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Theorem poimirlem18 33851
Description: Lemma for poimir 33866 stating that, given a face not on a front face of the main cube and a simplex in which it's opposite the first vertex on the walk, there exists exactly one other simplex containing it. (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)))
poimirlem22.2 (𝜑𝑇𝑆)
poimirlem18.3 ((𝜑𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 𝐾)
poimirlem18.4 (𝜑 → (2nd𝑇) = 0)
Assertion
Ref Expression
poimirlem18 (𝜑 → ∃!𝑧𝑆 𝑧𝑇)
Distinct variable groups:   𝑓,𝑗,𝑛,𝑝,𝑡,𝑦,𝑧   𝜑,𝑗,𝑛,𝑦   𝑗,𝐹,𝑛,𝑦   𝑗,𝑁,𝑛,𝑦   𝑇,𝑗,𝑛,𝑦   𝜑,𝑝,𝑡   𝑓,𝐾,𝑗,𝑛,𝑝,𝑡   𝑓,𝑁,𝑝,𝑡   𝑇,𝑓,𝑝   𝜑,𝑧   𝑓,𝐹,𝑝,𝑡,𝑧   𝑧,𝐾   𝑧,𝑁   𝑡,𝑇,𝑧   𝑆,𝑗,𝑛,𝑝,𝑡,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑓)   𝑆(𝑓)   𝐾(𝑦)

Proof of Theorem poimirlem18
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 poimir.0 . . 3 (𝜑𝑁 ∈ ℕ)
2 poimirlem22.s . . 3 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}))))}
3 poimirlem22.1 . . 3 (𝜑𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)))
4 poimirlem22.2 . . 3 (𝜑𝑇𝑆)
5 poimirlem18.3 . . 3 ((𝜑𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 𝐾)
6 poimirlem18.4 . . 3 (𝜑 → (2nd𝑇) = 0)
71, 2, 3, 4, 5, 6poimirlem17 33850 . 2 (𝜑 → ∃𝑧𝑆 𝑧𝑇)
86adantr 472 . . . . . . . 8 ((𝜑𝑧𝑆) → (2nd𝑇) = 0)
9 0nnn 11309 . . . . . . . . . . . . 13 ¬ 0 ∈ ℕ
10 elfznn 12577 . . . . . . . . . . . . 13 (0 ∈ (1...(𝑁 − 1)) → 0 ∈ ℕ)
119, 10mto 188 . . . . . . . . . . . 12 ¬ 0 ∈ (1...(𝑁 − 1))
12 eleq1 2832 . . . . . . . . . . . 12 ((2nd𝑧) = 0 → ((2nd𝑧) ∈ (1...(𝑁 − 1)) ↔ 0 ∈ (1...(𝑁 − 1))))
1311, 12mtbiri 318 . . . . . . . . . . 11 ((2nd𝑧) = 0 → ¬ (2nd𝑧) ∈ (1...(𝑁 − 1)))
1413necon2ai 2966 . . . . . . . . . 10 ((2nd𝑧) ∈ (1...(𝑁 − 1)) → (2nd𝑧) ≠ 0)
151ad2antrr 717 . . . . . . . . . . . . . 14 (((𝜑𝑧𝑆) ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → 𝑁 ∈ ℕ)
16 fveq2 6375 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑧 → (2nd𝑡) = (2nd𝑧))
1716breq2d 4821 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = 𝑧 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑧)))
1817ifbid 4265 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = 𝑧 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑧), 𝑦, (𝑦 + 1)))
1918csbeq1d 3698 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑧if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑧), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))
20 2fveq3 6380 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = 𝑧 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑧)))
21 2fveq3 6380 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑧 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑧)))
2221imaeq1d 5647 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑧 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑧)) “ (1...𝑗)))
2322xpeq1d 5306 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑧 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑧)) “ (1...𝑗)) × {1}))
2421imaeq1d 5647 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑧 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)))
2524xpeq1d 5306 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑧 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))
2623, 25uneq12d 3930 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = 𝑧 → ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) × {0})))
