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Theorem poimirlem22 34906
Description: Lemma for poimir 34917, that a given face belongs to exactly two simplices, provided it's not on the boundary of the cube. (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)
poimirlem22.3 ((𝜑𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 0)
poimirlem22.4 ((𝜑𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 𝐾)
Assertion
Ref Expression
poimirlem22 (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 poimirlem22
StepHypRef Expression
1 poimir.0 . . . . 5 (𝜑𝑁 ∈ ℕ)
21adantr 483 . . . 4 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → 𝑁 ∈ ℕ)
3 poimirlem22.s . . . 4 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
4 poimirlem22.1 . . . . 5 (𝜑𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))
54adantr 483 . . . 4 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))
6 poimirlem22.2 . . . . 5 (𝜑𝑇𝑆)
76adantr 483 . . . 4 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → 𝑇𝑆)
8 simpr 487 . . . 4 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (2nd𝑇) ∈ (1...(𝑁 − 1)))
92, 3, 5, 7, 8poimirlem15 34899 . . 3 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → ⟨⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩, (2nd𝑇)⟩ ∈ 𝑆)
10 fveq2 6663 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑇 → (2nd𝑡) = (2nd𝑇))
1110breq2d 5069 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑇 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑇)))
1211ifbid 4487 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = 𝑇 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)))
1312csbeq1d 3885 . . . . . . . . . . . . . . . . . . . . 21 (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}))))
14 2fveq3 6668 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑇 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
15 2fveq3 6668 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = 𝑇 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑇)))
1615imaeq1d 5921 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑗)))
1716xpeq1d 5577 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}))
1815imaeq1d 5921 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)))
1918xpeq1d 5577 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
2017, 19uneq12d 4138 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑇 → ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))
2114, 20oveq12d 7166 . . . . . . . . . . . . . . . . . . . . . 22 (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}))))
2221csbeq2dv 3888 . . . . . . . . . . . . . . . . . . . . 21 (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}))))
2313, 22eqtrd 2854 . . . . . . . . . . . . . . . . . . . 20 (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}))))
2423mpteq2dv 5153 . . . . . . . . . . . . . . . . . . 19 (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})))))
2524eqeq2d 2830 . . . . . . . . . . . . . . . . . 18 (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}))))))
2625, 3elrab2 3681 . . . . . . . . . . . . . . . . 17 (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}))))))
2726simprbi 499 . . . . . . . . . . . . . . . 16 (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})))))
286, 27syl 17 . . . . . . . . . . . . . . 15 (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})))))
2928adantr 483 . . . . . . . . . . . . . 14 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → 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})))))
30 elrabi 3673 . . . . . . . . . . . . . . . . . . . . 21 (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)))
3130, 3eleq2s 2929 . . . . . . . . . . . . . . . . . . . 20 (𝑇𝑆𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
326, 31syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
33 xp1st 7713 . . . . . . . . . . . . . . . . . . 19 (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)}))
3432, 33syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1st𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
35 xp1st 7713 . . . . . . . . . . . . . . . . . 18 ((1st𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
3634, 35syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
37 elmapi 8420 . . . . . . . . . . . . . . . . 17 ((1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
3836, 37syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
39 elfzoelz 13030 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (0..^𝐾) → 𝑛 ∈ ℤ)
4039ssriv 3969 . . . . . . . . . . . . . . . 16 (0..^𝐾) ⊆ ℤ
41 fss 6520 . . . . . . . . . . . . . . . 16 (((1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾) ∧ (0..^𝐾) ⊆ ℤ) → (1st ‘(1st𝑇)):(1...𝑁)⟶ℤ)
4238, 40, 41sylancl 588 . . . . . . . . . . . . . . 15 (𝜑 → (1st ‘(1st𝑇)):(1...𝑁)⟶ℤ)
4342adantr 483 . . . . . . . . . . . . . 14 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (1st ‘(1st𝑇)):(1...𝑁)⟶ℤ)
44 xp2nd 7714 . . . . . . . . . . . . . . . . 17 ((1st𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
4534, 44syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
46 fvex 6676 . . . . . . . . . . . . . . . . 17 (2nd ‘(1st𝑇)) ∈ V
47 f1oeq1 6597 . . . . . . . . . . . . . . . . 17 (𝑓 = (2nd ‘(1st𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
4846, 47elab 3665 . . . . . . . . . . . . . . . 16 ((2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
4945, 48sylib 220 . . . . . . . . . . . . . . 15 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
5049adantr 483 . . . . . . . . . . . . . 14 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
512, 29, 43, 50, 8poimirlem1 34885 . . . . . . . . . . . . 13 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑇))‘𝑛))
5251adantr 483 . . . . . . . . . . . 12 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑇))‘𝑛))
531ad3antrrr 728 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ (2nd𝑧) ≠ (2nd𝑇)) → 𝑁 ∈ ℕ)
54 fveq2 6663 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑧 → (2nd𝑡) = (2nd𝑧))
5554breq2d 5069 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑧 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑧)))
5655ifbid 4487 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = 𝑧 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑧), 𝑦, (𝑦 + 1)))
