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Theorem poimirlem18 33098
Description: Lemma for poimir 33113 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 33097 . 2 (𝜑 → ∃𝑧𝑆 𝑧𝑇)
86adantr 481 . . . . . . . 8 ((𝜑𝑧𝑆) → (2nd𝑇) = 0)
9 0nnn 11012 . . . . . . . . . . . . 13 ¬ 0 ∈ ℕ
10 elfznn 12328 . . . . . . . . . . . . 13 (0 ∈ (1...(𝑁 − 1)) → 0 ∈ ℕ)
119, 10mto 188 . . . . . . . . . . . 12 ¬ 0 ∈ (1...(𝑁 − 1))
12 eleq1 2686 . . . . . . . . . . . 12 ((2nd𝑧) = 0 → ((2nd𝑧) ∈ (1...(𝑁 − 1)) ↔ 0 ∈ (1...(𝑁 − 1))))
1311, 12mtbiri 317 . . . . . . . . . . 11 ((2nd𝑧) = 0 → ¬ (2nd𝑧) ∈ (1...(𝑁 − 1)))
1413necon2ai 2819 . . . . . . . . . 10 ((2nd𝑧) ∈ (1...(𝑁 − 1)) → (2nd𝑧) ≠ 0)
151ad2antrr 761 . . . . . . . . . . . . . 14 (((𝜑𝑧𝑆) ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → 𝑁 ∈ ℕ)
16 fveq2 6158 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑧 → (2nd𝑡) = (2nd𝑧))
1716breq2d 4635 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = 𝑧 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑧)))
1817ifbid 4086 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = 𝑧 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑧), 𝑦, (𝑦 + 1)))
1918csbeq1d 3526 . . . . . . . . . . . . . . . . . . . 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 fveq2 6158 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑧 → (1st𝑡) = (1st𝑧))
2120fveq2d 6162 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = 𝑧 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑧)))
2220fveq2d 6162 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑧 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑧)))
2322imaeq1d 5434 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑧 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑧)) “ (1...𝑗)))
2423xpeq1d 5108 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑧 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑧)) “ (1...𝑗)) × {1}))
2522imaeq1d 5434 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑧 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)))
2625xpeq1d 5108 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑧 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))
2724, 26uneq12d 3752 . . . . . . . . . . . . . . . . . . . . . 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})))
2821, 27oveq12d 6633 . . . . . . . . . . . . . . . . . . . . 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}))))
2928csbeq2dv 3970 . . . . . . . . . . . . . . . . . . . 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}))))
3019, 29eqtrd 2655 . . . . . . . . . . . . . . . . . . 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}))))
3130mpteq2dv 4715 . . . . . . . . . . . . . . . . . 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})))))
3231eqeq2d 2631 . . . . . . . . . . . . . . . . 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}))))))
3332, 2elrab2 3353 . . . . . . . . . . . . . . . 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}))))))
3433simprbi 480 . . . . . . . . . . . . . . 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})))))