2720, 26oveq12d 6860 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = 𝑧 → ((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑧)) ∘𝑓 + ((((2nd ‘(1st𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))))
2827csbeq2dv 4153 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑧if(𝑦 < (2nd𝑧), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑧), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑧)) ∘𝑓 + ((((2nd ‘(1st𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))))
2919, 28eqtrd 2799 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑧if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑧), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑧)) ∘𝑓 + ((((2nd ‘(1st𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))))
3029mpteq2dv 4904 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑧 → (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})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑧), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑧)) ∘𝑓 + ((((2nd ‘(1st𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) × {0})))))
3130eqeq2d 2775 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑧 → (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})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑧), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑧)) ∘𝑓 + ((((2nd ‘(1st𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
3231, 2elrab2 3523 . . . . . . . . . . . . . . . 16 (𝑧𝑆 ↔ (𝑧 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑧), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑧)) ∘𝑓 + ((((2nd ‘(1st𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
3332simprbi 490 . . . . . . . . . . . . . . 15 (𝑧𝑆𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑧), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑧)) ∘𝑓 + ((((2nd ‘(1st𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) × {0})))))
3433ad2antlr 718 . . . . . . . . . . . . . 14 (((𝜑𝑧𝑆) ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑧), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑧)) ∘𝑓 + ((((2nd ‘(1st𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) × {0})))))
35 elrabi 3514 . . . . . . . . . . . . . . . . . . . 20 (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)))
3635, 2eleq2s 2862 . . . . . . . . . . . . . . . . . . 19 (𝑧𝑆𝑧 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
37 xp1st 7398 . . . . . . . . . . . . . . . . . . 19 (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)}))
3836, 37syl 17 . . . . . . . . . . . . . . . . . 18 (𝑧𝑆 → (1st𝑧) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
39 xp1st 7398 . . . . . . . . . . . . . . . . . 18 ((1st𝑧) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st𝑧)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
4038, 39syl 17 . . . . . . . . . . . . . . . . 17 (𝑧𝑆 → (1st ‘(1st𝑧)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
41 elmapi 8082 . . . . . . . . . . . . . . . . 17 ((1st ‘(1st𝑧)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st ‘(1st𝑧)):(1...𝑁)⟶(0..^𝐾))
4240, 41syl 17 . . . . . . . . . . . . . . . 16 (𝑧𝑆 → (1st ‘(1st𝑧)):(1...𝑁)⟶(0..^𝐾))
43 elfzoelz 12678 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (0..^𝐾) → 𝑛 ∈ ℤ)
4443ssriv 3765 . . . . . . . . . . . . . . . 16 (0..^𝐾) ⊆ ℤ
45 fss 6236 . . . . . . . . . . . . . . . 16 (((1st ‘(1st𝑧)):(1...𝑁)⟶(0..^𝐾) ∧ (0..^𝐾) ⊆ ℤ) → (1st ‘(1st𝑧)):(1...𝑁)⟶ℤ)
