (1...𝑁) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁)))
6462, 63syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑛 ∈ (1...𝑁) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁)))
6564anbi1d 741 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑛 ∈ (1...𝑁) ∧ 𝑛𝑁) ↔ ((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁) ∧ 𝑛𝑁)))
6665biimpa 500 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛𝑁)) → ((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁) ∧ 𝑛𝑁))
67 df-ne 2824 . . . . . . . . . . . . . . . . . 18 (𝑛𝑁 ↔ ¬ 𝑛 = 𝑁)
6867anbi2i 730 . . . . . . . . . . . . . . . . 17 (((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁) ∧ 𝑛𝑁) ↔ ((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁) ∧ ¬ 𝑛 = 𝑁))
69 pm5.61 749 . . . . . . . . . . . . . . . . 17 (((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁) ∧ ¬ 𝑛 = 𝑁) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∧ ¬ 𝑛 = 𝑁))
7068, 69bitri 264 . . . . . . . . . . . . . . . 16 (((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁) ∧ 𝑛𝑁) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∧ ¬ 𝑛 = 𝑁))
7166, 70sylib 208 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛𝑁)) → (𝑛 ∈ (1...(𝑁 − 1)) ∧ ¬ 𝑛 = 𝑁))
72 fz1ssfz0 12474 . . . . . . . . . . . . . . . . 17 (1...(𝑁 − 1)) ⊆ (0...(𝑁 − 1))
7372sseli 3632 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ (0...(𝑁 − 1)))
7473adantr 480 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (1...(𝑁 − 1)) ∧ ¬ 𝑛 = 𝑁) → 𝑛 ∈ (0...(𝑁 − 1)))
7571, 74syl 17 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛𝑁)) → 𝑛 ∈ (0...(𝑁 − 1)))
7660, 75sylan 487 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛𝑁)) → 𝑛 ∈ (0...(𝑁 − 1)))
77 eleq1 2718 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑛 → (𝑚 ∈ (0...(𝑁 − 1)) ↔ 𝑛 ∈ (0...(𝑁 − 1))))
7877anbi2d 740 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) ↔ (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 1)))))
79 oveq2 6698 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
8079imaeq2d 5501 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑛 → ((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑧)) “ (1...𝑛)))
8179imaeq2d 5501 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑛 → ((2nd ‘(1st𝑘)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...𝑛)))
8280, 81eqeq12d 2666 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → (((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...𝑚)) ↔ ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑛))))
8378, 82imbi12d 333 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → (((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...𝑚))) ↔ ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑛)))))
841ad3antrrr 766 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℕ)
85 poimirlem22.1 . . . . . . . . . . . . . . . . 17 (𝜑𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)))
8685ad3antrrr 766 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)))
87 simpl 472 . . . . . . . . . . . . . . . . 17 ((𝑧𝑆𝑘𝑆) → 𝑧𝑆)
8887ad3antlr 767 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → 𝑧𝑆)
89 simplrl 817 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → (2nd𝑧) = 𝑁)
90 simpr 476 . . . . . . . . . . . . . . . . 17 ((𝑧𝑆𝑘𝑆) → 𝑘𝑆)
9190ad3antlr 767 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → 𝑘𝑆)
92 simplrr 818 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → (2nd𝑘) = 𝑁)
93 simpr 476 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → 𝑚 ∈ (0...(𝑁 − 1)))