3534ad2antlr 762 . . . . . . . . . . . . . 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})))))
36 elrabi 3347 . . . . . . . . . . . . . . . . . . . 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)))
3736, 2eleq2s 2716 . . . . . . . . . . . . . . . . . . 19 (𝑧𝑆𝑧 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
38 xp1st 7158 . . . . . . . . . . . . . . . . . . 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)}))
3937, 38syl 17 . . . . . . . . . . . . . . . . . 18 (𝑧𝑆 → (1st𝑧) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
40 xp1st 7158 . . . . . . . . . . . . . . . . . 18 ((1st𝑧) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st𝑧)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
4139, 40syl 17 . . . . . . . . . . . . . . . . 17 (𝑧𝑆 → (1st ‘(1st𝑧)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
42 elmapi 7839 . . . . . . . . . . . . . . . . 17 ((1st ‘(1st𝑧)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st ‘(1st𝑧)):(1...𝑁)⟶(0..^𝐾))
4341, 42syl 17 . . . . . . . . . . . . . . . 16 (𝑧𝑆 → (1st ‘(1st𝑧)):(1...𝑁)⟶(0..^𝐾))
44 elfzoelz 12427 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (0..^𝐾) → 𝑛 ∈ ℤ)
4544ssriv 3592 . . . . . . . . . . . . . . . 16 (0..^𝐾) ⊆ ℤ
46 fss 6023 . . . . . . . . . . . . . . . 16 (((1st ‘(1st𝑧)):(1...𝑁)⟶(0..^𝐾) ∧ (0..^𝐾) ⊆ ℤ) → (1st ‘(1st𝑧)):(1...𝑁)⟶ℤ)
4743, 45, 46sylancl 693 . . . . . . . . . . . . . . 15 (𝑧𝑆 → (1st ‘(1st𝑧)):(1...𝑁)⟶ℤ)
4847ad2antlr 762 . . . . . . . . . . . . . 14 (((𝜑𝑧𝑆) ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → (1st ‘(1st𝑧)):(1...𝑁)⟶ℤ)
49 xp2nd 7159 . . . . . . . . . . . . . . . . 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)})
5039, 49syl 17 . . . . . . . . . . . . . . . 16 (𝑧𝑆 → (2nd ‘(1st𝑧)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
51 fvex 6168 . . . . . . . . . . . . . . . . 17 (2nd ‘(1st𝑧)) ∈ V
52 f1oeq1 6094 . . . . . . . . . . . . . . . . 17 (𝑓 = (2nd ‘(1st𝑧)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁)))
5351, 52elab 3338 . . . . . . . . . . . . . . . 16 ((2nd ‘(1st𝑧)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
5450, 53sylib 208 . . . . . . . . . . . . . . 15 (𝑧𝑆 → (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
5554ad2antlr 762 . . . . . . . . . . . . . 14 (((𝜑𝑧𝑆) ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
56 simpr 477 . . . . . . . . . . . . . 14 (((𝜑𝑧𝑆) ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → (2nd𝑧) ∈ (1...(𝑁 − 1)))
5715, 35, 48, 55, 56poimirlem1 33081 . . . . . . . . . . . . 13 (((𝜑𝑧𝑆) ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑧) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑧))‘𝑛))
581ad2antrr 761 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd𝑇) ≠ (2nd𝑧)) → 𝑁 ∈ ℕ)
59 fveq2 6158 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 = 𝑇 → (2nd𝑡) = (2nd𝑇))
6059breq2d 4635 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = 𝑇 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑇)))
6160ifbid 4086 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑇 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)))