4642, 44, 45sylancl 580 . . . . . . . . . . . . . . 15 (𝑧𝑆 → (1st ‘(1st𝑧)):(1...𝑁)⟶ℤ)
4746ad2antlr 718 . . . . . . . . . . . . . 14 (((𝜑𝑧𝑆) ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → (1st ‘(1st𝑧)):(1...𝑁)⟶ℤ)
48 xp2nd 7399 . . . . . . . . . . . . . . . . 17 ((1st𝑧) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘(1st𝑧)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
4938, 48syl 17 . . . . . . . . . . . . . . . 16 (𝑧𝑆 → (2nd ‘(1st𝑧)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
50 fvex 6388 . . . . . . . . . . . . . . . . 17 (2nd ‘(1st𝑧)) ∈ V
51 f1oeq1 6310 . . . . . . . . . . . . . . . . 17 (𝑓 = (2nd ‘(1st𝑧)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁)))
5250, 51elab 3504 . . . . . . . . . . . . . . . 16 ((2nd ‘(1st𝑧)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
5349, 52sylib 209 . . . . . . . . . . . . . . 15 (𝑧𝑆 → (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
5453ad2antlr 718 . . . . . . . . . . . . . 14 (((𝜑𝑧𝑆) ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
55 simpr 477 . . . . . . . . . . . . . 14 (((𝜑𝑧𝑆) ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → (2nd𝑧) ∈ (1...(𝑁 − 1)))
5615, 34, 47, 54, 55poimirlem1 33834 . . . . . . . . . . . . 13 (((𝜑𝑧𝑆) ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑧) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑧))‘𝑛))
571ad2antrr 717 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd𝑇) ≠ (2nd𝑧)) → 𝑁 ∈ ℕ)
58 fveq2 6375 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 = 𝑇 → (2nd𝑡) = (2nd𝑇))
5958breq2d 4821 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = 𝑇 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑇)))
6059ifbid 4265 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑇 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)))
6160csbeq1d 3698 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑇if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))
62 2fveq3 6380 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = 𝑇 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
63 2fveq3 6380 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑡 = 𝑇 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑇)))
6463imaeq1d 5647 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑗)))
6564xpeq1d 5306 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}))
6663imaeq1d 5647 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)))
6766xpeq1d 5306 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
6865, 67uneq12d 3930 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = 𝑇 → ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))
6962, 68oveq12d 6860 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑇 → ((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
7069csbeq2dv 4153 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑇if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
7161, 70eqtrd 2799 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑇if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
7271mpteq2dv 4904 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = 𝑇 → (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})))) = (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})))))
7372eqeq2d 2775 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = 𝑇 → (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})))) ↔ 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}))))))
