Users' Mathboxes Mathbox for Brendan Leahy < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  poimirlem22 Structured version   Visualization version   GIF version

Theorem poimirlem22 34066
Description: Lemma for poimir 34077, 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 474 . . . 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 474 . . . 4 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)))
6 poimirlem22.2 . . . . 5 (𝜑𝑇𝑆)
76adantr 474 . . . 4 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → 𝑇𝑆)
8 simpr 479 . . . 4 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (2nd𝑇) ∈ (1...(𝑁 − 1)))
92, 3, 5, 7, 8poimirlem15 34059 . . 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 6448 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑇 → (2nd𝑡) = (2nd𝑇))
1110breq2d 4900 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑇 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑇)))
1211ifbid 4329 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = 𝑇 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)))
1312csbeq1d 3758 . . . . . . . . . . . . . . . . . . . . 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 6453 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑇 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
15 2fveq3 6453 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = 𝑇 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑇)))
1615imaeq1d 5721 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑗)))
1716xpeq1d 5386 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}))
1815imaeq1d 5721 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)))
1918xpeq1d 5386 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
2017, 19uneq12d 3991 . . . . . . . . . . . . . . . . . . . . . . 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 6942 . . . . . . . . . . . . . . . . . . . . . 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 4217 . . . . . . . . . . . . . . . . . . . . 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 2814 . . . . . . . . . . . . . . . . . . . 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 4982 . . . . . . . . . . . . . . . . . . 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 2788 . . . . . . . . . . . . . . . . . 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 3576 . . . . . . . . . . . . . . . . 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 492 . . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . . 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 3567 . . . . . . . . . . . . . . . . . . . . 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 2877 . . . . . . . . . . . . . . . . . . . 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 7479 . . . . . . . . . . . . . . . . . . 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 7479 . . . . . . . . . . . . . . . . . 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 8164 . . . . . . . . . . . . . . . . 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 12794 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (0..^𝐾) → 𝑛 ∈ ℤ)
4039ssriv 3825 . . . . . . . . . . . . . . . 16 (0..^𝐾) ⊆ ℤ
41 fss 6306 . . . . . . . . . . . . . . . 16 (((1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾) ∧ (0..^𝐾) ⊆ ℤ) → (1st ‘(1st𝑇)):(1...𝑁)⟶ℤ)
4238, 40, 41sylancl 580 . . . . . . . . . . . . . . 15 (𝜑 → (1st ‘(1st𝑇)):(1...𝑁)⟶ℤ)
4342adantr 474 . . . . . . . . . . . . . 14 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (1st ‘(1st𝑇)):(1...𝑁)⟶ℤ)
44 xp2nd 7480 . . . . . . . . . . . . . . . . 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 6461 . . . . . . . . . . . . . . . . 17 (2nd ‘(1st𝑇)) ∈ V
47 f1oeq1 6382 . . . . . . . . . . . . . . . . 17 (𝑓 = (2nd ‘(1st𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
