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Theorem poimirlem18 35074
Description: Lemma for poimir 35089 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 35073 . 2 (𝜑 → ∃𝑧𝑆 𝑧𝑇)
86adantr 484 . . . . . . . 8 ((𝜑𝑧𝑆) → (2nd𝑇) = 0)
9 0nnn 11665 . . . . . . . . . . . . 13 ¬ 0 ∈ ℕ
10 elfznn 12935 . . . . . . . . . . . . 13 (0 ∈ (1...(𝑁 − 1)) → 0 ∈ ℕ)
119, 10mto 200 . . . . . . . . . . . 12 ¬ 0 ∈ (1...(𝑁 − 1))
12 eleq1 2880 . . . . . . . . . . . 12 ((2nd𝑧) = 0 → ((2nd𝑧) ∈ (1...(𝑁 − 1)) ↔ 0 ∈ (1...(𝑁 − 1))))
1311, 12mtbiri 330 . . . . . . . . . . 11 ((2nd𝑧) = 0 → ¬ (2nd𝑧) ∈ (1...(𝑁 − 1)))
1413necon2ai 3019 . . . . . . . . . 10 ((2nd𝑧) ∈ (1...(𝑁 − 1)) → (2nd𝑧) ≠ 0)
151ad2antrr 725 . . . . . . . . . . . . . 14 (((𝜑𝑧𝑆) ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → 𝑁 ∈ ℕ)
16 fveq2 6649 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑧 → (2nd𝑡) = (2nd𝑧))
1716breq2d 5045 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = 𝑧 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑧)))
1817ifbid 4450 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = 𝑧 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑧), 𝑦, (𝑦 + 1)))
1918csbeq1d 3835 . . . . . . . . . . . . . . . . . . . 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 6654 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = 𝑧 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑧)))
21 2fveq3 6654 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑧 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑧)))
2221imaeq1d 5899 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑧 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑧)) “ (1...𝑗)))
2322xpeq1d 5552 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑧 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑧)) “ (1...𝑗)) × {1}))
2421imaeq1d 5899 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑧 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)))
2524xpeq1d 5552 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑧 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))
2623, 25uneq12d 4094 . . . . . . . . . . . . . . . . . . . . . 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 7157 . . . . . . . . . . . . . . . . . . . . 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 3838 . . . . . . . . . . . . . . . . . . . 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 2836 . . . . . . . . . . . . . . . . . . 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 5129 . . . . . . . . . . . . . . . . . 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 2812 . . . . . . . . . . . . . . . . 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 3634 . . . . . . . . . . . . . . . 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 500 . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . 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 3626 . . . . . . . . . . . . . . . . . . . 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 2911 . . . . . . . . . . . . . . . . . . 19 (𝑧𝑆𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