7473, 2elrab2 3523 . . . . . . . . . . . . . . . . . . . 20 (𝑇𝑆 ↔ (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}))))))
7574simprbi 490 . . . . . . . . . . . . . . . . . . 19 (𝑇𝑆𝐹 = (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})))))
764, 75syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑𝐹 = (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})))))
7776ad2antrr 717 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd𝑇) ≠ (2nd𝑧)) → 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})))))
78 elrabi 3514 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑇 ∈ {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}))))} → 𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
7978, 2eleq2s 2862 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑇𝑆𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
804, 79syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
81 xp1st 7398 . . . . . . . . . . . . . . . . . . . . . 22 (𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
8280, 81syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (1st𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
83 xp1st 7398 . . . . . . . . . . . . . . . . . . . . 21 ((1st𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
8482, 83syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
85 elmapi 8082 . . . . . . . . . . . . . . . . . . . 20 ((1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
8684, 85syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
87 fss 6236 . . . . . . . . . . . . . . . . . . 19 (((1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾) ∧ (0..^𝐾) ⊆ ℤ) → (1st ‘(1st𝑇)):(1...𝑁)⟶ℤ)
8886, 44, 87sylancl 580 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1st ‘(1st𝑇)):(1...𝑁)⟶ℤ)
8988ad2antrr 717 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd𝑇) ≠ (2nd𝑧)) → (1st ‘(1st𝑇)):(1...𝑁)⟶ℤ)
90 xp2nd 7399 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
9182, 90syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
92 fvex 6388 . . . . . . . . . . . . . . . . . . . 20 (2nd ‘(1st𝑇)) ∈ V
93 f1oeq1 6310 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = (2nd ‘(1st𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
9492, 93elab 3504 . . . . . . . . . . . . . . . . . . 19 ((2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
9591, 94sylib 209 . . . . . . . . . . . . . . . . . 18 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
9695ad2antrr 717 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd𝑇) ≠ (2nd𝑧)) → (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
97 simplr 785 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd𝑇) ≠ (2nd𝑧)) → (2nd𝑧) ∈ (1...(𝑁 − 1)))
98 xp2nd 7399 . . . . . . . . . . . . . . . . . . . 20 (𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd𝑇) ∈ (0...𝑁))
9980, 98syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (2nd𝑇) ∈ (0...𝑁))
10099adantr 472 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → (2nd𝑇) ∈ (0...𝑁))
101 eldifsn 4472 . . . . . . . . . . . . . . . . . . 19 ((2nd𝑇) ∈ ((0...𝑁) ∖ {(2nd𝑧)}) ↔ ((2nd𝑇) ∈ (0...𝑁) ∧ (2nd𝑇) ≠ (2nd𝑧)))
102101biimpri 219 . . . . . . . . . . . . . . . . . 18 (((2nd𝑇) ∈ (0...𝑁) ∧ (2nd𝑇) ≠ (2nd𝑧)) → (2nd𝑇) ∈ ((0...𝑁) ∖ {(2nd𝑧)}))
103100, 102sylan 575 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd𝑇) ≠ (2nd𝑧)) → (2nd𝑇) ∈ ((0...𝑁) ∖ {(2nd𝑧)}))