4846, 47elab 3558 . . . . . . . . . . . . . . . 16 ((2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
4945, 48sylib 210 . . . . . . . . . . . . . . 15 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
5049adantr 474 . . . . . . . . . . . . . 14 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
512, 29, 43, 50, 8poimirlem1 34045 . . . . . . . . . . . . 13 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑇))‘𝑛))
5251adantr 474 . . . . . . . . . . . 12 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑇))‘𝑛))
531ad3antrrr 720 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ (2nd𝑧) ≠ (2nd𝑇)) → 𝑁 ∈ ℕ)
54 fveq2 6448 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑧 → (2nd𝑡) = (2nd𝑧))
5554breq2d 4900 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑧 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑧)))
5655ifbid 4329 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = 𝑧 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑧), 𝑦, (𝑦 + 1)))
5756csbeq1d 3758 . . . . . . . . . . . . . . . . . . . . 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 6453 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑧 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑧)))
59 2fveq3 6453 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = 𝑧 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑧)))
6059imaeq1d 5721 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑧 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑧)) “ (1...𝑗)))
6160xpeq1d 5386 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑧 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑧)) “ (1...𝑗)) × {1}))
6259imaeq1d 5721 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑧 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)))
6362xpeq1d 5386 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑧 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))
6461, 63uneq12d 3991 . . . . . . . . . . . . . . . . . . . . . . 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 6942 . . . . . . . . . . . . . . . . . . . . . 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 4217 . . . . . . . . . . . . . . . . . . . . 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 2814 . . . . . . . . . . . . . . . . . . . 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 4982 . . . . . . . . . . . . . . . . . . 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 2788 . . . . . . . . . . . . . . . . . 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 3576 . . . . . . . . . . . . . . . . 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 492 . . . . . . . . . . . . . . . 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 717 . . . . . . . . . . . . . . 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 3567 . . . . . . . . . . . . . . . . . . . . 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 2877 . . . . . . . . . . . . . . . . . . . 20 (𝑧𝑆𝑧 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
75 xp1st 7479 . . . . . . . . . . . . . . . . . . . 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 7479 . . . . . . . . . . . . . . . . . . 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 8164 . . . . . . . . . . . . . . . . . 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 6306 . . . . . . . . . . . . . . . . 17 (((1st ‘(1st𝑧)):(1...𝑁)⟶(0..^𝐾) ∧ (0..^𝐾) ⊆ ℤ) → (1st ‘(1st𝑧)):(1...𝑁)⟶ℤ)
8280, 40, 81sylancl 580 . . . . . . . . . . . . . . . 16 (𝑧𝑆 → (1st ‘(1st𝑧)):(1...𝑁)⟶ℤ)
8382ad2antlr 717 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ (2nd𝑧) ≠ (2nd𝑇)) → (1st ‘(1st𝑧)):(1...𝑁)⟶ℤ)
84 xp2nd 7480 . . . . . . . . . . . . . . . . . 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 6461 . . . . . . . . . . . . . . . . . 18 (2nd ‘(1st𝑧)) ∈ V
87 f1oeq1 6382 . . . . . . . . . . . . . . . . . 18 (𝑓 = (2nd ‘(1st𝑧)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁)))