9484, 3, 86, 88, 89, 91, 92, 93poimirlem12 33551 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑧)) “ (1...𝑚)) ⊆ ((2nd ‘(1st𝑘)) “ (1...𝑚)))
9584, 3, 86, 91, 92, 88, 89, 93poimirlem12 33551 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑘)) “ (1...𝑚)) ⊆ ((2nd ‘(1st𝑧)) “ (1...𝑚)))
9694, 95eqssd 3653 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...𝑚)))
9783, 96chvarv 2299 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑛)))
9876, 97syldan 486 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑛𝑁)) → ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑛)))
9998anassrs 681 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑛𝑁) → ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑛)))
10059, 99pm2.61dane 2910 . . . . . . . . . 10 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑛)))
101 simpr 476 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...𝑁)) → 𝑛 ∈ (1...𝑁))
102 elfzelz 12380 . . . . . . . . . . . . . 14 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℤ)
1031nnzd 11519 . . . . . . . . . . . . . 14 (𝜑𝑁 ∈ ℤ)
104 elfzm1b 12456 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑛 ∈ (1...𝑁) ↔ (𝑛 − 1) ∈ (0...(𝑁 − 1))))
105102, 103, 104syl2anr 494 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑛 ∈ (1...𝑁) ↔ (𝑛 − 1) ∈ (0...(𝑁 − 1))))
106101, 105mpbid 222 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ∈ (0...(𝑁 − 1)))
10760, 106sylan 487 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (𝑛 − 1) ∈ (0...(𝑁 − 1)))
108 ovex 6718 . . . . . . . . . . . 12 (𝑛 − 1) ∈ V
109 eleq1 2718 . . . . . . . . . . . . . 14 (𝑚 = (𝑛 − 1) → (𝑚 ∈ (0...(𝑁 − 1)) ↔ (𝑛 − 1) ∈ (0...(𝑁 − 1))))
110109anbi2d 740 . . . . . . . . . . . . 13 (𝑚 = (𝑛 − 1) → ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) ↔ (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ (𝑛 − 1) ∈ (0...(𝑁 − 1)))))
111 oveq2 6698 . . . . . . . . . . . . . . 15 (𝑚 = (𝑛 − 1) → (1...𝑚) = (1...(𝑛 − 1)))
112111imaeq2d 5501 . . . . . . . . . . . . . 14 (𝑚 = (𝑛 − 1) → ((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))))
113111imaeq2d 5501 . . . . . . . . . . . . . 14 (𝑚 = (𝑛 − 1) → ((2nd ‘(1st𝑘)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))
114112, 113eqeq12d 2666 . . . . . . . . . . . . 13 (𝑚 = (𝑛 − 1) → (((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...𝑚)) ↔ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1)))))
115110, 114imbi12d 333 . . . . . . . . . . . 12 (𝑚 = (𝑛 − 1) → (((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑚 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑧)) “ (1...𝑚)) = ((2nd ‘(1st𝑘)) “ (1...𝑚))) ↔ ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ (𝑛 − 1) ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))))
116108, 115, 96vtocl 3290 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ (𝑛 − 1) ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))
117107, 116syldan 486 . . . . . . . . . 10 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))
118100, 117difeq12d 3762 . . . . . . . . 9 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (((2nd ‘(1st𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1)))) = (((2nd ‘(1st𝑘)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1)))))
119 fnsnfv 6297 . . . . . . . . . . . 12 (((2nd ‘(1st𝑧)) Fn (1...𝑁) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑧))‘𝑛)} = ((2nd ‘(1st𝑧)) “ {𝑛}))
12028, 119sylan 487 . . . . . . . . . . 11 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑧))‘𝑛)} = ((2nd ‘(1st𝑧)) “ {𝑛}))