6261csbeq1d 3526 . . . . . . . . . . . . . . . . . . . . . . . 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}))))
63 fveq2 6158 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 = 𝑇 → (1st𝑡) = (1st𝑇))
6463fveq2d 6162 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = 𝑇 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
6563fveq2d 6162 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑡 = 𝑇 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑇)))
6665imaeq1d 5434 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑗)))
6766xpeq1d 5108 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}))
6865imaeq1d 5434 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)))
6968xpeq1d 5108 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
7067, 69uneq12d 3752 . . . . . . . . . . . . . . . . . . . . . . . . . 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})))
7164, 70oveq12d 6633 . . . . . . . . . . . . . . . . . . . . . . . . 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}))))
7271csbeq2dv 3970 . . . . . . . . . . . . . . . . . . . . . . . 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}))))
7362, 72eqtrd 2655 . . . . . . . . . . . . . . . . . . . . . . 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}))))
7473mpteq2dv 4715 . . . . . . . . . . . . . . . . . . . . . 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})))))
7574eqeq2d 2631 . . . . . . . . . . . . . . . . . . . . 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}))))))
7675, 2elrab2 3353 . . . . . . . . . . . . . . . . . . . 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}))))))
7776simprbi 480 . . . . . . . . . . . . . . . . . . 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})))))
784, 77syl 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})))))
7978ad2antrr 761 . . . . . . . . . . . . . . . . 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})))))
80 elrabi 3347 . . . . . . . . . . . . . . . . . . . . . . . 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)))
8180, 2eleq2s 2716 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑇𝑆𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
824, 81syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
83 xp1st 7158 . . . . . . . . . . . . . . . . . . . . . 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)}))
8482, 83syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (1st𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
85 xp1st 7158 . . . . . . . . . . . . . . . . . . . . 21 ((1st𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
8684, 85syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
87 elmapi 7839 . . . . . . . . . . . . . . . . . . . 20 ((1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
8886, 87syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
89 fss 6023 . . . . . . . . . . . . . . . . . . 19 (((1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾) ∧ (0..^𝐾) ⊆ ℤ) → (1st ‘(1st𝑇)):(1...𝑁)⟶ℤ)