8886, 87elab 3558 . . . . . . . . . . . . . . . . 17 ((2nd ‘(1st𝑧)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
8985, 88sylib 210 . . . . . . . . . . . . . . . 16 (𝑧𝑆 → (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
9089ad2antlr 717 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ (2nd𝑧) ≠ (2nd𝑇)) → (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
91 simpllr 766 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ (2nd𝑧) ≠ (2nd𝑇)) → (2nd𝑇) ∈ (1...(𝑁 − 1)))
92 xp2nd 7480 . . . . . . . . . . . . . . . . . 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 475 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (2nd𝑧) ∈ (0...𝑁))
95 eldifsn 4550 . . . . . . . . . . . . . . . . 17 ((2nd𝑧) ∈ ((0...𝑁) ∖ {(2nd𝑇)}) ↔ ((2nd𝑧) ∈ (0...𝑁) ∧ (2nd𝑧) ≠ (2nd𝑇)))
9695biimpri 220 . . . . . . . . . . . . . . . 16 (((2nd𝑧) ∈ (0...𝑁) ∧ (2nd𝑧) ≠ (2nd𝑇)) → (2nd𝑧) ∈ ((0...𝑁) ∖ {(2nd𝑇)}))
9794, 96sylan 575 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ (2nd𝑧) ≠ (2nd𝑇)) → (2nd𝑧) ∈ ((0...𝑁) ∖ {(2nd𝑇)}))
9853, 72, 83, 90, 91, 97poimirlem2 34046 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ (2nd𝑧) ≠ (2nd𝑇)) → ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑇))‘𝑛))
9998ex 403 . . . . . . . . . . . . 13 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → ((2nd𝑧) ≠ (2nd𝑇) → ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑇))‘𝑛)))
10099necon1bd 2987 . . . . . . . . . . . 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 2847 . . . . . . . . . . . . . . . 16 ((2nd𝑧) = (2nd𝑇) → ((2nd𝑧) ∈ (1...(𝑁 − 1)) ↔ (2nd𝑇) ∈ (1...(𝑁 − 1))))
103102biimparc 473 . . . . . . . . . . . . . . 15 (((2nd𝑇) ∈ (1...(𝑁 − 1)) ∧ (2nd𝑧) = (2nd𝑇)) → (2nd𝑧) ∈ (1...(𝑁 − 1)))
104103anim2i 610 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((2nd𝑇) ∈ (1...(𝑁 − 1)) ∧ (2nd𝑧) = (2nd𝑇))) → (𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))))
105104anassrs 461 . . . . . . . . . . . . 13 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ (2nd𝑧) = (2nd𝑇)) → (𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))))
10671adantl 475 . . . . . . . . . . . . . 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 4891 . . . . . . . . . . . . . . . . . 18 (𝑦 = 0 → (𝑦 < (2nd𝑧) ↔ 0 < (2nd𝑧)))
108 id 22 . . . . . . . . . . . . . . . . . 18 (𝑦 = 0 → 𝑦 = 0)
109107, 108ifbieq1d 4330 . . . . . . . . . . . . . . . . 17 (𝑦 = 0 → if(𝑦 < (2nd𝑧), 𝑦, (𝑦 + 1)) = if(0 < (2nd𝑧), 0, (𝑦 + 1)))
110 elfznn 12692 . . . . . . . . . . . . . . . . . . . 20 ((2nd𝑧) ∈ (1...(𝑁 − 1)) → (2nd𝑧) ∈ ℕ)
111110nngt0d 11429 . . . . . . . . . . . . . . . . . . 19 ((2nd𝑧) ∈ (1...(𝑁 − 1)) → 0 < (2nd𝑧))
112111iftrued 4315 . . . . . . . . . . . . . . . . . 18 ((2nd𝑧) ∈ (1...(𝑁 − 1)) → if(0 < (2nd𝑧), 0, (𝑦 + 1)) = 0)
113112ad2antlr 717 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → if(0 < (2nd𝑧), 0, (𝑦 + 1)) = 0)
114109, 113sylan9eqr 2836 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ 𝑦 = 0) → if(𝑦 < (2nd𝑧), 𝑦, (𝑦 + 1)) = 0)
115114csbeq1d 3758 . . . . . . . . . . . . . . 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 10372 . . . . . . . . . . . . . . . . . 18 0 ∈ V
117 oveq2 6932 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 = 0 → (1...𝑗) = (1...0))
118 fz10 12684 . . . . . . . . . . . . . . . . . . . . . . . 24 (1...0) = ∅
119117, 118syl6eq 2830 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = 0 → (1...𝑗) = ∅)
120119imaeq2d 5722 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 0 → ((2nd ‘(1st𝑧)) “ (1...𝑗)) = ((2nd ‘(1st𝑧)) “ ∅))
121120xpeq1d 5386 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 0 → (((2nd ‘(1st𝑧)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑧)) “ ∅) × {1}))
122 oveq1 6931 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 = 0 → (𝑗 + 1) = (0 + 1))