5756csbeq1d 3885 . . . . . . . . . . . . . . . . . . . . 21 (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}))))
58 2fveq3 6668 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑧 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑧)))
59 2fveq3 6668 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = 𝑧 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑧)))
6059imaeq1d 5921 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑧 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑧)) “ (1...𝑗)))
6160xpeq1d 5577 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑧 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑧)) “ (1...𝑗)) × {1}))
6259imaeq1d 5921 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑧 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)))
6362xpeq1d 5577 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑧 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))
6461, 63uneq12d 4138 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑧 → ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) × {0})))
6558, 64oveq12d 7166 . . . . . . . . . . . . . . . . . . . . . 22 (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}))))
6665csbeq2dv 3888 . . . . . . . . . . . . . . . . . . . . 21 (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}))))
6757, 66eqtrd 2854 . . . . . . . . . . . . . . . . . . . 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 ‘(1st𝑧)) ∘f + ((((2nd ‘(1st𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))))
6867mpteq2dv 5153 . . . . . . . . . . . . . . . . . . 19 (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})))))
6968eqeq2d 2830 . . . . . . . . . . . . . . . . . 18 (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}))))))
7069, 3elrab2 3681 . . . . . . . . . . . . . . . . 17 (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}))))))
7170simprbi 499 . . . . . . . . . . . . . . . 16 (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})))))
7271ad2antlr 725 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ (2nd𝑧) ≠ (2nd𝑇)) → 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})))))
73 elrabi 3673 . . . . . . . . . . . . . . . . . . . . 21 (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)))
7473, 3eleq2s 2929 . . . . . . . . . . . . . . . . . . . 20 (𝑧𝑆𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
75 xp1st 7713 . . . . . . . . . . . . . . . . . . . 20 (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)}))
7674, 75syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑧𝑆 → (1st𝑧) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
77 xp1st 7713 . . . . . . . . . . . . . . . . . . 19 ((1st𝑧) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st𝑧)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
7876, 77syl 17 . . . . . . . . . . . . . . . . . 18 (𝑧𝑆 → (1st ‘(1st𝑧)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
79 elmapi 8420 . . . . . . . . . . . . . . . . . 18 ((1st ‘(1st𝑧)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st ‘(1st𝑧)):(1...𝑁)⟶(0..^𝐾))
8078, 79syl 17 . . . . . . . . . . . . . . . . 17 (𝑧𝑆 → (1st ‘(1st𝑧)):(1...𝑁)⟶(0..^𝐾))
81 fss 6520 . . . . . . . . . . . . . . . . 17 (((1st ‘(1st𝑧)):(1...𝑁)⟶(0..^𝐾) ∧ (0..^𝐾) ⊆ ℤ) → (1st ‘(1st𝑧)):(1...𝑁)⟶ℤ)
8280, 40, 81sylancl 588 . . . . . . . . . . . . . . . 16 (𝑧𝑆 → (1st ‘(1st𝑧)):(1...𝑁)⟶ℤ)
8382ad2antlr 725 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ (2nd𝑧) ≠ (2nd𝑇)) → (1st ‘(1st𝑧)):(1...𝑁)⟶ℤ)
84 xp2nd 7714 . . . . . . . . . . . . . . . . . 18 ((1st𝑧) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘(1st𝑧)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
8576, 84syl 17 . . . . . . . . . . . . . . . . 17 (𝑧𝑆 → (2nd ‘(1st𝑧)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
86 fvex 6676 . . . . . . . . . . . . . . . . . 18 (2nd ‘(1st𝑧)) ∈ V
87 f1oeq1 6597 . . . . . . . . . . . . . . . . . 18 (𝑓 = (2nd ‘(1st𝑧)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁)))
8886, 87elab 3665 . . . . . . . . . . . . . . . . 17 ((2nd ‘(1st𝑧)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
8985, 88sylib 220 . . . . . . . . . . . . . . . 16 (𝑧𝑆 → (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
9089ad2antlr 725 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ (2nd𝑧) ≠ (2nd𝑇)) → (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
91 simpllr 774 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ (2nd𝑧) ≠ (2nd𝑇)) → (2nd𝑇) ∈ (1...(𝑁 − 1)))
92 xp2nd 7714 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd𝑧) ∈ (0...𝑁))
9374, 92syl 17 . . . . . . . . . . . . . . . . 17 (𝑧𝑆 → (2nd𝑧) ∈ (0...𝑁))
9493adantl 484 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (2nd𝑧) ∈ (0...𝑁))
95 eldifsn 4711 . . . . . . . . . . . . . . . . 17 ((2nd𝑧) ∈ ((0...𝑁) ∖ {(2nd𝑇)}) ↔ ((2nd𝑧) ∈ (0...𝑁) ∧ (2nd𝑧) ≠ (2nd𝑇)))
9695biimpri 230 . . . . . . . . . . . . . . . 16 (((2nd𝑧) ∈ (0...𝑁) ∧ (2nd𝑧) ≠ (2nd𝑇)) → (2nd𝑧) ∈ ((0...𝑁) ∖ {(2nd𝑇)}))
9794, 96sylan 582 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ (2nd𝑧) ≠ (2nd𝑇)) → (2nd𝑧) ∈ ((0...𝑁) ∖ {(2nd𝑇)}))
9853, 72, 83, 90, 91, 97poimirlem2 34886 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ (2nd𝑧) ≠ (2nd𝑇)) → ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑇))‘𝑛))
9998ex 415 . . . . . . . . . . . . 13 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → ((2nd𝑧) ≠ (2nd𝑇) → ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑇))‘𝑛)))
10099necon1bd 3032 . . . . . . . . . . . 12 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑇))‘𝑛) → (2nd𝑧) = (2nd𝑇)))
10152, 100mpd 15 . . . . . . . . . . 11 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (2nd𝑧) = (2nd𝑇))
102 eleq1 2898 . . . . . . . . . . . . . . . 16 ((2nd𝑧) = (2nd𝑇) → ((2nd𝑧) ∈ (1...(𝑁 − 1)) ↔ (2nd𝑇) ∈ (1...(𝑁 − 1))))
103102biimparc 482 . . . . . . . . . . . . . . 15 (((2nd𝑇) ∈ (1...(𝑁 − 1)) ∧ (2nd𝑧) = (2nd𝑇)) → (2nd𝑧) ∈ (1...(𝑁 − 1)))
104103anim2i 618 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((2nd𝑇) ∈ (1...(𝑁 − 1)) ∧ (2nd𝑧) = (2nd𝑇))) → (𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))))
105104anassrs 470 . . . . . . . . . . . . 13 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ (2nd𝑧) = (2nd𝑇)) → (𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))))