9088, 45, 89sylancl 693 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1st ‘(1st𝑇)):(1...𝑁)⟶ℤ)
9190ad2antrr 761 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd𝑇) ≠ (2nd𝑧)) → (1st ‘(1st𝑇)):(1...𝑁)⟶ℤ)
92 xp2nd 7159 . . . . . . . . . . . . . . . . . . . 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)})
9384, 92syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
94 fvex 6168 . . . . . . . . . . . . . . . . . . . 20 (2nd ‘(1st𝑇)) ∈ V
95 f1oeq1 6094 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = (2nd ‘(1st𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
9694, 95elab 3338 . . . . . . . . . . . . . . . . . . 19 ((2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
9793, 96sylib 208 . . . . . . . . . . . . . . . . . 18 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
9897ad2antrr 761 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd𝑇) ≠ (2nd𝑧)) → (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
99 simplr 791 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd𝑇) ≠ (2nd𝑧)) → (2nd𝑧) ∈ (1...(𝑁 − 1)))
100 xp2nd 7159 . . . . . . . . . . . . . . . . . . . 20 (𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd𝑇) ∈ (0...𝑁))
10182, 100syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (2nd𝑇) ∈ (0...𝑁))
102101adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → (2nd𝑇) ∈ (0...𝑁))
103 eldifsn 4294 . . . . . . . . . . . . . . . . . . 19 ((2nd𝑇) ∈ ((0...𝑁) ∖ {(2nd𝑧)}) ↔ ((2nd𝑇) ∈ (0...𝑁) ∧ (2nd𝑇) ≠ (2nd𝑧)))
104103biimpri 218 . . . . . . . . . . . . . . . . . 18 (((2nd𝑇) ∈ (0...𝑁) ∧ (2nd𝑇) ≠ (2nd𝑧)) → (2nd𝑇) ∈ ((0...𝑁) ∖ {(2nd𝑧)}))
105102, 104sylan 488 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd𝑇) ≠ (2nd𝑧)) → (2nd𝑇) ∈ ((0...𝑁) ∖ {(2nd𝑧)}))
10658, 79, 91, 98, 99, 105poimirlem2 33082 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd𝑇) ≠ (2nd𝑧)) → ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑧) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑧))‘𝑛))
107106ex 450 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → ((2nd𝑇) ≠ (2nd𝑧) → ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑧) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑧))‘𝑛)))
108107necon1bd 2808 . . . . . . . . . . . . . 14 ((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → (¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑧) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑧))‘𝑛) → (2nd𝑇) = (2nd𝑧)))
109108adantlr 750 . . . . . . . . . . . . 13 (((𝜑𝑧𝑆) ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → (¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑧) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑧))‘𝑛) → (2nd𝑇) = (2nd𝑧)))
11057, 109mpd 15 . . . . . . . . . . . 12 (((𝜑𝑧𝑆) ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → (2nd𝑇) = (2nd𝑧))
111110neeq1d 2849 . . . . . . . . . . 11 (((𝜑𝑧𝑆) ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → ((2nd𝑇) ≠ 0 ↔ (2nd𝑧) ≠ 0))
112111exbiri 651 . . . . . . . . . 10 ((𝜑𝑧𝑆) → ((2nd𝑧) ∈ (1...(𝑁 − 1)) → ((2nd𝑧) ≠ 0 → (2nd𝑇) ≠ 0)))
11314, 112mpdi 45 . . . . . . . . 9 ((𝜑𝑧𝑆) → ((2nd𝑧) ∈ (1...(𝑁 − 1)) → (2nd𝑇) ≠ 0))