123 0p1e1 11509 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0 + 1) = 1
124122, 123syl6eq 2830 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 = 0 → (𝑗 + 1) = 1)
125124oveq1d 6939 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = 0 → ((𝑗 + 1)...𝑁) = (1...𝑁))
126125imaeq2d 5722 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 0 → ((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑧)) “ (1...𝑁)))
127126xpeq1d 5386 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 0 → (((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0}))
128121, 127uneq12d 3991 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 0 → ((((2nd ‘(1st𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑧)) “ ∅) × {1}) ∪ (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0})))
129 ima0 5737 . . . . . . . . . . . . . . . . . . . . . . . 24 ((2nd ‘(1st𝑧)) “ ∅) = ∅
130129xpeq1i 5383 . . . . . . . . . . . . . . . . . . . . . . 23 (((2nd ‘(1st𝑧)) “ ∅) × {1}) = (∅ × {1})
131 0xp 5449 . . . . . . . . . . . . . . . . . . . . . . 23 (∅ × {1}) = ∅
132130, 131eqtri 2802 . . . . . . . . . . . . . . . . . . . . . 22 (((2nd ‘(1st𝑧)) “ ∅) × {1}) = ∅
133132uneq1i 3986 . . . . . . . . . . . . . . . . . . . . 21 ((((2nd ‘(1st𝑧)) “ ∅) × {1}) ∪ (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0})) = (∅ ∪ (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0}))
134 uncom 3980 . . . . . . . . . . . . . . . . . . . . 21 (∅ ∪ (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0})) = ((((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0}) ∪ ∅)
135 un0 4193 . . . . . . . . . . . . . . . . . . . . 21 ((((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0}) ∪ ∅) = (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0})
136133, 134, 1353eqtri 2806 . . . . . . . . . . . . . . . . . . . 20 ((((2nd ‘(1st𝑧)) “ ∅) × {1}) ∪ (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0})) = (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0})
137128, 136syl6eq 2830 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 0 → ((((2nd ‘(1st𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) × {0})) = (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0}))
138137oveq2d 6940 . . . . . . . . . . . . . . . . . 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 3777 . . . . . . . . . . . . . . . . 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 6400 . . . . . . . . . . . . . . . . . . . . . 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 6373 . . . . . . . . . . . . . . . . . . . . 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 5386 . . . . . . . . . . . . . . . . . . 19 (𝑧𝑆 → (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0}) = ((1...𝑁) × {0}))
145144oveq2d 6940 . . . . . . . . . . . . . . . . . 18 (𝑧𝑆 → ((1st ‘(1st𝑧)) ∘𝑓 + (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0})) = ((1st ‘(1st𝑧)) ∘𝑓 + ((1...𝑁) × {0})))
146 ovexd 6958 . . . . . . . . . . . . . . . . . . 19 (𝑧𝑆 → (1...𝑁) ∈ V)
14780ffnd 6294 . . . . . . . . . . . . . . . . . . 19 (𝑧𝑆 → (1st ‘(1st𝑧)) Fn (1...𝑁))
148 fnconstg 6345 . . . . . . . . . . . . . . . . . . . 20 (0 ∈ V → ((1...𝑁) × {0}) Fn (1...𝑁))
149116, 148mp1i 13 . . . . . . . . . . . . . . . . . . 19 (𝑧𝑆 → ((1...𝑁) × {0}) Fn (1...𝑁))
150 eqidd 2779 . . . . . . . . . . . . . . . . . . 19 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑧))‘𝑛) = ((1st ‘(1st𝑧))‘𝑛))
151116fvconst2 6743 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {0})‘𝑛) = 0)
152151adantl 475 . . . . . . . . . . . . . . . . . . 19 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → (((1...𝑁) × {0})‘𝑛) = 0)
15380ffvelrnda 6625 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑧))‘𝑛) ∈ (0..^𝐾))
154 elfzonn0 12837 . . . . . . . . . . . . . . . . . . . . . 22 (((1st ‘(1st𝑧))‘𝑛) ∈ (0..^𝐾) → ((1st ‘(1st𝑧))‘𝑛) ∈ ℕ0)