10457, 77, 89, 96, 97, 103poimirlem2 33835 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd𝑇) ≠ (2nd𝑧)) → ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑧) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑧))‘𝑛))
105104ex 401 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → ((2nd𝑇) ≠ (2nd𝑧) → ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑧) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑧))‘𝑛)))
106105necon1bd 2955 . . . . . . . . . . . . . 14 ((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → (¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑧) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑧))‘𝑛) → (2nd𝑇) = (2nd𝑧)))
107106adantlr 706 . . . . . . . . . . . . 13 (((𝜑𝑧𝑆) ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → (¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑧) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑧))‘𝑛) → (2nd𝑇) = (2nd𝑧)))
10856, 107mpd 15 . . . . . . . . . . . 12 (((𝜑𝑧𝑆) ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → (2nd𝑇) = (2nd𝑧))
109108neeq1d 2996 . . . . . . . . . . 11 (((𝜑𝑧𝑆) ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → ((2nd𝑇) ≠ 0 ↔ (2nd𝑧) ≠ 0))
110109exbiri 845 . . . . . . . . . 10 ((𝜑𝑧𝑆) → ((2nd𝑧) ∈ (1...(𝑁 − 1)) → ((2nd𝑧) ≠ 0 → (2nd𝑇) ≠ 0)))
11114, 110mpdi 45 . . . . . . . . 9 ((𝜑𝑧𝑆) → ((2nd𝑧) ∈ (1...(𝑁 − 1)) → (2nd𝑇) ≠ 0))
112111necon2bd 2953 . . . . . . . 8 ((𝜑𝑧𝑆) → ((2nd𝑇) = 0 → ¬ (2nd𝑧) ∈ (1...(𝑁 − 1))))
1138, 112mpd 15 . . . . . . 7 ((𝜑𝑧𝑆) → ¬ (2nd𝑧) ∈ (1...(𝑁 − 1)))
114 xp2nd 7399 . . . . . . . . 9 (𝑧 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd𝑧) ∈ (0...𝑁))
11536, 114syl 17 . . . . . . . 8 (𝑧𝑆 → (2nd𝑧) ∈ (0...𝑁))
1161nncnd 11292 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑁 ∈ ℂ)
117 npcan1 10709 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
118116, 117syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
119 nnuz 11923 . . . . . . . . . . . . . . . . . . 19 ℕ = (ℤ‘1)
1201, 119syl6eleq 2854 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ (ℤ‘1))
121118, 120eqeltrd 2844 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘1))
1221nnzd 11728 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑁 ∈ ℤ)
123 peano2zm 11667 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
124122, 123syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑁 − 1) ∈ ℤ)
125 uzid 11901 . . . . . . . . . . . . . . . . . . 19 ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)))
126 peano2uz 11941 . . . . . . . . . . . . . . . . . . 19 ((𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
127124, 125, 1263syl 18 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
128118, 127eqeltrrd 2845 . . . . . . . . . . . . . . . . 17 (𝜑𝑁 ∈ (ℤ‘(𝑁 − 1)))
129 fzsplit2 12573 . . . . . . . . . . . . . . . . 17 ((((𝑁 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ‘(𝑁 − 1))) → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
130121, 128, 129syl2anc 579 . . . . . . . . . . . . . . . 16 (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
131118oveq1d 6857 . . . . . . . . . . . . . . . . . 18 (𝜑 → (((𝑁 − 1) + 1)...𝑁) = (𝑁...𝑁))
132 fzsn 12590 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
133122, 132syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑁...𝑁) = {𝑁})
134131, 133eqtrd 2799 . . . . . . . . . . . . . . . . 17 (𝜑 → (((𝑁 − 1) + 1)...𝑁) = {𝑁})
135134uneq2d 3929 . . . . . . . . . . . . . . . 16 (𝜑 → ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = ((1...(𝑁 − 1)) ∪ {𝑁}))