10671adantl 484 . . . . . . . . . . . . . 14 (((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → 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})))))
107 breq1 5060 . . . . . . . . . . . . . . . . . 18 (𝑦 = 0 → (𝑦 < (2nd𝑧) ↔ 0 < (2nd𝑧)))
108 id 22 . . . . . . . . . . . . . . . . . 18 (𝑦 = 0 → 𝑦 = 0)
109107, 108ifbieq1d 4488 . . . . . . . . . . . . . . . . 17 (𝑦 = 0 → if(𝑦 < (2nd𝑧), 𝑦, (𝑦 + 1)) = if(0 < (2nd𝑧), 0, (𝑦 + 1)))
110 elfznn 12928 . . . . . . . . . . . . . . . . . . . 20 ((2nd𝑧) ∈ (1...(𝑁 − 1)) → (2nd𝑧) ∈ ℕ)
111110nngt0d 11678 . . . . . . . . . . . . . . . . . . 19 ((2nd𝑧) ∈ (1...(𝑁 − 1)) → 0 < (2nd𝑧))
112111iftrued 4473 . . . . . . . . . . . . . . . . . 18 ((2nd𝑧) ∈ (1...(𝑁 − 1)) → if(0 < (2nd𝑧), 0, (𝑦 + 1)) = 0)
113112ad2antlr 725 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → if(0 < (2nd𝑧), 0, (𝑦 + 1)) = 0)
114109, 113sylan9eqr 2876 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ 𝑦 = 0) → if(𝑦 < (2nd𝑧), 𝑦, (𝑦 + 1)) = 0)
115114csbeq1d 3885 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ 𝑦 = 0) → if(𝑦 < (2nd𝑧), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑧)) ∘f + ((((2nd ‘(1st𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))) = 0 / 𝑗((1st ‘(1st𝑧)) ∘f + ((((2nd ‘(1st𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))))
116 c0ex 10627 . . . . . . . . . . . . . . . . . 18 0 ∈ V
117 oveq2 7156 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 = 0 → (1...𝑗) = (1...0))
118 fz10 12920 . . . . . . . . . . . . . . . . . . . . . . . 24 (1...0) = ∅
119117, 118syl6eq 2870 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = 0 → (1...𝑗) = ∅)
120119imaeq2d 5922 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 0 → ((2nd ‘(1st𝑧)) “ (1...𝑗)) = ((2nd ‘(1st𝑧)) “ ∅))
121120xpeq1d 5577 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 0 → (((2nd ‘(1st𝑧)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑧)) “ ∅) × {1}))
122 oveq1 7155 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 = 0 → (𝑗 + 1) = (0 + 1))
123 0p1e1 11751 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0 + 1) = 1
124122, 123syl6eq 2870 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 = 0 → (𝑗 + 1) = 1)
125124oveq1d 7163 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = 0 → ((𝑗 + 1)...𝑁) = (1...𝑁))
126125imaeq2d 5922 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 0 → ((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑧)) “ (1...𝑁)))
127126xpeq1d 5577 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 0 → (((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0}))
128121, 127uneq12d 4138 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 0 → ((((2nd ‘(1st𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑧)) “ ∅) × {1}) ∪ (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0})))
129 ima0 5938 . . . . . . . . . . . . . . . . . . . . . . . 24 ((2nd ‘(1st𝑧)) “ ∅) = ∅
130129xpeq1i 5574 . . . . . . . . . . . . . . . . . . . . . . 23 (((2nd ‘(1st𝑧)) “ ∅) × {1}) = (∅ × {1})
131 0xp 5642 . . . . . . . . . . . . . . . . . . . . . . 23 (∅ × {1}) = ∅
132130, 131eqtri 2842 . . . . . . . . . . . . . . . . . . . . . 22 (((2nd ‘(1st𝑧)) “ ∅) × {1}) = ∅
133132uneq1i 4133 . . . . . . . . . . . . . . . . . . . . 21 ((((2nd ‘(1st𝑧)) “ ∅) × {1}) ∪ (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0})) = (∅ ∪ (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0}))
134 uncom 4127 . . . . . . . . . . . . . . . . . . . . 21 (∅ ∪ (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0})) = ((((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0}) ∪ ∅)
135 un0 4342 . . . . . . . . . . . . . . . . . . . . 21 ((((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0}) ∪ ∅) = (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0})
136133, 134, 1353eqtri 2846 . . . . . . . . . . . . . . . . . . . 20 ((((2nd ‘(1st𝑧)) “ ∅) × {1}) ∪ (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0})) = (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0})
137128, 136syl6eq 2870 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 0 → ((((2nd ‘(1st𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) × {0})) = (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0}))
138137oveq2d 7164 . . . . . . . . . . . . . . . . . 18 (𝑗 = 0 → ((1st ‘(1st𝑧)) ∘f + ((((2nd ‘(1st𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑧)) ∘f + (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0})))
139116, 138csbie 3916 . . . . . . . . . . . . . . . . 17 0 / 𝑗((1st ‘(1st𝑧)) ∘f + ((((2nd ‘(1st𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑧)) ∘f + (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0}))
140 f1ofo 6615 . . . . . . . . . . . . . . . . . . . . . 22 ((2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑧)):(1...𝑁)–onto→(1...𝑁))
14189, 140syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑧𝑆 → (2nd ‘(1st𝑧)):(1...𝑁)–onto→(1...𝑁))
142 foima 6588 . . . . . . . . . . . . . . . . . . . . 21 ((2nd ‘(1st𝑧)):(1...𝑁)–onto→(1...𝑁) → ((2nd ‘(1st𝑧)) “ (1...𝑁)) = (1...𝑁))
143141, 142syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑧𝑆 → ((2nd ‘(1st𝑧)) “ (1...𝑁)) = (1...𝑁))
144143xpeq1d 5577 . . . . . . . . . . . . . . . . . . 19 (𝑧𝑆 → (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0}) = ((1...𝑁) × {0}))
145144oveq2d 7164 . . . . . . . . . . . . . . . . . 18 (𝑧𝑆 → ((1st ‘(1st𝑧)) ∘f + (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0})) = ((1st ‘(1st𝑧)) ∘f + ((1...𝑁) × {0})))
146 ovexd 7183 . . . . . . . . . . . . . . . . . . 19 (𝑧𝑆 → (1...𝑁) ∈ V)
14780ffnd 6508 . . . . . . . . . . . . . . . . . . 19 (𝑧𝑆 → (1st ‘(1st𝑧)) Fn (1...𝑁))
148 fnconstg 6560 . . . . . . . . . . . . . . . . . . . 20 (0 ∈ V → ((1...𝑁) × {0}) Fn (1...𝑁))
149116, 148mp1i 13 . . . . . . . . . . . . . . . . . . 19 (𝑧𝑆 → ((1...𝑁) × {0}) Fn (1...𝑁))
150 eqidd 2820 . . . . . . . . . . . . . . . . . . 19 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑧))‘𝑛) = ((1st ‘(1st𝑧))‘𝑛))
151116fvconst2 6959 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {0})‘𝑛) = 0)
152151adantl 484 . . . . . . . . . . . . . . . . . . 19 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → (((1...𝑁) × {0})‘𝑛) = 0)
15380ffvelrnda 6844 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑧))‘𝑛) ∈ (0..^𝐾))