114113necon2bd 2806 . . . . . . . 8 ((𝜑𝑧𝑆) → ((2nd𝑇) = 0 → ¬ (2nd𝑧) ∈ (1...(𝑁 − 1))))
1158, 114mpd 15 . . . . . . 7 ((𝜑𝑧𝑆) → ¬ (2nd𝑧) ∈ (1...(𝑁 − 1)))
116 xp2nd 7159 . . . . . . . . 9 (𝑧 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd𝑧) ∈ (0...𝑁))
11737, 116syl 17 . . . . . . . 8 (𝑧𝑆 → (2nd𝑧) ∈ (0...𝑁))
1181nncnd 10996 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑁 ∈ ℂ)
119 npcan1 10415 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
120118, 119syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
121 nnuz 11683 . . . . . . . . . . . . . . . . . . 19 ℕ = (ℤ‘1)
1221, 121syl6eleq 2708 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ (ℤ‘1))
123120, 122eqeltrd 2698 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘1))
1241nnzd 11441 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑁 ∈ ℤ)
125 peano2zm 11380 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
126124, 125syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑁 − 1) ∈ ℤ)
127 uzid 11662 . . . . . . . . . . . . . . . . . . 19 ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)))
128 peano2uz 11701 . . . . . . . . . . . . . . . . . . 19 ((𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
129126, 127, 1283syl 18 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
130120, 129eqeltrrd 2699 . . . . . . . . . . . . . . . . 17 (𝜑𝑁 ∈ (ℤ‘(𝑁 − 1)))
131 fzsplit2 12324 . . . . . . . . . . . . . . . . 17 ((((𝑁 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ‘(𝑁 − 1))) → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
132123, 130, 131syl2anc 692 . . . . . . . . . . . . . . . 16 (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
133120oveq1d 6630 . . . . . . . . . . . . . . . . . 18 (𝜑 → (((𝑁 − 1) + 1)...𝑁) = (𝑁...𝑁))
134 fzsn 12341 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
135124, 134syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑁...𝑁) = {𝑁})
136133, 135eqtrd 2655 . . . . . . . . . . . . . . . . 17 (𝜑 → (((𝑁 − 1) + 1)...𝑁) = {𝑁})
137136uneq2d 3751 . . . . . . . . . . . . . . . 16 (𝜑 → ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = ((1...(𝑁 − 1)) ∪ {𝑁}))
138132, 137eqtrd 2655 . . . . . . . . . . . . . . 15 (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁}))
139138eleq2d 2684 . . . . . . . . . . . . . 14 (𝜑 → ((2nd𝑧) ∈ (1...𝑁) ↔ (2nd𝑧) ∈ ((1...(𝑁 − 1)) ∪ {𝑁})))
140139notbid 308 . . . . . . . . . . . . 13 (𝜑 → (¬ (2nd𝑧) ∈ (1...𝑁) ↔ ¬ (2nd𝑧) ∈ ((1...(𝑁 − 1)) ∪ {𝑁})))
141 ioran 511 . . . . . . . . . . . . . 14 (¬ ((2nd𝑧) ∈ (1...(𝑁 − 1)) ∨ (2nd𝑧) = 𝑁) ↔ (¬ (2nd𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd𝑧) = 𝑁))
142 elun 3737 . . . . . . . . . . . . . . 15 ((2nd𝑧) ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ ((2nd𝑧) ∈ (1...(𝑁 − 1)) ∨ (2nd𝑧) ∈ {𝑁}))
143 fvex 6168 . . . . . . . . . . . . . . . . 17 (2nd𝑧) ∈ V
144143elsn 4170 . . . . . . . . . . . . . . . 16 ((2nd𝑧) ∈ {𝑁} ↔ (2nd𝑧) = 𝑁)
145144orbi2i 541 . . . . . . . . . . . . . . 15 (((2nd𝑧) ∈ (1...(𝑁 − 1)) ∨ (2nd𝑧) ∈ {𝑁}) ↔ ((2nd𝑧) ∈ (1...(𝑁 − 1)) ∨ (2nd𝑧) = 𝑁))