136130, 135eqtrd 2799 . . . . . . . . . . . . . . 15 (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁}))
137136eleq2d 2830 . . . . . . . . . . . . . 14 (𝜑 → ((2nd𝑧) ∈ (1...𝑁) ↔ (2nd𝑧) ∈ ((1...(𝑁 − 1)) ∪ {𝑁})))
138137notbid 309 . . . . . . . . . . . . 13 (𝜑 → (¬ (2nd𝑧) ∈ (1...𝑁) ↔ ¬ (2nd𝑧) ∈ ((1...(𝑁 − 1)) ∪ {𝑁})))
139 ioran 1006 . . . . . . . . . . . . . 14 (¬ ((2nd𝑧) ∈ (1...(𝑁 − 1)) ∨ (2nd𝑧) = 𝑁) ↔ (¬ (2nd𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd𝑧) = 𝑁))
140 elun 3915 . . . . . . . . . . . . . . 15 ((2nd𝑧) ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ ((2nd𝑧) ∈ (1...(𝑁 − 1)) ∨ (2nd𝑧) ∈ {𝑁}))
141 fvex 6388 . . . . . . . . . . . . . . . . 17 (2nd𝑧) ∈ V
142141elsn 4349 . . . . . . . . . . . . . . . 16 ((2nd𝑧) ∈ {𝑁} ↔ (2nd𝑧) = 𝑁)
143142orbi2i 936 . . . . . . . . . . . . . . 15 (((2nd𝑧) ∈ (1...(𝑁 − 1)) ∨ (2nd𝑧) ∈ {𝑁}) ↔ ((2nd𝑧) ∈ (1...(𝑁 − 1)) ∨ (2nd𝑧) = 𝑁))
144140, 143bitri 266 . . . . . . . . . . . . . 14 ((2nd𝑧) ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ ((2nd𝑧) ∈ (1...(𝑁 − 1)) ∨ (2nd𝑧) = 𝑁))
145139, 144xchnxbir 324 . . . . . . . . . . . . 13 (¬ (2nd𝑧) ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ (¬ (2nd𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd𝑧) = 𝑁))
146138, 145syl6bb 278 . . . . . . . . . . . 12 (𝜑 → (¬ (2nd𝑧) ∈ (1...𝑁) ↔ (¬ (2nd𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd𝑧) = 𝑁)))
147146anbi2d 622 . . . . . . . . . . 11 (𝜑 → (((2nd𝑧) ∈ (0...𝑁) ∧ ¬ (2nd𝑧) ∈ (1...𝑁)) ↔ ((2nd𝑧) ∈ (0...𝑁) ∧ (¬ (2nd𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd𝑧) = 𝑁))))
1481nnnn0d 11598 . . . . . . . . . . . . . . . . 17 (𝜑𝑁 ∈ ℕ0)
149 nn0uz 11922 . . . . . . . . . . . . . . . . 17 0 = (ℤ‘0)
150148, 149syl6eleq 2854 . . . . . . . . . . . . . . . 16 (𝜑𝑁 ∈ (ℤ‘0))
151 fzpred 12596 . . . . . . . . . . . . . . . 16 (𝑁 ∈ (ℤ‘0) → (0...𝑁) = ({0} ∪ ((0 + 1)...𝑁)))
152150, 151syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (0...𝑁) = ({0} ∪ ((0 + 1)...𝑁)))
153152difeq1d 3889 . . . . . . . . . . . . . 14 (𝜑 → ((0...𝑁) ∖ (1...𝑁)) = (({0} ∪ ((0 + 1)...𝑁)) ∖ (1...𝑁)))
154 difun2 4208 . . . . . . . . . . . . . . 15 (({0} ∪ (1...𝑁)) ∖ (1...𝑁)) = ({0} ∖ (1...𝑁))
155 0p1e1 11401 . . . . . . . . . . . . . . . . . 18 (0 + 1) = 1
156155oveq1i 6852 . . . . . . . . . . . . . . . . 17 ((0 + 1)...𝑁) = (1...𝑁)
157156uneq2i 3926 . . . . . . . . . . . . . . . 16 ({0} ∪ ((0 + 1)...𝑁)) = ({0} ∪ (1...𝑁))
158157difeq1i 3886 . . . . . . . . . . . . . . 15 (({0} ∪ ((0 + 1)...𝑁)) ∖ (1...𝑁)) = (({0} ∪ (1...𝑁)) ∖ (1...𝑁))
159 incom 3967 . . . . . . . . . . . . . . . . 17 ({0} ∩ (1...𝑁)) = ((1...𝑁) ∩ {0})
160 elfznn 12577 . . . . . . . . . . . . . . . . . . 19 (0 ∈ (1...𝑁) → 0 ∈ ℕ)
1619, 160mto 188 . . . . . . . . . . . . . . . . . 18 ¬ 0 ∈ (1...𝑁)
162 disjsn 4402 . . . . . . . . . . . . . . . . . 18 (((1...𝑁) ∩ {0}) = ∅ ↔ ¬ 0 ∈ (1...𝑁))
163161, 162mpbir 222 . . . . . . . . . . . . . . . . 17 ((1...𝑁) ∩ {0}) = ∅
164159, 163eqtri 2787 . . . . . . . . . . . . . . . 16 ({0} ∩ (1...𝑁)) = ∅
165 disj3 4182 . . . . . . . . . . . . . . . 16 (({0} ∩ (1...𝑁)) = ∅ ↔ {0} = ({0} ∖ (1...𝑁)))
166164, 165mpbi 221 . . . . . . . . . . . . . . 15 {0} = ({0} ∖ (1...𝑁))
167154, 158, 1663eqtr4i 2797 . . . . . . . . . . . . . 14 (({0} ∪ ((0 + 1)...𝑁)) ∖ (1...𝑁)) = {0}
168153, 167syl6eq 2815 . . . . . . . . . . . . 13 (𝜑 → ((0...𝑁) ∖ (1...𝑁)) = {0})
169168eleq2d 2830 . . . . . . . . . . . 12 (𝜑 → ((2nd𝑧) ∈ ((0...𝑁) ∖ (1...𝑁)) ↔ (2nd𝑧) ∈ {0}))