37 xp1st 7707 . . . . . . . . . . . . . . . . . . 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 7707 . . . . . . . . . . . . . . . . . 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 8415 . . . . . . . . . . . . . . . . 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 13037 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (0..^𝐾) → 𝑛 ∈ ℤ)
4443ssriv 3922 . . . . . . . . . . . . . . . 16 (0..^𝐾) ⊆ ℤ
45 fss 6505 . . . . . . . . . . . . . . . 16 (((1st ‘(1st𝑧)):(1...𝑁)⟶(0..^𝐾) ∧ (0..^𝐾) ⊆ ℤ) → (1st ‘(1st𝑧)):(1...𝑁)⟶ℤ)
4642, 44, 45sylancl 589 . . . . . . . . . . . . . . 15 (𝑧𝑆 → (1st ‘(1st𝑧)):(1...𝑁)⟶ℤ)
4746ad2antlr 726 . . . . . . . . . . . . . 14 (((𝜑𝑧𝑆) ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → (1st ‘(1st𝑧)):(1...𝑁)⟶ℤ)
48 xp2nd 7708 . . . . . . . . . . . . . . . . 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 6662 . . . . . . . . . . . . . . . . 17 (2nd ‘(1st𝑧)) ∈ V
51 f1oeq1 6583 . . . . . . . . . . . . . . . . 17 (𝑓 = (2nd ‘(1st𝑧)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁)))
5250, 51elab 3618 . . . . . . . . . . . . . . . 16 ((2nd ‘(1st𝑧)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
5349, 52sylib 221 . . . . . . . . . . . . . . 15 (𝑧𝑆 → (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
5453ad2antlr 726 . . . . . . . . . . . . . 14 (((𝜑𝑧𝑆) ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
55 simpr 488 . . . . . . . . . . . . . 14 (((𝜑𝑧𝑆) ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → (2nd𝑧) ∈ (1...(𝑁 − 1)))
5615, 34, 47, 54, 55poimirlem1 35057 . . . . . . . . . . . . 13 (((𝜑𝑧𝑆) ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑧) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑧))‘𝑛))
571ad2antrr 725 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd𝑇) ≠ (2nd𝑧)) → 𝑁 ∈ ℕ)
58 fveq2 6649 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 = 𝑇 → (2nd𝑡) = (2nd𝑇))
5958breq2d 5045 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = 𝑇 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑇)))
6059ifbid 4450 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑇 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)))
6160csbeq1d 3835 . . . . . . . . . . . . . . . . . . . . . . . 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 6654 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = 𝑇 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
63 2fveq3 6654 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑡 = 𝑇 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑇)))
6463imaeq1d 5899 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑗)))
6564xpeq1d 5552 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}))
6663imaeq1d 5899 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)))
6766xpeq1d 5552 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