154 elfzonn0 13074 . . . . . . . . . . . . . . . . . . . . . 22 (((1st ‘(1st𝑧))‘𝑛) ∈ (0..^𝐾) → ((1st ‘(1st𝑧))‘𝑛) ∈ ℕ0)
155153, 154syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑧))‘𝑛) ∈ ℕ0)
156155nn0cnd 11949 . . . . . . . . . . . . . . . . . . . 20 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑧))‘𝑛) ∈ ℂ)
157156addid1d 10832 . . . . . . . . . . . . . . . . . . 19 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑧))‘𝑛) + 0) = ((1st ‘(1st𝑧))‘𝑛))
158146, 147, 149, 147, 150, 152, 157offveq 7422 . . . . . . . . . . . . . . . . . 18 (𝑧𝑆 → ((1st ‘(1st𝑧)) ∘f + ((1...𝑁) × {0})) = (1st ‘(1st𝑧)))
159145, 158eqtrd 2854 . . . . . . . . . . . . . . . . 17 (𝑧𝑆 → ((1st ‘(1st𝑧)) ∘f + (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0})) = (1st ‘(1st𝑧)))
160139, 159syl5eq 2866 . . . . . . . . . . . . . . . 16 (𝑧𝑆0 / 𝑗((1st ‘(1st𝑧)) ∘f + ((((2nd ‘(1st𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))) = (1st ‘(1st𝑧)))
161160ad2antlr 725 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ 𝑦 = 0) → 0 / 𝑗((1st ‘(1st𝑧)) ∘f + ((((2nd ‘(1st𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))) = (1st ‘(1st𝑧)))
162115, 161eqtrd 2854 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ 𝑦 = 0) → if(𝑦 < (2nd𝑧), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑧)) ∘f + ((((2nd ‘(1st𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))) = (1st ‘(1st𝑧)))
163 nnm1nn0 11930 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
1641, 163syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑁 − 1) ∈ ℕ0)
165 0elfz 12996 . . . . . . . . . . . . . . . 16 ((𝑁 − 1) ∈ ℕ0 → 0 ∈ (0...(𝑁 − 1)))
166164, 165syl 17 . . . . . . . . . . . . . . 15 (𝜑 → 0 ∈ (0...(𝑁 − 1)))
167166ad2antrr 724 . . . . . . . . . . . . . 14 (((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → 0 ∈ (0...(𝑁 − 1)))
168 fvexd 6678 . . . . . . . . . . . . . 14 (((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (1st ‘(1st𝑧)) ∈ V)
169106, 162, 167, 168fvmptd 6768 . . . . . . . . . . . . 13 (((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (𝐹‘0) = (1st ‘(1st𝑧)))
170105, 169sylan 582 . . . . . . . . . . . 12 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ (2nd𝑧) = (2nd𝑇)) ∧ 𝑧𝑆) → (𝐹‘0) = (1st ‘(1st𝑧)))
171170an32s 650 . . . . . . . . . . 11 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ (2nd𝑧) = (2nd𝑇)) → (𝐹‘0) = (1st ‘(1st𝑧)))
172101, 171mpdan 685 . . . . . . . . . 10 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (𝐹‘0) = (1st ‘(1st𝑧)))
173 fveq2 6663 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑇 → (2nd𝑧) = (2nd𝑇))
174173eleq1d 2895 . . . . . . . . . . . . . . 15 (𝑧 = 𝑇 → ((2nd𝑧) ∈ (1...(𝑁 − 1)) ↔ (2nd𝑇) ∈ (1...(𝑁 − 1))))
175174anbi2d 630 . . . . . . . . . . . . . 14 (𝑧 = 𝑇 → ((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ↔ (𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1)))))
176 2fveq3 6668 . . . . . . . . . . . . . . 15 (𝑧 = 𝑇 → (1st ‘(1st𝑧)) = (1st ‘(1st𝑇)))
177176eqeq2d 2830 . . . . . . . . . . . . . 14 (𝑧 = 𝑇 → ((𝐹‘0) = (1st ‘(1st𝑧)) ↔ (𝐹‘0) = (1st ‘(1st𝑇))))
178175, 177imbi12d 347 . . . . . . . . . . . . 13 (𝑧 = 𝑇 → (((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → (𝐹‘0) = (1st ‘(1st𝑧))) ↔ ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (𝐹‘0) = (1st ‘(1st𝑇)))))
179169expcom 416 . . . . . . . . . . . . 13 (𝑧𝑆 → ((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → (𝐹‘0) = (1st ‘(1st𝑧))))
180178, 179vtoclga 3572 . . . . . . . . . . . 12 (𝑇𝑆 → ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (𝐹‘0) = (1st ‘(1st𝑇))))
1817, 180mpcom 38 . . . . . . . . . . 11 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (𝐹‘0) = (1st ‘(1st𝑇)))
182181adantr 483 . . . . . . . . . 10 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (𝐹‘0) = (1st ‘(1st𝑇)))
183172, 182eqtr3d 2856 . . . . . . . . 9 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (1st ‘(1st𝑧)) = (1st ‘(1st𝑇)))
184183adantr 483 . . . . . . . 8 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ 𝑧𝑇) → (1st ‘(1st𝑧)) = (1st ‘(1st𝑇)))
1851ad3antrrr 728 . . . . . . . . 9 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ 𝑧𝑇) → 𝑁 ∈ ℕ)
1866ad3antrrr 728 . . . . . . . . 9 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ 𝑧𝑇) → 𝑇𝑆)
187 simpllr 774 . . . . . . . . 9 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ 𝑧𝑇) → (2nd𝑇) ∈ (1...(𝑁 − 1)))
188 simplr 767 . . . . . . . . 9 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ 𝑧𝑇) → 𝑧𝑆)
18934adantr 483 . . . . . . . . . . . . . 14 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (1st𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
190 xpopth 7722 . . . . . . . . . . . . . 14 (((1st𝑧) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (1st𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) → (((1st ‘(1st𝑧)) = (1st ‘(1st𝑇)) ∧ (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑇))) ↔ (1st𝑧) = (1st𝑇)))
19176, 189, 190syl2anr 598 . . . . . . . . . . . . 13 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (((1st ‘(1st𝑧)) = (1st ‘(1st𝑇)) ∧ (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑇))) ↔ (1st𝑧) = (1st𝑇)))
19232adantr 483 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
193 xpopth 7722 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (((1st𝑧) = (1st𝑇) ∧ (2nd𝑧) = (2nd𝑇)) ↔ 𝑧 = 𝑇))
194193biimpd 231 . . . . . . . . . . . . . . 15 ((𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (((1st𝑧) = (1st𝑇) ∧ (2nd𝑧) = (2nd𝑇)) → 𝑧 = 𝑇))
19574, 192, 194syl2anr 598 . . . . . . . . . . . . . 14 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (((1st𝑧) = (1st𝑇) ∧ (2nd𝑧) = (2nd𝑇)) → 𝑧 = 𝑇))
196101, 195mpan2d 692 . . . . . . . . . . . . 13 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → ((1st𝑧) = (1st𝑇) → 𝑧 = 𝑇))
197191, 196sylbid 242 . . . . . . . . . . . 12 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (((1st ‘(1st𝑧)) = (1st ‘(1st𝑇)) ∧ (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑇))) → 𝑧 = 𝑇))
198183, 197mpand 693 . . . . . . . . . . 11 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → ((2nd ‘(1st𝑧)) = (2nd ‘(1st𝑇)) → 𝑧 = 𝑇))
199198necon3d 3035 . . . . . . . . . 10 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (𝑧𝑇 → (2nd ‘(1st𝑧)) ≠ (2nd ‘(1st𝑇))))
200199imp 409 . . . . . . . . 9 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ 𝑧𝑇) → (2nd ‘(1st𝑧)) ≠ (2nd ‘(1st𝑇)))