121 elfznn 12408 . . . . . . . . . . . . . 14 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℕ)
122 uncom 3790 . . . . . . . . . . . . . . . . 17 ((1...(𝑛 − 1)) ∪ {𝑛}) = ({𝑛} ∪ (1...(𝑛 − 1)))
123122difeq1i 3757 . . . . . . . . . . . . . . . 16 (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))) = (({𝑛} ∪ (1...(𝑛 − 1))) ∖ (1...(𝑛 − 1)))
124 difun2 4081 . . . . . . . . . . . . . . . 16 (({𝑛} ∪ (1...(𝑛 − 1))) ∖ (1...(𝑛 − 1))) = ({𝑛} ∖ (1...(𝑛 − 1)))
125123, 124eqtri 2673 . . . . . . . . . . . . . . 15 (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))) = ({𝑛} ∖ (1...(𝑛 − 1)))
126 nncn 11066 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
127 npcan1 10493 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℂ → ((𝑛 − 1) + 1) = 𝑛)
128126, 127syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) = 𝑛)
129 elnnuz 11762 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ‘1))
130129biimpi 206 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ‘1))
131128, 130eqeltrd 2730 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) ∈ (ℤ‘1))
132 nnm1nn0 11372 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
133132nn0zd 11518 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℤ)
134 uzid 11740 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 − 1) ∈ ℤ → (𝑛 − 1) ∈ (ℤ‘(𝑛 − 1)))
135 peano2uz 11779 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 − 1) ∈ (ℤ‘(𝑛 − 1)) → ((𝑛 − 1) + 1) ∈ (ℤ‘(𝑛 − 1)))
136133, 134, 1353syl 18 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → ((𝑛 − 1) + 1) ∈ (ℤ‘(𝑛 − 1)))
137128, 136eqeltrrd 2731 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ‘(𝑛 − 1)))
138 fzsplit2 12404 . . . . . . . . . . . . . . . . . 18 ((((𝑛 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑛 ∈ (ℤ‘(𝑛 − 1))) → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)))
139131, 137, 138syl2anc 694 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ → (1...𝑛) = ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)))
140128oveq1d 6705 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → (((𝑛 − 1) + 1)...𝑛) = (𝑛...𝑛))
141 nnz 11437 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
142 fzsn 12421 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℤ → (𝑛...𝑛) = {𝑛})
143141, 142syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → (𝑛...𝑛) = {𝑛})
144140, 143eqtrd 2685 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → (((𝑛 − 1) + 1)...𝑛) = {𝑛})
145144uneq2d 3800 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ → ((1...(𝑛 − 1)) ∪ (((𝑛 − 1) + 1)...𝑛)) = ((1...(𝑛 − 1)) ∪ {𝑛}))
146139, 145eqtrd 2685 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → (1...𝑛) = ((1...(𝑛 − 1)) ∪ {𝑛}))
147146difeq1d 3760 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → ((1...𝑛) ∖ (1...(𝑛 − 1))) = (((1...(𝑛 − 1)) ∪ {𝑛}) ∖ (1...(𝑛 − 1))))
148 nnre 11065 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
149 ltm1 10901 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℝ → (𝑛 − 1) < 𝑛)
150 peano2rem 10386 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℝ → (𝑛 − 1) ∈ ℝ)
151 ltnle 10155 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 − 1) ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((𝑛 − 1) < 𝑛 ↔ ¬ 𝑛 ≤ (𝑛 − 1)))
152150, 151mpancom 704 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℝ → ((𝑛 − 1) < 𝑛 ↔ ¬ 𝑛 ≤ (𝑛 − 1)))
153149, 152mpbid 222 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℝ → ¬ 𝑛 ≤ (𝑛 − 1))
154 elfzle2 12383 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...(𝑛 − 1)) → 𝑛 ≤ (𝑛 − 1))