6865, 67uneq12d 4094 . . . . . . . . . . . . . . . . . . . . . . . . . 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 7157 . . . . . . . . . . . . . . . . . . . . . . . . 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 3838 . . . . . . . . . . . . . . . . . . . . . . . 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 2836 . . . . . . . . . . . . . . . . . . . . . . 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 5129 . . . . . . . . . . . . . . . . . . . . . 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 2812 . . . . . . . . . . . . . . . . . . . . 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 3634 . . . . . . . . . . . . . . . . . . . 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 500 . . . . . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . . . . 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 3626 . . . . . . . . . . . . . . . . . . . . . . . 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 2911 . . . . . . . . . . . . . . . . . . . . . . 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 7707 . . . . . . . . . . . . . . . . . . . . . 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 7707 . . . . . . . . . . . . . . . . . . . . 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 8415 . . . . . . . . . . . . . . . . . . . 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 6505 . . . . . . . . . . . . . . . . . . 19 (((1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾) ∧ (0..^𝐾) ⊆ ℤ) → (1st ‘(1st𝑇)):(1...𝑁)⟶ℤ)
8886, 44, 87sylancl 589 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1st ‘(1st𝑇)):(1...𝑁)⟶ℤ)
8988ad2antrr 725 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd𝑇) ≠ (2nd𝑧)) → (1st ‘(1st𝑇)):(1...𝑁)⟶ℤ)
90 xp2nd 7708 . . . . . . . . . . . . . . . . . . . 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 6662 . . . . . . . . . . . . . . . . . . . 20 (2nd ‘(1st𝑇)) ∈ V
93 f1oeq1 6583 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = (2nd ‘(1st𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
9492, 93elab 3618 . . . . . . . . . . . . . . . . . . 19 ((2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
9591, 94sylib 221 . . . . . . . . . . . . . . . . . 18 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
9695ad2antrr 725 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd𝑇) ≠ (2nd𝑧)) → (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
97 simplr 768 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd𝑇) ≠ (2nd𝑧)) → (2nd𝑧) ∈ (1...(𝑁 − 1)))
98 xp2nd 7708 . . . . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → (2nd𝑇) ∈ (0...𝑁))
101 eldifsn 4683 . . . . . . . . . . . . . . . . . . 19 ((2nd𝑇) ∈ ((0...𝑁) ∖ {(2nd𝑧)}) ↔ ((2nd𝑇) ∈ (0...𝑁) ∧ (2nd𝑇) ≠ (2nd𝑧)))
102101biimpri 231 . . . . . . . . . . . . . . . . . 18 (((2nd𝑇) ∈ (0...𝑁) ∧ (2nd𝑇) ≠ (2nd𝑧)) → (2nd𝑇) ∈ ((0...𝑁) ∖ {(2nd𝑧)}))
103100, 102sylan 583 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd𝑇) ≠ (2nd𝑧)) → (2nd𝑇) ∈ ((0...𝑁) ∖ {(2nd𝑧)}))
10457, 77, 89, 96, 97, 103poimirlem2 35058 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd𝑇) ≠ (2nd𝑧)) → ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑧) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑧))‘𝑛))
105104ex 416 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → ((2nd𝑇) ≠ (2nd𝑧) → ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑧) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑧))‘𝑛)))