170 eldif 3742 . . . . . . . . . . . 12 ((2nd𝑧) ∈ ((0...𝑁) ∖ (1...𝑁)) ↔ ((2nd𝑧) ∈ (0...𝑁) ∧ ¬ (2nd𝑧) ∈ (1...𝑁)))
171141elsn 4349 . . . . . . . . . . . 12 ((2nd𝑧) ∈ {0} ↔ (2nd𝑧) = 0)
172169, 170, 1713bitr3g 304 . . . . . . . . . . 11 (𝜑 → (((2nd𝑧) ∈ (0...𝑁) ∧ ¬ (2nd𝑧) ∈ (1...𝑁)) ↔ (2nd𝑧) = 0))
173147, 172bitr3d 272 . . . . . . . . . 10 (𝜑 → (((2nd𝑧) ∈ (0...𝑁) ∧ (¬ (2nd𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd𝑧) = 𝑁)) ↔ (2nd𝑧) = 0))
174173biimpd 220 . . . . . . . . 9 (𝜑 → (((2nd𝑧) ∈ (0...𝑁) ∧ (¬ (2nd𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd𝑧) = 𝑁)) → (2nd𝑧) = 0))
175174expdimp 444 . . . . . . . 8 ((𝜑 ∧ (2nd𝑧) ∈ (0...𝑁)) → ((¬ (2nd𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd𝑧) = 𝑁) → (2nd𝑧) = 0))
176115, 175sylan2 586 . . . . . . 7 ((𝜑𝑧𝑆) → ((¬ (2nd𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd𝑧) = 𝑁) → (2nd𝑧) = 0))
177113, 176mpand 686 . . . . . 6 ((𝜑𝑧𝑆) → (¬ (2nd𝑧) = 𝑁 → (2nd𝑧) = 0))
1781, 2, 3poimirlem13 33846 . . . . . . . . . 10 (𝜑 → ∃*𝑧𝑆 (2nd𝑧) = 0)
179 fveqeq2 6384 . . . . . . . . . . 11 (𝑧 = 𝑠 → ((2nd𝑧) = 0 ↔ (2nd𝑠) = 0))
180179rmo4 3558 . . . . . . . . . 10 (∃*𝑧𝑆 (2nd𝑧) = 0 ↔ ∀𝑧𝑆𝑠𝑆 (((2nd𝑧) = 0 ∧ (2nd𝑠) = 0) → 𝑧 = 𝑠))
181178, 180sylib 209 . . . . . . . . 9 (𝜑 → ∀𝑧𝑆𝑠𝑆 (((2nd𝑧) = 0 ∧ (2nd𝑠) = 0) → 𝑧 = 𝑠))
182181r19.21bi 3079 . . . . . . . 8 ((𝜑𝑧𝑆) → ∀𝑠𝑆 (((2nd𝑧) = 0 ∧ (2nd𝑠) = 0) → 𝑧 = 𝑠))
1834adantr 472 . . . . . . . 8 ((𝜑𝑧𝑆) → 𝑇𝑆)
184 fveqeq2 6384 . . . . . . . . . . 11 (𝑠 = 𝑇 → ((2nd𝑠) = 0 ↔ (2nd𝑇) = 0))
185184anbi2d 622 . . . . . . . . . 10 (𝑠 = 𝑇 → (((2nd𝑧) = 0 ∧ (2nd𝑠) = 0) ↔ ((2nd𝑧) = 0 ∧ (2nd𝑇) = 0)))
186 eqeq2 2776 . . . . . . . . . 10 (𝑠 = 𝑇 → (𝑧 = 𝑠𝑧 = 𝑇))
187185, 186imbi12d 335 . . . . . . . . 9 (𝑠 = 𝑇 → ((((2nd𝑧) = 0 ∧ (2nd𝑠) = 0) → 𝑧 = 𝑠) ↔ (((2nd𝑧) = 0 ∧ (2nd𝑇) = 0) → 𝑧 = 𝑇)))
188187rspccv 3458 . . . . . . . 8 (∀𝑠𝑆 (((2nd𝑧) = 0 ∧ (2nd𝑠) = 0) → 𝑧 = 𝑠) → (𝑇𝑆 → (((2nd𝑧) = 0 ∧ (2nd𝑇) = 0) → 𝑧 = 𝑇)))
189182, 183, 188sylc 65 . . . . . . 7 ((𝜑𝑧𝑆) → (((2nd𝑧) = 0 ∧ (2nd𝑇) = 0) → 𝑧 = 𝑇))
1908, 189mpan2d 685 . . . . . 6 ((𝜑𝑧𝑆) → ((2nd𝑧) = 0 → 𝑧 = 𝑇))
191177, 190syld 47 . . . . 5 ((𝜑𝑧𝑆) → (¬ (2nd𝑧) = 𝑁𝑧 = 𝑇))
192191necon1ad 2954 . . . 4 ((𝜑𝑧𝑆) → (𝑧𝑇 → (2nd𝑧) = 𝑁))
193192ralrimiva 3113 . . 3 (𝜑 → ∀𝑧𝑆 (𝑧𝑇 → (2nd𝑧) = 𝑁))
1941, 2, 3poimirlem14 33847 . . 3 (𝜑 → ∃*𝑧𝑆 (2nd𝑧) = 𝑁)
195 rmoim 3568 . . 3 (∀𝑧𝑆 (𝑧𝑇 → (2nd𝑧) = 𝑁) → (∃*𝑧𝑆 (2nd𝑧) = 𝑁 → ∃*𝑧𝑆 𝑧𝑇))
196193, 194, 195sylc 65 . 2 (𝜑 → ∃*𝑧𝑆 𝑧𝑇)
197 reu5 3307 . 2 (∃!𝑧𝑆 𝑧𝑇 ↔ (∃𝑧𝑆 𝑧𝑇 ∧ ∃*𝑧𝑆 𝑧𝑇))
1987, 196, 197sylanbrc 578 1 (𝜑 → ∃!𝑧𝑆 𝑧𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wo 873   = wceq 1652  wcel 2155  {cab 2751  wne 2937  wral 3055  wrex 3056  ∃!wreu 3057  ∃*wrmo 3058  {crab 3059  csb 3691  cdif 3729  cun 3730  cin 3731  wss 3732  c0 4079  ifcif 4243  {csn 4334   class class class wbr 4809  cmpt 4888   × cxp 5275  ran crn 5278  cima 5280  wf 6064  1-1-ontowf1o 6067  cfv 6068  (class class class)co 6842  𝑓 cof 7093  1st c1st 7364  2nd c2nd 7365  𝑚 cmap 8060  cc 10187  0cc0 10189  1c1 10190   + caddc 10192   < clt 10328  cmin 10520  cn 11274  0cn0 11538  cz 11624  cuz 11886  ...cfz 12533  ..^cfzo 12673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-er 7947  df-map 8062  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-n0 11539  df-z 11625  df-uz 11887  df-fz 12534  df-fzo 12674
This theorem is referenced by:  poimirlem22  33855
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