155153, 154nsyl 135 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℝ → ¬ 𝑛 ∈ (1...(𝑛 − 1)))
156148, 155syl 17 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → ¬ 𝑛 ∈ (1...(𝑛 − 1)))
157 incom 3838 . . . . . . . . . . . . . . . . . 18 ((1...(𝑛 − 1)) ∩ {𝑛}) = ({𝑛} ∩ (1...(𝑛 − 1)))
158157eqeq1i 2656 . . . . . . . . . . . . . . . . 17 (((1...(𝑛 − 1)) ∩ {𝑛}) = ∅ ↔ ({𝑛} ∩ (1...(𝑛 − 1))) = ∅)
159 disjsn 4278 . . . . . . . . . . . . . . . . 17 (((1...(𝑛 − 1)) ∩ {𝑛}) = ∅ ↔ ¬ 𝑛 ∈ (1...(𝑛 − 1)))
160 disj3 4054 . . . . . . . . . . . . . . . . 17 (({𝑛} ∩ (1...(𝑛 − 1))) = ∅ ↔ {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1))))
161158, 159, 1603bitr3i 290 . . . . . . . . . . . . . . . 16 𝑛 ∈ (1...(𝑛 − 1)) ↔ {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1))))
162156, 161sylib 208 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → {𝑛} = ({𝑛} ∖ (1...(𝑛 − 1))))
163125, 147, 1623eqtr4a 2711 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → ((1...𝑛) ∖ (1...(𝑛 − 1))) = {𝑛})
164121, 163syl 17 . . . . . . . . . . . . 13 (𝑛 ∈ (1...𝑁) → ((1...𝑛) ∖ (1...(𝑛 − 1))) = {𝑛})
165164imaeq2d 5501 . . . . . . . . . . . 12 (𝑛 ∈ (1...𝑁) → ((2nd ‘(1st𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = ((2nd ‘(1st𝑧)) “ {𝑛}))
166165adantl 481 . . . . . . . . . . 11 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = ((2nd ‘(1st𝑧)) “ {𝑛}))
167 dff1o3 6181 . . . . . . . . . . . . . 14 ((2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd ‘(1st𝑧)):(1...𝑁)–onto→(1...𝑁) ∧ Fun (2nd ‘(1st𝑧))))
168167simprbi 479 . . . . . . . . . . . . 13 ((2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun (2nd ‘(1st𝑧)))
169 imadif 6011 . . . . . . . . . . . . 13 (Fun (2nd ‘(1st𝑧)) → ((2nd ‘(1st𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = (((2nd ‘(1st𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1)))))
17026, 168, 1693syl 18 . . . . . . . . . . . 12 (𝑧𝑆 → ((2nd ‘(1st𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = (((2nd ‘(1st𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1)))))
171170adantr 480 . . . . . . . . . . 11 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧)) “ ((1...𝑛) ∖ (1...(𝑛 − 1)))) = (((2nd ‘(1st𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1)))))
172120, 166, 1713eqtr2d 2691 . . . . . . . . . 10 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑧))‘𝑛)} = (((2nd ‘(1st𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1)))))
1734, 172sylan 487 . . . . . . . . 9 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑧))‘𝑛)} = (((2nd ‘(1st𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1)))))
174 eleq1 2718 . . . . . . . . . . . . 13 (𝑧 = 𝑘 → (𝑧𝑆𝑘𝑆))
175174anbi1d 741 . . . . . . . . . . . 12 (𝑧 = 𝑘 → ((𝑧𝑆𝑛 ∈ (1...𝑁)) ↔ (𝑘𝑆𝑛 ∈ (1...𝑁))))
176 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑘 → (1st𝑧) = (1st𝑘))
177176fveq2d 6233 . . . . . . . . . . . . . . 15 (𝑧 = 𝑘 → (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑘)))
178177fveq1d 6231 . . . . . . . . . . . . . 14 (𝑧 = 𝑘 → ((2nd ‘(1st𝑧))‘𝑛) = ((2nd ‘(1st𝑘))‘𝑛))
179178sneqd 4222 . . . . . . . . . . . . 13 (𝑧 = 𝑘 → {((2nd ‘(1st𝑧))‘𝑛)} = {((2nd ‘(1st𝑘))‘𝑛)})
180177imaeq1d 5500 . . . . . . . . . . . . . 14 (𝑧 = 𝑘 → ((2nd ‘(1st𝑧)) “ (1...𝑛)) = ((2nd ‘(1st𝑘)) “ (1...𝑛)))
181177imaeq1d 5500 . . . . . . . . . . . . . 14 (𝑧 = 𝑘 → ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1))) = ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1))))