106105necon1bd 3008 . . . . . . . . . . . . . 14 ((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → (¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑧) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑧))‘𝑛) → (2nd𝑇) = (2nd𝑧)))
107106adantlr 714 . . . . . . . . . . . . 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 3049 . . . . . . . . . . 11 (((𝜑𝑧𝑆) ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → ((2nd𝑇) ≠ 0 ↔ (2nd𝑧) ≠ 0))
110109exbiri 810 . . . . . . . . . 10 ((𝜑𝑧𝑆) → ((2nd𝑧) ∈ (1...(𝑁 − 1)) → ((2nd𝑧) ≠ 0 → (2nd𝑇) ≠ 0)))
11114, 110mpdi 45 . . . . . . . . 9 ((𝜑𝑧𝑆) → ((2nd𝑧) ∈ (1...(𝑁 − 1)) → (2nd𝑇) ≠ 0))
112111necon2bd 3006 . . . . . . . 8 ((𝜑𝑧𝑆) → ((2nd𝑇) = 0 → ¬ (2nd𝑧) ∈ (1...(𝑁 − 1))))
1138, 112mpd 15 . . . . . . 7 ((𝜑𝑧𝑆) → ¬ (2nd𝑧) ∈ (1...(𝑁 − 1)))
114 xp2nd 7708 . . . . . . . . 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 11645 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑁 ∈ ℂ)
117 npcan1 11058 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
118116, 117syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
119 nnuz 12273 . . . . . . . . . . . . . . . . . . 19 ℕ = (ℤ‘1)
1201, 119eleqtrdi 2903 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ (ℤ‘1))
121118, 120eqeltrd 2893 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘1))
1221nnzd 12078 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑁 ∈ ℤ)
123 peano2zm 12017 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
124122, 123syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑁 − 1) ∈ ℤ)
125 uzid 12250 . . . . . . . . . . . . . . . . . . 19 ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)))
126 peano2uz 12293 . . . . . . . . . . . . . . . . . . 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 2894 . . . . . . . . . . . . . . . . 17 (𝜑𝑁 ∈ (ℤ‘(𝑁 − 1)))
129 fzsplit2 12931 . . . . . . . . . . . . . . . . 17 ((((𝑁 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ‘(𝑁 − 1))) → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
130121, 128, 129syl2anc 587 . . . . . . . . . . . . . . . 16 (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
131118oveq1d 7154 . . . . . . . . . . . . . . . . . 18 (𝜑 → (((𝑁 − 1) + 1)...𝑁) = (𝑁...𝑁))
132 fzsn 12948 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
133122, 132syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑁...𝑁) = {𝑁})
134131, 133eqtrd 2836 . . . . . . . . . . . . . . . . 17 (𝜑 → (((𝑁 − 1) + 1)...𝑁) = {𝑁})
135134uneq2d 4093 . . . . . . . . . . . . . . . 16 (𝜑 → ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = ((1...(𝑁 − 1)) ∪ {𝑁}))
136130, 135eqtrd 2836 . . . . . . . . . . . . . . 15 (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁}))
137136eleq2d 2878 . . . . . . . . . . . . . 14 (𝜑 → ((2nd𝑧) ∈ (1...𝑁) ↔ (2nd𝑧) ∈ ((1...(𝑁 − 1)) ∪ {𝑁})))
138137notbid 321 . . . . . . . . . . . . 13 (𝜑 → (¬ (2nd𝑧) ∈ (1...𝑁) ↔ ¬ (2nd𝑧) ∈ ((1...(𝑁 − 1)) ∪ {𝑁})))