201185, 3, 186, 187, 188, 200poimirlem9 34893 . . . . . . . 8 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ 𝑧𝑇) → (2nd ‘(1st𝑧)) = ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))))
202101adantr 483 . . . . . . . 8 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ 𝑧𝑇) → (2nd𝑧) = (2nd𝑇))
203184, 201, 202jca31 517 . . . . . . 7 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ 𝑧𝑇) → (((1st ‘(1st𝑧)) = (1st ‘(1st𝑇)) ∧ (2nd ‘(1st𝑧)) = ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))) ∧ (2nd𝑧) = (2nd𝑇)))
204203ex 415 . . . . . 6 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (𝑧𝑇 → (((1st ‘(1st𝑧)) = (1st ‘(1st𝑇)) ∧ (2nd ‘(1st𝑧)) = ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))) ∧ (2nd𝑧) = (2nd𝑇))))
205 simplr 767 . . . . . . . 8 ((((1st ‘(1st𝑧)) = (1st ‘(1st𝑇)) ∧ (2nd ‘(1st𝑧)) = ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))) ∧ (2nd𝑧) = (2nd𝑇)) → (2nd ‘(1st𝑧)) = ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))))
206 elfznn 12928 . . . . . . . . . . . . . 14 ((2nd𝑇) ∈ (1...(𝑁 − 1)) → (2nd𝑇) ∈ ℕ)
207206nnred 11645 . . . . . . . . . . . . 13 ((2nd𝑇) ∈ (1...(𝑁 − 1)) → (2nd𝑇) ∈ ℝ)
208207ltp1d 11562 . . . . . . . . . . . . 13 ((2nd𝑇) ∈ (1...(𝑁 − 1)) → (2nd𝑇) < ((2nd𝑇) + 1))
209207, 208ltned 10768 . . . . . . . . . . . 12 ((2nd𝑇) ∈ (1...(𝑁 − 1)) → (2nd𝑇) ≠ ((2nd𝑇) + 1))
210209adantl 484 . . . . . . . . . . 11 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (2nd𝑇) ≠ ((2nd𝑇) + 1))
211 fveq1 6662 . . . . . . . . . . . . 13 ((2nd ‘(1st𝑇)) = ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) → ((2nd ‘(1st𝑇))‘(2nd𝑇)) = (((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))‘(2nd𝑇)))
212 id 22 . . . . . . . . . . . . . . . . . . . . . 22 ((2nd𝑇) ∈ ℝ → (2nd𝑇) ∈ ℝ)
213 ltp1 11472 . . . . . . . . . . . . . . . . . . . . . 22 ((2nd𝑇) ∈ ℝ → (2nd𝑇) < ((2nd𝑇) + 1))
214212, 213ltned 10768 . . . . . . . . . . . . . . . . . . . . 21 ((2nd𝑇) ∈ ℝ → (2nd𝑇) ≠ ((2nd𝑇) + 1))
215 fvex 6676 . . . . . . . . . . . . . . . . . . . . . 22 (2nd𝑇) ∈ V
216 ovex 7181 . . . . . . . . . . . . . . . . . . . . . 22 ((2nd𝑇) + 1) ∈ V
217215, 216, 216, 215fpr 6909 . . . . . . . . . . . . . . . . . . . . 21 ((2nd𝑇) ≠ ((2nd𝑇) + 1) → {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩}:{(2nd𝑇), ((2nd𝑇) + 1)}⟶{((2nd𝑇) + 1), (2nd𝑇)})
218214, 217syl 17 . . . . . . . . . . . . . . . . . . . 20 ((2nd𝑇) ∈ ℝ → {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩}:{(2nd𝑇), ((2nd𝑇) + 1)}⟶{((2nd𝑇) + 1), (2nd𝑇)})
219 f1oi 6645 . . . . . . . . . . . . . . . . . . . . 21 ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})):((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})–1-1-onto→((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})
220 f1of 6608 . . . . . . . . . . . . . . . . . . . . 21 (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})):((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})–1-1-onto→((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) → ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})):((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})⟶((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
221219, 220ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})):((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})⟶((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})
222 disjdif 4419 . . . . . . . . . . . . . . . . . . . . 21 ({(2nd𝑇), ((2nd𝑇) + 1)} ∩ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = ∅
223 fun 6533 . . . . . . . . . . . . . . . . . . . . 21 ((({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩}:{(2nd𝑇), ((2nd𝑇) + 1)}⟶{((2nd𝑇) + 1), (2nd𝑇)} ∧ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})):((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})⟶((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) ∧ ({(2nd𝑇), ((2nd𝑇) + 1)} ∩ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = ∅) → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))):({(2nd𝑇), ((2nd𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))⟶({((2nd𝑇) + 1), (2nd𝑇)} ∪ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))
224222, 223mpan2 689 . . . . . . . . . . . . . . . . . . . 20 (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩}:{(2nd𝑇), ((2nd𝑇) + 1)}⟶{((2nd𝑇) + 1), (2nd𝑇)} ∧ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})):((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})⟶((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))):({(2nd𝑇), ((2nd𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))⟶({((2nd𝑇) + 1), (2nd𝑇)} ∪ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))
225218, 221, 224sylancl 588 . . . . . . . . . . . . . . . . . . 19 ((2nd𝑇) ∈ ℝ → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))):({(2nd𝑇), ((2nd𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))⟶({((2nd𝑇) + 1), (2nd𝑇)} ∪ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))
226215prid1 4690 . . . . . . . . . . . . . . . . . . . 20 (2nd𝑇) ∈ {(2nd𝑇), ((2nd𝑇) + 1)}
227 elun1 4150 . . . . . . . . . . . . . . . . . . . 20 ((2nd𝑇) ∈ {(2nd𝑇), ((2nd𝑇) + 1)} → (2nd𝑇) ∈ ({(2nd𝑇), ((2nd𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))
228226, 227ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (2nd𝑇) ∈ ({(2nd𝑇), ((2nd𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
229 fvco3 6753 . . . . . . . . . . . . . . . . . . 19 ((({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))):({(2nd𝑇), ((2nd𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))⟶({((2nd𝑇) + 1), (2nd𝑇)} ∪ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) ∧ (2nd𝑇) ∈ ({(2nd𝑇), ((2nd𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))) → (((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))‘(2nd𝑇)) = ((2nd ‘(1st𝑇))‘(({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))‘(2nd𝑇))))
230225, 228, 229sylancl 588 . . . . . . . . . . . . . . . . . 18 ((2nd𝑇) ∈ ℝ → (((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))‘(2nd𝑇)) = ((2nd ‘(1st𝑇))‘(({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))‘(2nd𝑇))))
231218ffnd 6508 . . . . . . . . . . . . . . . . . . . . 21 ((2nd𝑇) ∈ ℝ → {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} Fn {(2nd𝑇), ((2nd𝑇) + 1)})