146142, 145bitri 264 . . . . . . . . . . . . . 14 ((2nd𝑧) ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ ((2nd𝑧) ∈ (1...(𝑁 − 1)) ∨ (2nd𝑧) = 𝑁))
147141, 146xchnxbir 323 . . . . . . . . . . . . 13 (¬ (2nd𝑧) ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ (¬ (2nd𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd𝑧) = 𝑁))
148140, 147syl6bb 276 . . . . . . . . . . . 12 (𝜑 → (¬ (2nd𝑧) ∈ (1...𝑁) ↔ (¬ (2nd𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd𝑧) = 𝑁)))
149148anbi2d 739 . . . . . . . . . . 11 (𝜑 → (((2nd𝑧) ∈ (0...𝑁) ∧ ¬ (2nd𝑧) ∈ (1...𝑁)) ↔ ((2nd𝑧) ∈ (0...𝑁) ∧ (¬ (2nd𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd𝑧) = 𝑁))))
1501nnnn0d 11311 . . . . . . . . . . . . . . . . 17 (𝜑𝑁 ∈ ℕ0)
151 nn0uz 11682 . . . . . . . . . . . . . . . . 17 0 = (ℤ‘0)
152150, 151syl6eleq 2708 . . . . . . . . . . . . . . . 16 (𝜑𝑁 ∈ (ℤ‘0))
153 fzpred 12347 . . . . . . . . . . . . . . . 16 (𝑁 ∈ (ℤ‘0) → (0...𝑁) = ({0} ∪ ((0 + 1)...𝑁)))
154152, 153syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (0...𝑁) = ({0} ∪ ((0 + 1)...𝑁)))
155154difeq1d 3711 . . . . . . . . . . . . . 14 (𝜑 → ((0...𝑁) ∖ (1...𝑁)) = (({0} ∪ ((0 + 1)...𝑁)) ∖ (1...𝑁)))
156 difun2 4026 . . . . . . . . . . . . . . 15 (({0} ∪ (1...𝑁)) ∖ (1...𝑁)) = ({0} ∖ (1...𝑁))
157 0p1e1 11092 . . . . . . . . . . . . . . . . . 18 (0 + 1) = 1
158157oveq1i 6625 . . . . . . . . . . . . . . . . 17 ((0 + 1)...𝑁) = (1...𝑁)
159158uneq2i 3748 . . . . . . . . . . . . . . . 16 ({0} ∪ ((0 + 1)...𝑁)) = ({0} ∪ (1...𝑁))
160159difeq1i 3708 . . . . . . . . . . . . . . 15 (({0} ∪ ((0 + 1)...𝑁)) ∖ (1...𝑁)) = (({0} ∪ (1...𝑁)) ∖ (1...𝑁))
161 incom 3789 . . . . . . . . . . . . . . . . 17 ({0} ∩ (1...𝑁)) = ((1...𝑁) ∩ {0})
162 elfznn 12328 . . . . . . . . . . . . . . . . . . 19 (0 ∈ (1...𝑁) → 0 ∈ ℕ)
1639, 162mto 188 . . . . . . . . . . . . . . . . . 18 ¬ 0 ∈ (1...𝑁)
164 disjsn 4223 . . . . . . . . . . . . . . . . . 18 (((1...𝑁) ∩ {0}) = ∅ ↔ ¬ 0 ∈ (1...𝑁))
165163, 164mpbir 221 . . . . . . . . . . . . . . . . 17 ((1...𝑁) ∩ {0}) = ∅
166161, 165eqtri 2643 . . . . . . . . . . . . . . . 16 ({0} ∩ (1...𝑁)) = ∅
167 disj3 3999 . . . . . . . . . . . . . . . 16 (({0} ∩ (1...𝑁)) = ∅ ↔ {0} = ({0} ∖ (1...𝑁)))
168166, 167mpbi 220 . . . . . . . . . . . . . . 15 {0} = ({0} ∖ (1...𝑁))
169156, 160, 1683eqtr4i 2653 . . . . . . . . . . . . . 14 (({0} ∪ ((0 + 1)...𝑁)) ∖ (1...𝑁)) = {0}
170155, 169syl6eq 2671 . . . . . . . . . . . . 13 (𝜑 → ((0...𝑁) ∖ (1...𝑁)) = {0})
171170eleq2d 2684 . . . . . . . . . . . 12 (𝜑 → ((2nd𝑧) ∈ ((0...𝑁) ∖ (1...𝑁)) ↔ (2nd𝑧) ∈ {0}))
172 eldif 3570 . . . . . . . . . . . 12 ((2nd𝑧) ∈ ((0...𝑁) ∖ (1...𝑁)) ↔ ((2nd𝑧) ∈ (0...𝑁) ∧ ¬ (2nd𝑧) ∈ (1...𝑁)))
173143elsn 4170 . . . . . . . . . . . 12 ((2nd𝑧) ∈ {0} ↔ (2nd𝑧) = 0)
174171, 172, 1733bitr3g 302 . . . . . . . . . . 11 (𝜑 → (((2nd𝑧) ∈ (0...𝑁) ∧ ¬ (2nd𝑧) ∈ (1...𝑁)) ↔ (2nd𝑧) = 0))
175149, 174bitr3d 270 . . . . . . . . . 10 (𝜑 → (((2nd𝑧) ∈ (0...𝑁) ∧ (¬ (2nd𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd𝑧) = 𝑁)) ↔ (2nd𝑧) = 0))