139 ioran 981 . . . . . . . . . . . . . 14 (¬ ((2nd𝑧) ∈ (1...(𝑁 − 1)) ∨ (2nd𝑧) = 𝑁) ↔ (¬ (2nd𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd𝑧) = 𝑁))
140 elun 4079 . . . . . . . . . . . . . . 15 ((2nd𝑧) ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ ((2nd𝑧) ∈ (1...(𝑁 − 1)) ∨ (2nd𝑧) ∈ {𝑁}))
141 fvex 6662 . . . . . . . . . . . . . . . . 17 (2nd𝑧) ∈ V
142141elsn 4543 . . . . . . . . . . . . . . . 16 ((2nd𝑧) ∈ {𝑁} ↔ (2nd𝑧) = 𝑁)
143142orbi2i 910 . . . . . . . . . . . . . . 15 (((2nd𝑧) ∈ (1...(𝑁 − 1)) ∨ (2nd𝑧) ∈ {𝑁}) ↔ ((2nd𝑧) ∈ (1...(𝑁 − 1)) ∨ (2nd𝑧) = 𝑁))
144140, 143bitri 278 . . . . . . . . . . . . . 14 ((2nd𝑧) ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ ((2nd𝑧) ∈ (1...(𝑁 − 1)) ∨ (2nd𝑧) = 𝑁))
145139, 144xchnxbir 336 . . . . . . . . . . . . 13 (¬ (2nd𝑧) ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ (¬ (2nd𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd𝑧) = 𝑁))
146138, 145syl6bb 290 . . . . . . . . . . . 12 (𝜑 → (¬ (2nd𝑧) ∈ (1...𝑁) ↔ (¬ (2nd𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd𝑧) = 𝑁)))
147146anbi2d 631 . . . . . . . . . . 11 (𝜑 → (((2nd𝑧) ∈ (0...𝑁) ∧ ¬ (2nd𝑧) ∈ (1...𝑁)) ↔ ((2nd𝑧) ∈ (0...𝑁) ∧ (¬ (2nd𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd𝑧) = 𝑁))))
1481nnnn0d 11947 . . . . . . . . . . . . . . . . 17 (𝜑𝑁 ∈ ℕ0)
149 nn0uz 12272 . . . . . . . . . . . . . . . . 17 0 = (ℤ‘0)
150148, 149eleqtrdi 2903 . . . . . . . . . . . . . . . 16 (𝜑𝑁 ∈ (ℤ‘0))
151 fzpred 12954 . . . . . . . . . . . . . . . 16 (𝑁 ∈ (ℤ‘0) → (0...𝑁) = ({0} ∪ ((0 + 1)...𝑁)))
152150, 151syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (0...𝑁) = ({0} ∪ ((0 + 1)...𝑁)))
153152difeq1d 4052 . . . . . . . . . . . . . 14 (𝜑 → ((0...𝑁) ∖ (1...𝑁)) = (({0} ∪ ((0 + 1)...𝑁)) ∖ (1...𝑁)))
154 difun2 4390 . . . . . . . . . . . . . . 15 (({0} ∪ (1...𝑁)) ∖ (1...𝑁)) = ({0} ∖ (1...𝑁))
155 0p1e1 11751 . . . . . . . . . . . . . . . . . 18 (0 + 1) = 1
156155oveq1i 7149 . . . . . . . . . . . . . . . . 17 ((0 + 1)...𝑁) = (1...𝑁)
157156uneq2i 4090 . . . . . . . . . . . . . . . 16 ({0} ∪ ((0 + 1)...𝑁)) = ({0} ∪ (1...𝑁))
158157difeq1i 4049 . . . . . . . . . . . . . . 15 (({0} ∪ ((0 + 1)...𝑁)) ∖ (1...𝑁)) = (({0} ∪ (1...𝑁)) ∖ (1...𝑁))
159 incom 4131 . . . . . . . . . . . . . . . . 17 ({0} ∩ (1...𝑁)) = ((1...𝑁) ∩ {0})
160 elfznn 12935 . . . . . . . . . . . . . . . . . . 19 (0 ∈ (1...𝑁) → 0 ∈ ℕ)
1619, 160mto 200 . . . . . . . . . . . . . . . . . 18 ¬ 0 ∈ (1...𝑁)
162 disjsn 4610 . . . . . . . . . . . . . . . . . 18 (((1...𝑁) ∩ {0}) = ∅ ↔ ¬ 0 ∈ (1...𝑁))
163161, 162mpbir 234 . . . . . . . . . . . . . . . . 17 ((1...𝑁) ∩ {0}) = ∅
164159, 163eqtri 2824 . . . . . . . . . . . . . . . 16 ({0} ∩ (1...𝑁)) = ∅
165 disj3 4364 . . . . . . . . . . . . . . . 16 (({0} ∩ (1...𝑁)) = ∅ ↔ {0} = ({0} ∖ (1...𝑁)))
166164, 165mpbi 233 . . . . . . . . . . . . . . 15 {0} = ({0} ∖ (1...𝑁))
167154, 158, 1663eqtr4i 2834 . . . . . . . . . . . . . 14 (({0} ∪ ((0 + 1)...𝑁)) ∖ (1...𝑁)) = {0}
168153, 167eqtrdi 2852 . . . . . . . . . . . . 13 (𝜑 → ((0...𝑁) ∖ (1...𝑁)) = {0})
169168eleq2d 2878 . . . . . . . . . . . 12 (𝜑 → ((2nd𝑧) ∈ ((0...𝑁) ∖ (1...𝑁)) ↔ (2nd𝑧) ∈ {0}))