232 fnresi 6469 . . . . . . . . . . . . . . . . . . . . . 22 ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) Fn ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})
233222, 226pm3.2i 473 . . . . . . . . . . . . . . . . . . . . . 22 (({(2nd𝑇), ((2nd𝑇) + 1)} ∩ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = ∅ ∧ (2nd𝑇) ∈ {(2nd𝑇), ((2nd𝑇) + 1)})
234 fvun1 6747 . . . . . . . . . . . . . . . . . . . . . 22 (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} Fn {(2nd𝑇), ((2nd𝑇) + 1)} ∧ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) Fn ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ∧ (({(2nd𝑇), ((2nd𝑇) + 1)} ∩ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = ∅ ∧ (2nd𝑇) ∈ {(2nd𝑇), ((2nd𝑇) + 1)})) → (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))‘(2nd𝑇)) = ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩}‘(2nd𝑇)))
235232, 233, 234mp3an23 1446 . . . . . . . . . . . . . . . . . . . . 21 ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} Fn {(2nd𝑇), ((2nd𝑇) + 1)} → (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))‘(2nd𝑇)) = ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩}‘(2nd𝑇)))
236231, 235syl 17 . . . . . . . . . . . . . . . . . . . 20 ((2nd𝑇) ∈ ℝ → (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))‘(2nd𝑇)) = ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩}‘(2nd𝑇)))
237215, 216fvpr1 6945 . . . . . . . . . . . . . . . . . . . . 21 ((2nd𝑇) ≠ ((2nd𝑇) + 1) → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩}‘(2nd𝑇)) = ((2nd𝑇) + 1))
238214, 237syl 17 . . . . . . . . . . . . . . . . . . . 20 ((2nd𝑇) ∈ ℝ → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩}‘(2nd𝑇)) = ((2nd𝑇) + 1))
239236, 238eqtrd 2854 . . . . . . . . . . . . . . . . . . 19 ((2nd𝑇) ∈ ℝ → (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))‘(2nd𝑇)) = ((2nd𝑇) + 1))
240239fveq2d 6667 . . . . . . . . . . . . . . . . . 18 ((2nd𝑇) ∈ ℝ → ((2nd ‘(1st𝑇))‘(({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))‘(2nd𝑇))) = ((2nd ‘(1st𝑇))‘((2nd𝑇) + 1)))
241230, 240eqtrd 2854 . . . . . . . . . . . . . . . . 17 ((2nd𝑇) ∈ ℝ → (((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))‘(2nd𝑇)) = ((2nd ‘(1st𝑇))‘((2nd𝑇) + 1)))
242207, 241syl 17 . . . . . . . . . . . . . . . 16 ((2nd𝑇) ∈ (1...(𝑁 − 1)) → (((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))‘(2nd𝑇)) = ((2nd ‘(1st𝑇))‘((2nd𝑇) + 1)))
243242eqeq2d 2830 . . . . . . . . . . . . . . 15 ((2nd𝑇) ∈ (1...(𝑁 − 1)) → (((2nd ‘(1st𝑇))‘(2nd𝑇)) = (((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))‘(2nd𝑇)) ↔ ((2nd ‘(1st𝑇))‘(2nd𝑇)) = ((2nd ‘(1st𝑇))‘((2nd𝑇) + 1))))
244243adantl 484 . . . . . . . . . . . . . 14 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (((2nd ‘(1st𝑇))‘(2nd𝑇)) = (((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))‘(2nd𝑇)) ↔ ((2nd ‘(1st𝑇))‘(2nd𝑇)) = ((2nd ‘(1st𝑇))‘((2nd𝑇) + 1))))
245 f1of1 6607 . . . . . . . . . . . . . . . . 17 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)):(1...𝑁)–1-1→(1...𝑁))
24649, 245syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–1-1→(1...𝑁))
247246adantr 483 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (2nd ‘(1st𝑇)):(1...𝑁)–1-1→(1...𝑁))
2481nncnd 11646 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑁 ∈ ℂ)
249 npcan1 11057 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
250248, 249syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
251164nn0zd 12077 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑁 − 1) ∈ ℤ)
252 uzid 12250 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)))
253251, 252syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)))
254 peano2uz 12293 . . . . . . . . . . . . . . . . . . 19 ((𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
255253, 254syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
256250, 255eqeltrrd 2912 . . . . . . . . . . . . . . . . 17 (𝜑𝑁 ∈ (ℤ‘(𝑁 − 1)))
257 fzss2 12939 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ (ℤ‘(𝑁 − 1)) → (1...(𝑁 − 1)) ⊆ (1...𝑁))
258256, 257syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
259258sselda 3965 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (2nd𝑇) ∈ (1...𝑁))
260 fzp1elp1 12952 . . . . . . . . . . . . . . . . 17 ((2nd𝑇) ∈ (1...(𝑁 − 1)) → ((2nd𝑇) + 1) ∈ (1...((𝑁 − 1) + 1)))
261260adantl 484 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → ((2nd𝑇) + 1) ∈ (1...((𝑁 − 1) + 1)))
262250oveq2d 7164 . . . . . . . . . . . . . . . . 17 (𝜑 → (1...((𝑁 − 1) + 1)) = (1...𝑁))
263262adantr 483 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (1...((𝑁 − 1) + 1)) = (1...𝑁))
264261, 263eleqtrd 2913 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → ((2nd𝑇) + 1) ∈ (1...𝑁))
265 f1veqaeq 7007 . . . . . . . . . . . . . . 15 (((2nd ‘(1st𝑇)):(1...𝑁)–1-1→(1...𝑁) ∧ ((2nd𝑇) ∈ (1...𝑁) ∧ ((2nd𝑇) + 1) ∈ (1...𝑁))) → (((2nd ‘(1st𝑇))‘(2nd𝑇)) = ((2nd ‘(1st𝑇))‘((2nd𝑇) + 1)) → (2nd𝑇) = ((2nd𝑇) + 1)))
266247, 259, 264, 265syl12anc 834 . . . . . . . . . . . . . 14 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (((2nd ‘(1st𝑇))‘(2nd𝑇)) = ((2nd ‘(1st𝑇))‘((2nd𝑇) + 1)) → (2nd𝑇) = ((2nd𝑇) + 1)))
267244, 266sylbid 242 . . . . . . . . . . . . 13 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (((2nd ‘(1st𝑇))‘(2nd𝑇)) = (((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))‘(2nd𝑇)) → (2nd𝑇) = ((2nd𝑇) + 1)))
268211, 267syl5 34 . . . . . . . . . . . 12 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) = ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) → (2nd𝑇) = ((2nd𝑇) + 1)))
269268necon3d 3035 . . . . . . . . . . 11 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → ((2nd𝑇) ≠ ((2nd𝑇) + 1) → (2nd ‘(1st𝑇)) ≠ ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))))
270210, 269mpd 15 . . . . . . . . . 10 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (2nd ‘(1st𝑇)) ≠ ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))))
271 2fveq3 6668 . . . . . . . . . . 11 (𝑧 = 𝑇 → (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑇)))
272271neeq1d 3073 . . . . . . . . . 10 (𝑧 = 𝑇 → ((2nd ‘(1st𝑧)) ≠ ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) ↔ (2nd ‘(1st𝑇)) ≠ ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))))
273270, 272syl5ibrcom 249 . . . . . . . . 9 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (𝑧 = 𝑇 → (2nd ‘(1st𝑧)) ≠ ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))))