182180, 181difeq12d 3762 . . . . . . . . . . . . 13 (𝑧 = 𝑘 → (((2nd ‘(1st𝑧)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑧)) “ (1...(𝑛 − 1)))) = (((2nd ‘(1st𝑘)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1)))))
183179, 182eqeq12d 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))))))
184175, 183imbi12d 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)))))))
185184, 172chvarv 2299 . . . . . . . . . 10 ((𝑘𝑆𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑘))‘𝑛)} = (((2nd ‘(1st𝑘)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1)))))
18611, 185sylan 487 . . . . . . . . 9 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑘))‘𝑛)} = (((2nd ‘(1st𝑘)) “ (1...𝑛)) ∖ ((2nd ‘(1st𝑘)) “ (1...(𝑛 − 1)))))
187118, 173, 1863eqtr4d 2695 . . . . . . . 8 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → {((2nd ‘(1st𝑧))‘𝑛)} = {((2nd ‘(1st𝑘))‘𝑛)})
188 fvex 6239 . . . . . . . . 9 ((2nd ‘(1st𝑧))‘𝑛) ∈ V
189188sneqr 4403 . . . . . . . 8 ({((2nd ‘(1st𝑧))‘𝑛)} = {((2nd ‘(1st𝑘))‘𝑛)} → ((2nd ‘(1st𝑧))‘𝑛) = ((2nd ‘(1st𝑘))‘𝑛))
190187, 189syl 17 . . . . . . 7 ((((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → ((2nd ‘(1st𝑧))‘𝑛) = ((2nd ‘(1st𝑘))‘𝑛))
19130, 43, 190eqfnfvd 6354 . . . . . 6 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑘)))
19219, 20syl 17 . . . . . . . 8 (𝑧𝑆 → (1st𝑧) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
19332, 33syl 17 . . . . . . . 8 (𝑘𝑆 → (1st𝑘) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
194 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)))
195192, 193, 194syl2an 493 . . . . . . 7 ((𝑧𝑆𝑘𝑆) → (((1st ‘(1st𝑧)) = (1st ‘(1st𝑘)) ∧ (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑘))) ↔ (1st𝑧) = (1st𝑘)))
196195ad2antlr 763 . . . . . 6 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → (((1st ‘(1st𝑧)) = (1st ‘(1st𝑘)) ∧ (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑘))) ↔ (1st𝑧) = (1st𝑘)))
19717, 191, 196mpbi2and 976 . . . . 5 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → (1st𝑧) = (1st𝑘))
198 eqtr3 2672 . . . . . 6 (((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁) → (2nd𝑧) = (2nd𝑘))
199198adantl 481 . . . . 5 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → (2nd𝑧) = (2nd𝑘))
200 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))
20119, 32, 200syl2an 493 . . . . . 6 ((𝑧𝑆𝑘𝑆) → (((1st𝑧) = (1st𝑘) ∧ (2nd𝑧) = (2nd𝑘)) ↔ 𝑧 = 𝑘))
202201ad2antlr 763 . . . . 5 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → (((1st𝑧) = (1st𝑘) ∧ (2nd𝑧) = (2nd𝑘)) ↔ 𝑧 = 𝑘))
203197, 199, 202mpbi2and 976 . . . 4 (((𝜑 ∧ (𝑧𝑆𝑘𝑆)) ∧ ((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁)) → 𝑧 = 𝑘)
204203ex 449 . . 3 ((𝜑 ∧ (𝑧𝑆𝑘𝑆)) → (((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁) → 𝑧 = 𝑘))
205204ralrimivva 3000 . 2 (𝜑 → ∀𝑧𝑆𝑘𝑆 (((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁) → 𝑧 = 𝑘))
206 fveq2 6229 . . . 4 (𝑧 = 𝑘 → (2nd𝑧) = (2nd𝑘))
207206eqeq1d 2653 . . 3 (𝑧 = 𝑘 → ((2nd𝑧) = 𝑁 ↔ (2nd𝑘) = 𝑁))
208207rmo4 3432 . 2 (∃*𝑧𝑆 (2nd𝑧) = 𝑁 ↔ ∀𝑧𝑆𝑘𝑆 (((2nd𝑧) = 𝑁 ∧ (2nd𝑘) = 𝑁) → 𝑧 = 𝑘))
209205, 208sylibr 224 1 (𝜑 → ∃*𝑧𝑆 (2nd𝑧) = 𝑁)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 382   ∧ 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  –onto→wfo 5924  –1-1-onto→wf1o 5925  ‘cfv 5926  (class class class)co 6690   ∘𝑓 cof 6937  1st c1st 7208  2nd c2nd 7209   ↑𝑚 cmap 7899  ℂcc 9972  ℝcr 9973  0cc0 9974  1c1 9975   + caddc 9977   < clt 10112   ≤ cle 10113   − cmin 10304  ℕcn 11058  ℤcz 11415  ℤ≥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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