170 eldif 3894 . . . . . . . . . . . 12 ((2nd𝑧) ∈ ((0...𝑁) ∖ (1...𝑁)) ↔ ((2nd𝑧) ∈ (0...𝑁) ∧ ¬ (2nd𝑧) ∈ (1...𝑁)))
171141elsn 4543 . . . . . . . . . . . 12 ((2nd𝑧) ∈ {0} ↔ (2nd𝑧) = 0)
172169, 170, 1713bitr3g 316 . . . . . . . . . . 11 (𝜑 → (((2nd𝑧) ∈ (0...𝑁) ∧ ¬ (2nd𝑧) ∈ (1...𝑁)) ↔ (2nd𝑧) = 0))
173147, 172bitr3d 284 . . . . . . . . . 10 (𝜑 → (((2nd𝑧) ∈ (0...𝑁) ∧ (¬ (2nd𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd𝑧) = 𝑁)) ↔ (2nd𝑧) = 0))
174173biimpd 232 . . . . . . . . 9 (𝜑 → (((2nd𝑧) ∈ (0...𝑁) ∧ (¬ (2nd𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd𝑧) = 𝑁)) → (2nd𝑧) = 0))
175174expdimp 456 . . . . . . . 8 ((𝜑 ∧ (2nd𝑧) ∈ (0...𝑁)) → ((¬ (2nd𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd𝑧) = 𝑁) → (2nd𝑧) = 0))
176115, 175sylan2 595 . . . . . . 7 ((𝜑𝑧𝑆) → ((¬ (2nd𝑧) ∈ (1...(𝑁 − 1)) ∧ ¬ (2nd𝑧) = 𝑁) → (2nd𝑧) = 0))
177113, 176mpand 694 . . . . . 6 ((𝜑𝑧𝑆) → (¬ (2nd𝑧) = 𝑁 → (2nd𝑧) = 0))
1781, 2, 3poimirlem13 35069 . . . . . . . . . 10 (𝜑 → ∃*𝑧𝑆 (2nd𝑧) = 0)
179 fveqeq2 6658 . . . . . . . . . . 11 (𝑧 = 𝑠 → ((2nd𝑧) = 0 ↔ (2nd𝑠) = 0))
180179rmo4 3672 . . . . . . . . . 10 (∃*𝑧𝑆 (2nd𝑧) = 0 ↔ ∀𝑧𝑆𝑠𝑆 (((2nd𝑧) = 0 ∧ (2nd𝑠) = 0) → 𝑧 = 𝑠))
181178, 180sylib 221 . . . . . . . . 9 (𝜑 → ∀𝑧𝑆𝑠𝑆 (((2nd𝑧) = 0 ∧ (2nd𝑠) = 0) → 𝑧 = 𝑠))
182181r19.21bi 3176 . . . . . . . 8 ((𝜑𝑧𝑆) → ∀𝑠𝑆 (((2nd𝑧) = 0 ∧ (2nd𝑠) = 0) → 𝑧 = 𝑠))
1834adantr 484 . . . . . . . 8 ((𝜑𝑧𝑆) → 𝑇𝑆)
184 fveqeq2 6658 . . . . . . . . . . 11 (𝑠 = 𝑇 → ((2nd𝑠) = 0 ↔ (2nd𝑇) = 0))
185184anbi2d 631 . . . . . . . . . 10 (𝑠 = 𝑇 → (((2nd𝑧) = 0 ∧ (2nd𝑠) = 0) ↔ ((2nd𝑧) = 0 ∧ (2nd𝑇) = 0)))
186 eqeq2 2813 . . . . . . . . . 10 (𝑠 = 𝑇 → (𝑧 = 𝑠𝑧 = 𝑇))
187185, 186imbi12d 348 . . . . . . . . 9 (𝑠 = 𝑇 → ((((2nd𝑧) = 0 ∧ (2nd𝑠) = 0) → 𝑧 = 𝑠) ↔ (((2nd𝑧) = 0 ∧ (2nd𝑇) = 0) → 𝑧 = 𝑇)))
188187rspccv 3571 . . . . . . . 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 693 . . . . . 6 ((𝜑𝑧𝑆) → ((2nd𝑧) = 0 → 𝑧 = 𝑇))
191177, 190syld 47 . . . . 5 ((𝜑𝑧𝑆) → (¬ (2nd𝑧) = 𝑁𝑧 = 𝑇))
192191necon1ad 3007 . . . 4 ((𝜑𝑧𝑆) → (𝑧𝑇 → (2nd𝑧) = 𝑁))
193192ralrimiva 3152 . . 3 (𝜑 → ∀𝑧𝑆 (𝑧𝑇 → (2nd𝑧) = 𝑁))
1941, 2, 3poimirlem14 35070 . . 3 (𝜑 → ∃*𝑧𝑆 (2nd𝑧) = 𝑁)
195 rmoim 3682 . . 3 (∀𝑧𝑆 (𝑧𝑇 → (2nd𝑧) = 𝑁) → (∃*𝑧𝑆 (2nd𝑧) = 𝑁 → ∃*𝑧𝑆 𝑧𝑇))
196193, 194, 195sylc 65 . 2 (𝜑 → ∃*𝑧𝑆 𝑧𝑇)
197 reu5 3378 . 2 (∃!𝑧𝑆 𝑧𝑇 ↔ (∃𝑧𝑆 𝑧𝑇 ∧ ∃*𝑧𝑆 𝑧𝑇))
1987, 196, 197sylanbrc 586 1 (𝜑 → ∃!𝑧𝑆 𝑧𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wo 844   = wceq 1538  wcel 2112  {cab 2779  wne 2990  wral 3109  wrex 3110  ∃!wreu 3111  ∃*wrmo 3112  {crab 3113  csb 3831  cdif 3881  cun 3882  cin 3883  wss 3884  c0 4246  ifcif 4428  {csn 4528   class class class wbr 5033  cmpt 5113   × cxp 5521  ran crn 5524  cima 5526  wf 6324  1-1-ontowf1o 6327  cfv 6328  (class class class)co 7139  f cof 7391  1st c1st 7673  2nd c2nd 7674  m cmap 8393  cc 10528  0cc0 10530  1c1 10531   + caddc 10533   < clt 10668  cmin 10863  cn 11629  0cn0 11889  cz 11973  cuz 12235  ...cfz 12889  ..^cfzo 13032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12890  df-fzo 13033
This theorem is referenced by:  poimirlem22  35078
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