274273necon2d 3037 . . . . . . . 8 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → ((2nd ‘(1st𝑧)) = ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) → 𝑧𝑇))
275205, 274syl5 34 . . . . . . 7 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → ((((1st ‘(1st𝑧)) = (1st ‘(1st𝑇)) ∧ (2nd ‘(1st𝑧)) = ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))) ∧ (2nd𝑧) = (2nd𝑇)) → 𝑧𝑇))
276275adantr 483 . . . . . 6 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → ((((1st ‘(1st𝑧)) = (1st ‘(1st𝑇)) ∧ (2nd ‘(1st𝑧)) = ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))) ∧ (2nd𝑧) = (2nd𝑇)) → 𝑧𝑇))
277204, 276impbid 214 . . . . 5 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (𝑧𝑇 ↔ (((1st ‘(1st𝑧)) = (1st ‘(1st𝑇)) ∧ (2nd ‘(1st𝑧)) = ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))) ∧ (2nd𝑧) = (2nd𝑇))))
278 eqop 7723 . . . . . . . 8 (𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (𝑧 = ⟨⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩, (2nd𝑇)⟩ ↔ ((1st𝑧) = ⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩ ∧ (2nd𝑧) = (2nd𝑇))))
279 eqop 7723 . . . . . . . . . 10 ((1st𝑧) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → ((1st𝑧) = ⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩ ↔ ((1st ‘(1st𝑧)) = (1st ‘(1st𝑇)) ∧ (2nd ‘(1st𝑧)) = ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))))))
28075, 279syl 17 . . . . . . . . 9 (𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → ((1st𝑧) = ⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩ ↔ ((1st ‘(1st𝑧)) = (1st ‘(1st𝑇)) ∧ (2nd ‘(1st𝑧)) = ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))))))
281280anbi1d 631 . . . . . . . 8 (𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (((1st𝑧) = ⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩ ∧ (2nd𝑧) = (2nd𝑇)) ↔ (((1st ‘(1st𝑧)) = (1st ‘(1st𝑇)) ∧ (2nd ‘(1st𝑧)) = ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))) ∧ (2nd𝑧) = (2nd𝑇))))
282278, 281bitrd 281 . . . . . . 7 (𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (𝑧 = ⟨⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩, (2nd𝑇)⟩ ↔ (((1st ‘(1st𝑧)) = (1st ‘(1st𝑇)) ∧ (2nd ‘(1st𝑧)) = ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))) ∧ (2nd𝑧) = (2nd𝑇))))
28374, 282syl 17 . . . . . 6 (𝑧𝑆 → (𝑧 = ⟨⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩, (2nd𝑇)⟩ ↔ (((1st ‘(1st𝑧)) = (1st ‘(1st𝑇)) ∧ (2nd ‘(1st𝑧)) = ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))) ∧ (2nd𝑧) = (2nd𝑇))))
284283adantl 484 . . . . 5 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (𝑧 = ⟨⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩, (2nd𝑇)⟩ ↔ (((1st ‘(1st𝑧)) = (1st ‘(1st𝑇)) ∧ (2nd ‘(1st𝑧)) = ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))) ∧ (2nd𝑧) = (2nd𝑇))))
285277, 284bitr4d 284 . . . 4 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (𝑧𝑇𝑧 = ⟨⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩, (2nd𝑇)⟩))
286285ralrimiva 3180 . . 3 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → ∀𝑧𝑆 (𝑧𝑇𝑧 = ⟨⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩, (2nd𝑇)⟩))
287 reu6i 3717 . . 3 ((⟨⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩, (2nd𝑇)⟩ ∈ 𝑆 ∧ ∀𝑧𝑆 (𝑧𝑇𝑧 = ⟨⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩, (2nd𝑇)⟩)) → ∃!𝑧𝑆 𝑧𝑇)
2889, 286, 287syl2anc 586 . 2 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → ∃!𝑧𝑆 𝑧𝑇)
289 xp2nd 7714 . . . . . . 7 (𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd𝑇) ∈ (0...𝑁))
29032, 289syl 17 . . . . . 6 (𝜑 → (2nd𝑇) ∈ (0...𝑁))
291290biantrurd 535 . . . . 5 (𝜑 → (¬ (2nd𝑇) ∈ (1...(𝑁 − 1)) ↔ ((2nd𝑇) ∈ (0...𝑁) ∧ ¬ (2nd𝑇) ∈ (1...(𝑁 − 1)))))
2921nnnn0d 11947 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℕ0)
293 nn0uz 12272 . . . . . . . . . . . 12 0 = (ℤ‘0)
294292, 293eleqtrdi 2921 . . . . . . . . . . 11 (𝜑𝑁 ∈ (ℤ‘0))
295 fzpred 12947 . . . . . . . . . . 11 (𝑁 ∈ (ℤ‘0) → (0...𝑁) = ({0} ∪ ((0 + 1)...𝑁)))
296294, 295syl 17 . . . . . . . . . 10 (𝜑 → (0...𝑁) = ({0} ∪ ((0 + 1)...𝑁)))
297123oveq1i 7158 . . . . . . . . . . 11 ((0 + 1)...𝑁) = (1...𝑁)
298297uneq2i 4134 . . . . . . . . . 10 ({0} ∪ ((0 + 1)...𝑁)) = ({0} ∪ (1...𝑁))
299296, 298syl6eq 2870 . . . . . . . . 9 (𝜑 → (0...𝑁) = ({0} ∪ (1...𝑁)))
300299difeq1d 4096 . . . . . . . 8 (𝜑 → ((0...𝑁) ∖ (1...(𝑁 − 1))) = (({0} ∪ (1...𝑁)) ∖ (1...(𝑁 − 1))))
301 difundir 4255 . . . . . . . . . 10 (({0} ∪ (1...𝑁)) ∖ (1...(𝑁 − 1))) = (({0} ∖ (1...(𝑁 − 1))) ∪ ((1...𝑁) ∖ (1...(𝑁 − 1))))
302 0lt1 11154 . . . . . . . . . . . . . 14 0 < 1
303 0re 10635 . . . . . . . . . . . . . . 15 0 ∈ ℝ
304 1re 10633 . . . . . . . . . . . . . . 15 1 ∈ ℝ
305303, 304ltnlei 10753 . . . . . . . . . . . . . 14 (0 < 1 ↔ ¬ 1 ≤ 0)
306302, 305mpbi 232 . . . . . . . . . . . . 13 ¬ 1 ≤ 0
307 elfzle1 12902 . . . . . . . . . . . . 13 (0 ∈ (1...(𝑁 − 1)) → 1 ≤ 0)
308306, 307mto 199 . . . . . . . . . . . 12 ¬ 0 ∈ (1...(𝑁 − 1))
309 incom 4176 . . . . . . . . . . . . . 14 ((1...(𝑁 − 1)) ∩ {0}) = ({0} ∩ (1...(𝑁 − 1)))
310309eqeq1i 2824 . . . . . . . . . . . . 13 (((1...(𝑁 − 1)) ∩ {0}) = ∅ ↔ ({0} ∩ (1...(𝑁 − 1))) = ∅)
311 disjsn 4639 . . . . . . . . . . . . 13 (((1...(𝑁 − 1)) ∩ {0}) = ∅ ↔ ¬ 0 ∈ (1...(𝑁 − 1)))
312 disj3 4401 . . . . . . . . . . . . 13 (({0} ∩ (1...(𝑁 − 1))) = ∅ ↔ {0} = ({0} ∖ (1...(𝑁 − 1))))
313310, 311, 3123bitr3i 303 . . . . . . . . . . . 12 (¬ 0 ∈ (1...(𝑁 − 1)) ↔ {0} = ({0} ∖ (1...(𝑁 − 1))))
314308, 313mpbi 232 . . . . . . . . . . 11 {0} = ({0} ∖ (1...(𝑁 − 1)))
315314uneq1i 4133 . . . . . . . . . 10 ({0} ∪ ((1...𝑁) ∖ (1...(𝑁 − 1)))) = (({0} ∖ (1...(𝑁 − 1))) ∪ ((1...𝑁) ∖ (1...(𝑁 − 1))))
316301, 315eqtr4i 2845 . . . . . . . . 9 (({0} ∪ (1...𝑁)) ∖ (1...(𝑁 − 1))) = ({0} ∪ ((1...𝑁) ∖ (1...(𝑁 − 1))))
317 difundir 4255 . . . . . . . . . . . 12 (((1...(𝑁 − 1)) ∪ {𝑁}) ∖ (1...(𝑁 − 1))) = (((1...(𝑁 − 1)) ∖ (1...(𝑁 − 1))) ∪ ({𝑁} ∖ (1...(𝑁 − 1))))
318 difid 4328 . . . . . . . . . . . . 13 ((1...(𝑁 − 1)) ∖ (1...(𝑁 − 1))) = ∅
319318uneq1i 4133 . . . . . . . . . . . 12 (((1...(𝑁 − 1)) ∖ (1...(𝑁 − 1))) ∪ ({𝑁} ∖ (1...(𝑁 − 1)))) = (∅ ∪ ({𝑁} ∖ (1...(𝑁 − 1))))
320 uncom 4127 . . . . . . . . . . . . 13 (∅ ∪ ({𝑁} ∖ (1...(𝑁 − 1)))) = (({𝑁} ∖ (1...(𝑁 − 1))) ∪ ∅)
321 un0 4342 . . . . . . . . . . . . 13 (({𝑁} ∖ (1...(𝑁 − 1))) ∪ ∅) = ({𝑁} ∖ (1...(𝑁 − 1)))
322320, 321eqtri 2842 . . . . . . . . . . . 12 (∅ ∪ ({𝑁} ∖ (1...(𝑁 − 1)))) = ({𝑁} ∖ (1...(𝑁 − 1)))