176175biimpd 219 . . . . . . . . 9 (𝜑 → (((2nd𝑧) ∈ (0...𝑁) ∧ (¬ (2nd𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd𝑧) = 𝑁)) → (2nd𝑧) = 0))
177176expdimp 453 . . . . . . . 8 ((𝜑 ∧ (2nd𝑧) ∈ (0...𝑁)) → ((¬ (2nd𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd𝑧) = 𝑁) → (2nd𝑧) = 0))
178117, 177sylan2 491 . . . . . . 7 ((𝜑𝑧𝑆) → ((¬ (2nd𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd𝑧) = 𝑁) → (2nd𝑧) = 0))
179115, 178mpand 710 . . . . . 6 ((𝜑𝑧𝑆) → (¬ (2nd𝑧) = 𝑁 → (2nd𝑧) = 0))
1801, 2, 3poimirlem13 33093 . . . . . . . . . 10 (𝜑 → ∃*𝑧𝑆 (2nd𝑧) = 0)
181 fveq2 6158 . . . . . . . . . . . 12 (𝑧 = 𝑠 → (2nd𝑧) = (2nd𝑠))
182181eqeq1d 2623 . . . . . . . . . . 11 (𝑧 = 𝑠 → ((2nd𝑧) = 0 ↔ (2nd𝑠) = 0))
183182rmo4 3386 . . . . . . . . . 10 (∃*𝑧𝑆 (2nd𝑧) = 0 ↔ ∀𝑧𝑆𝑠𝑆 (((2nd𝑧) = 0 ∧ (2nd𝑠) = 0) → 𝑧 = 𝑠))
184180, 183sylib 208 . . . . . . . . 9 (𝜑 → ∀𝑧𝑆𝑠𝑆 (((2nd𝑧) = 0 ∧ (2nd𝑠) = 0) → 𝑧 = 𝑠))
185184r19.21bi 2928 . . . . . . . 8 ((𝜑𝑧𝑆) → ∀𝑠𝑆 (((2nd𝑧) = 0 ∧ (2nd𝑠) = 0) → 𝑧 = 𝑠))
1864adantr 481 . . . . . . . 8 ((𝜑𝑧𝑆) → 𝑇𝑆)
187 fveq2 6158 . . . . . . . . . . . 12 (𝑠 = 𝑇 → (2nd𝑠) = (2nd𝑇))
188187eqeq1d 2623 . . . . . . . . . . 11 (𝑠 = 𝑇 → ((2nd𝑠) = 0 ↔ (2nd𝑇) = 0))
189188anbi2d 739 . . . . . . . . . 10 (𝑠 = 𝑇 → (((2nd𝑧) = 0 ∧ (2nd𝑠) = 0) ↔ ((2nd𝑧) = 0 ∧ (2nd𝑇) = 0)))
190 eqeq2 2632 . . . . . . . . . 10 (𝑠 = 𝑇 → (𝑧 = 𝑠𝑧 = 𝑇))
191189, 190imbi12d 334 . . . . . . . . 9 (𝑠 = 𝑇 → ((((2nd𝑧) = 0 ∧ (2nd𝑠) = 0) → 𝑧 = 𝑠) ↔ (((2nd𝑧) = 0 ∧ (2nd𝑇) = 0) → 𝑧 = 𝑇)))
192191rspccv 3296 . . . . . . . 8 (∀𝑠𝑆 (((2nd𝑧) = 0 ∧ (2nd𝑠) = 0) → 𝑧 = 𝑠) → (𝑇𝑆 → (((2nd𝑧) = 0 ∧ (2nd𝑇) = 0) → 𝑧 = 𝑇)))
193185, 186, 192sylc 65 . . . . . . 7 ((𝜑𝑧𝑆) → (((2nd𝑧) = 0 ∧ (2nd𝑇) = 0) → 𝑧 = 𝑇))
1948, 193mpan2d 709 . . . . . 6 ((𝜑𝑧𝑆) → ((2nd𝑧) = 0 → 𝑧 = 𝑇))
195179, 194syld 47 . . . . 5 ((𝜑𝑧𝑆) → (¬ (2nd𝑧) = 𝑁𝑧 = 𝑇))
196195necon1ad 2807 . . . 4 ((𝜑𝑧𝑆) → (𝑧𝑇 → (2nd𝑧) = 𝑁))
197196ralrimiva 2962 . . 3 (𝜑 → ∀𝑧𝑆 (𝑧𝑇 → (2nd𝑧) = 𝑁))
1981, 2, 3poimirlem14 33094 . . 3 (𝜑 → ∃*𝑧𝑆 (2nd𝑧) = 𝑁)
199 rmoim 3394 . . 3 (∀𝑧𝑆 (𝑧𝑇 → (2nd𝑧) = 𝑁) → (∃*𝑧𝑆 (2nd𝑧) = 𝑁 → ∃*𝑧𝑆 𝑧𝑇))
200197, 198, 199sylc 65 . 2 (𝜑 → ∃*𝑧𝑆 𝑧𝑇)
201 reu5 3152 . 2 (∃!𝑧𝑆 𝑧𝑇 ↔ (∃𝑧𝑆 𝑧𝑇 ∧ ∃*𝑧𝑆 𝑧𝑇))
2027, 200, 201sylanbrc 697 1 (𝜑 → ∃!𝑧𝑆 𝑧𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 383  wa 384   = wceq 1480  wcel 1987  {cab 2607  wne 2790  wral 2908  wrex 2909  ∃!wreu 2910  ∃*wrmo 2911  {crab 2912  csb 3519  cdif 3557  cun 3558  cin 3559  wss 3560  c0 3897  ifcif 4064  {csn 4155   class class class wbr 4623  cmpt 4683   × cxp 5082  ran crn 5085  cima 5087  wf 5853  1-1-ontowf1o 5856  cfv 5857  (class class class)co 6615  𝑓 cof 6860  1st c1st 7126  2nd c2nd 7127  𝑚 cmap 7817  cc 9894  0cc0 9896  1c1 9897   + caddc 9899   < clt 10034  cmin 10226  cn 10980  0cn0 11252  cz 11337  cuz 11647  ...cfz 12284  ..^cfzo 12422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-of 6862  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-er 7702  df-map 7819  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-n0 11253  df-z 11338  df-uz 11648  df-fz 12285  df-fzo 12423
This theorem is referenced by:  poimirlem22  33102
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