323317, 319, 3223eqtri 2846 . . . . . . . . . . 11 (((1...(𝑁 − 1)) ∪ {𝑁}) ∖ (1...(𝑁 − 1))) = ({𝑁} ∖ (1...(𝑁 − 1)))
324 nnuz 12273 . . . . . . . . . . . . . . . 16 ℕ = (ℤ‘1)
3251, 324eleqtrdi 2921 . . . . . . . . . . . . . . 15 (𝜑𝑁 ∈ (ℤ‘1))
326250, 325eqeltrd 2911 . . . . . . . . . . . . . 14 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘1))
327 fzsplit2 12924 . . . . . . . . . . . . . 14 ((((𝑁 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ‘(𝑁 − 1))) → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
328326, 256, 327syl2anc 586 . . . . . . . . . . . . 13 (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
329250oveq1d 7163 . . . . . . . . . . . . . . 15 (𝜑 → (((𝑁 − 1) + 1)...𝑁) = (𝑁...𝑁))
3301nnzd 12078 . . . . . . . . . . . . . . . 16 (𝜑𝑁 ∈ ℤ)
331 fzsn 12941 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
332330, 331syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (𝑁...𝑁) = {𝑁})
333329, 332eqtrd 2854 . . . . . . . . . . . . . 14 (𝜑 → (((𝑁 − 1) + 1)...𝑁) = {𝑁})
334333uneq2d 4137 . . . . . . . . . . . . 13 (𝜑 → ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = ((1...(𝑁 − 1)) ∪ {𝑁}))
335328, 334eqtrd 2854 . . . . . . . . . . . 12 (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁}))
336335difeq1d 4096 . . . . . . . . . . 11 (𝜑 → ((1...𝑁) ∖ (1...(𝑁 − 1))) = (((1...(𝑁 − 1)) ∪ {𝑁}) ∖ (1...(𝑁 − 1))))
3371nnred 11645 . . . . . . . . . . . . . . 15 (𝜑𝑁 ∈ ℝ)
338337ltm1d 11564 . . . . . . . . . . . . . 14 (𝜑 → (𝑁 − 1) < 𝑁)
339164nn0red 11948 . . . . . . . . . . . . . . 15 (𝜑 → (𝑁 − 1) ∈ ℝ)
340339, 337ltnled 10779 . . . . . . . . . . . . . 14 (𝜑 → ((𝑁 − 1) < 𝑁 ↔ ¬ 𝑁 ≤ (𝑁 − 1)))
341338, 340mpbid 234 . . . . . . . . . . . . 13 (𝜑 → ¬ 𝑁 ≤ (𝑁 − 1))
342 elfzle2 12903 . . . . . . . . . . . . 13 (𝑁 ∈ (1...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1))
343341, 342nsyl 142 . . . . . . . . . . . 12 (𝜑 → ¬ 𝑁 ∈ (1...(𝑁 − 1)))
344 incom 4176 . . . . . . . . . . . . . 14 ((1...(𝑁 − 1)) ∩ {𝑁}) = ({𝑁} ∩ (1...(𝑁 − 1)))
345344eqeq1i 2824 . . . . . . . . . . . . 13 (((1...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ({𝑁} ∩ (1...(𝑁 − 1))) = ∅)
346 disjsn 4639 . . . . . . . . . . . . 13 (((1...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (1...(𝑁 − 1)))
347 disj3 4401 . . . . . . . . . . . . 13 (({𝑁} ∩ (1...(𝑁 − 1))) = ∅ ↔ {𝑁} = ({𝑁} ∖ (1...(𝑁 − 1))))
348345, 346, 3473bitr3i 303 . . . . . . . . . . . 12 𝑁 ∈ (1...(𝑁 − 1)) ↔ {𝑁} = ({𝑁} ∖ (1...(𝑁 − 1))))
349343, 348sylib 220 . . . . . . . . . . 11 (𝜑 → {𝑁} = ({𝑁} ∖ (1...(𝑁 − 1))))
350323, 336, 3493eqtr4a 2880 . . . . . . . . . 10 (𝜑 → ((1...𝑁) ∖ (1...(𝑁 − 1))) = {𝑁})
351350uneq2d 4137 . . . . . . . . 9 (𝜑 → ({0} ∪ ((1...𝑁) ∖ (1...(𝑁 − 1)))) = ({0} ∪ {𝑁}))
352316, 351syl5eq 2866 . . . . . . . 8 (𝜑 → (({0} ∪ (1...𝑁)) ∖ (1...(𝑁 − 1))) = ({0} ∪ {𝑁}))
353300, 352eqtrd 2854 . . . . . . 7 (𝜑 → ((0...𝑁) ∖ (1...(𝑁 − 1))) = ({0} ∪ {𝑁}))
354353eleq2d 2896 . . . . . 6 (𝜑 → ((2nd𝑇) ∈ ((0...𝑁) ∖ (1...(𝑁 − 1))) ↔ (2nd𝑇) ∈ ({0} ∪ {𝑁})))
355 eldif 3944 . . . . . 6 ((2nd𝑇) ∈ ((0...𝑁) ∖ (1...(𝑁 − 1))) ↔ ((2nd𝑇) ∈ (0...𝑁) ∧ ¬ (2nd𝑇) ∈ (1...(𝑁 − 1))))
356 elun 4123 . . . . . . 7 ((2nd𝑇) ∈ ({0} ∪ {𝑁}) ↔ ((2nd𝑇) ∈ {0} ∨ (2nd𝑇) ∈ {𝑁}))
357215elsn 4574 . . . . . . . 8 ((2nd𝑇) ∈ {0} ↔ (2nd𝑇) = 0)
358215elsn 4574 . . . . . . . 8 ((2nd𝑇) ∈ {𝑁} ↔ (2nd𝑇) = 𝑁)
359357, 358orbi12i 910 . . . . . . 7 (((2nd𝑇) ∈ {0} ∨ (2nd𝑇) ∈ {𝑁}) ↔ ((2nd𝑇) = 0 ∨ (2nd𝑇) = 𝑁))
360356, 359bitri 277 . . . . . 6 ((2nd𝑇) ∈ ({0} ∪ {𝑁}) ↔ ((2nd𝑇) = 0 ∨ (2nd𝑇) = 𝑁))
361354, 355, 3603bitr3g 315 . . . . 5 (𝜑 → (((2nd𝑇) ∈ (0...𝑁) ∧ ¬ (2nd𝑇) ∈ (1...(𝑁 − 1))) ↔ ((2nd𝑇) = 0 ∨ (2nd𝑇) = 𝑁)))
362291, 361bitrd 281 . . . 4 (𝜑 → (¬ (2nd𝑇) ∈ (1...(𝑁 − 1)) ↔ ((2nd𝑇) = 0 ∨ (2nd𝑇) = 𝑁)))
363362biimpa 479 . . 3 ((𝜑 ∧ ¬ (2nd𝑇) ∈ (1...(𝑁 − 1))) → ((2nd𝑇) = 0 ∨ (2nd𝑇) = 𝑁))
3641adantr 483 . . . . 5 ((𝜑 ∧ (2nd𝑇) = 0) → 𝑁 ∈ ℕ)
3654adantr 483 . . . . 5 ((𝜑 ∧ (2nd𝑇) = 0) → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))
3666adantr 483 . . . . 5 ((𝜑 ∧ (2nd𝑇) = 0) → 𝑇𝑆)
367 poimirlem22.4 . . . . . 6 ((𝜑𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 𝐾)
368367adantlr 713 . . . . 5 (((𝜑 ∧ (2nd𝑇) = 0) ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 𝐾)
369 simpr 487 . . . . 5 ((𝜑 ∧ (2nd𝑇) = 0) → (2nd𝑇) = 0)
370364, 3, 365, 366, 368, 369poimirlem18 34902 . . . 4 ((𝜑 ∧ (2nd𝑇) = 0) → ∃!𝑧𝑆 𝑧𝑇)
3711adantr 483 . . . . 5 ((𝜑 ∧ (2nd𝑇) = 𝑁) → 𝑁 ∈ ℕ)
3724adantr 483 . . . . 5 ((𝜑 ∧ (2nd𝑇) = 𝑁) → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))
3736adantr 483 . . . . 5 ((𝜑 ∧ (2nd𝑇) = 𝑁) → 𝑇𝑆)
374 poimirlem22.3 . . . . . 6 ((𝜑𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 0)
375374adantlr 713 . . . . 5 (((𝜑 ∧ (2nd𝑇) = 𝑁) ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 0)
376 simpr 487 . . . . 5 ((𝜑 ∧ (2nd𝑇) = 𝑁) → (2nd𝑇) = 𝑁)
377371, 3, 372, 373, 375, 376poimirlem21 34905 . . . 4 ((𝜑 ∧ (2nd𝑇) = 𝑁) → ∃!𝑧𝑆 𝑧𝑇)
378370, 377jaodan 953 . . 3 ((𝜑 ∧ ((2nd𝑇) = 0 ∨ (2nd𝑇) = 𝑁)) → ∃!𝑧𝑆 𝑧𝑇)
379363, 378syldan 593 . 2 ((𝜑 ∧ ¬ (2nd𝑇) ∈ (1...(𝑁 − 1))) → ∃!𝑧𝑆 𝑧𝑇)
380288, 379pm2.61dan 811 1 (𝜑 → ∃!𝑧𝑆 𝑧𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wo 843   = wceq 1530  wcel 2107  {cab 2797  wne 3014  wral 3136  wrex 3137  ∃!wreu 3138  ∃*wrmo 3139  {crab 3140  Vcvv 3493  csb 3881  cdif 3931  cun 3932  cin 3933  wss 3934  c0 4289  ifcif 4465  {csn 4559  {cpr 4561  cop 4565   class class class wbr 5057  cmpt 5137   I cid 5452   × cxp 5546  ran crn 5549  cres 5550  cima 5551  ccom 5552   Fn wfn 6343  wf 6344  1-1wf1 6345  ontowfo 6346  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7148  f cof 7399  1st c1st 7679  2nd c2nd 7680  m cmap 8398  cc 10527  cr 10528  0cc0 10529  1c1 10530   + caddc 10532   < clt 10667  cle 10668  cmin 10862  cn 11630  0cn0 11889  cz 11973  cuz 12235  ...cfz 12884  ..^cfzo 13025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-of 7401  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-2o 8095  df-oadd 8098  df-er 8281  df-map 8400  df-pm 8401  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-dju 9322  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-n0 11890  df-xnn0 11960  df-z 11974  df-uz 12236  df-fz 12885  df-fzo 13026  df-seq 13362  df-fac 13626  df-bc 13655  df-hash 13683
This theorem is referenced by:  poimirlem27  34911
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