155153, 154syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑧))‘𝑛) ∈ ℕ0)
156155nn0cnd 11709 . . . . . . . . . . . . . . . . . . . 20 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑧))‘𝑛) ∈ ℂ)
157156addid1d 10578 . . . . . . . . . . . . . . . . . . 19 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑧))‘𝑛) + 0) = ((1st ‘(1st𝑧))‘𝑛))
158146, 147, 149, 147, 150, 152, 157offveq 7197 . . . . . . . . . . . . . . . . . 18 (𝑧𝑆 → ((1st ‘(1st𝑧)) ∘𝑓 + ((1...𝑁) × {0})) = (1st ‘(1st𝑧)))
159145, 158eqtrd 2814 . . . . . . . . . . . . . . . . 17 (𝑧𝑆 → ((1st ‘(1st𝑧)) ∘𝑓 + (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0})) = (1st ‘(1st𝑧)))
160139, 159syl5eq 2826 . . . . . . . . . . . . . . . 16 (𝑧𝑆0 / 𝑗((1st ‘(1st𝑧)) ∘𝑓 + ((((2nd ‘(1st𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))) = (1st ‘(1st𝑧)))
161160ad2antlr 717 . . . . . . . . . . . . . . 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 2814 . . . . . . . . . . . . . 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 11690 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
1641, 163syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑁 − 1) ∈ ℕ0)
165 0elfz 12760 . . . . . . . . . . . . . . . 16 ((𝑁 − 1) ∈ ℕ0 → 0 ∈ (0...(𝑁 − 1)))
166164, 165syl 17 . . . . . . . . . . . . . . 15 (𝜑 → 0 ∈ (0...(𝑁 − 1)))
167166ad2antrr 716 . . . . . . . . . . . . . 14 (((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → 0 ∈ (0...(𝑁 − 1)))
168 fvexd 6463 . . . . . . . . . . . . . 14 (((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (1st ‘(1st𝑧)) ∈ V)
169106, 162, 167, 168fvmptd 6550 . . . . . . . . . . . . 13 (((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (𝐹‘0) = (1st ‘(1st𝑧)))
170105, 169sylan 575 . . . . . . . . . . . 12 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ (2nd𝑧) = (2nd𝑇)) ∧ 𝑧𝑆) → (𝐹‘0) = (1st ‘(1st𝑧)))
171170an32s 642 . . . . . . . . . . 11 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ (2nd𝑧) = (2nd𝑇)) → (𝐹‘0) = (1st ‘(1st𝑧)))
172101, 171mpdan 677 . . . . . . . . . 10 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (𝐹‘0) = (1st ‘(1st𝑧)))
173 fveq2 6448 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑇 → (2nd𝑧) = (2nd𝑇))
174173eleq1d 2844 . . . . . . . . . . . . . . 15 (𝑧 = 𝑇 → ((2nd𝑧) ∈ (1...(𝑁 − 1)) ↔ (2nd𝑇) ∈ (1...(𝑁 − 1))))
175174anbi2d 622 . . . . . . . . . . . . . 14 (𝑧 = 𝑇 → ((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ↔ (𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1)))))
176 2fveq3 6453 . . . . . . . . . . . . . . 15 (𝑧 = 𝑇 → (1st ‘(1st𝑧)) = (1st ‘(1st𝑇)))
177176eqeq2d 2788 . . . . . . . . . . . . . 14 (𝑧 = 𝑇 → ((𝐹‘0) = (1st ‘(1st𝑧)) ↔ (𝐹‘0) = (1st ‘(1st𝑇))))
178175, 177imbi12d 336 . . . . . . . . . . . . 13 (𝑧 = 𝑇 → (((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → (𝐹‘0) = (1st ‘(1st𝑧))) ↔ ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (𝐹‘0) = (1st ‘(1st𝑇)))))
179169expcom 404 . . . . . . . . . . . . 13 (𝑧𝑆 → ((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → (𝐹‘0) = (1st ‘(1st𝑧))))
180178, 179vtoclga 3474 . . . . . . . . . . . 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 474 . . . . . . . . . 10 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (𝐹‘0) = (1st ‘(1st𝑇)))
183172, 182eqtr3d 2816 . . . . . . . . 9 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (1st ‘(1st𝑧)) = (1st ‘(1st𝑇)))
184183adantr 474 . . . . . . . 8 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ 𝑧𝑇) → (1st ‘(1st𝑧)) = (1st ‘(1st𝑇)))
1851ad3antrrr 720 . . . . . . . . 9 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ 𝑧𝑇) → 𝑁 ∈ ℕ)
1866ad3antrrr 720 . . . . . . . . 9 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ 𝑧𝑇) → 𝑇𝑆)
187 simpllr 766 . . . . . . . . 9 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ 𝑧𝑇) → (2nd𝑇) ∈ (1...(𝑁 − 1)))
188 simplr 759 . . . . . . . . 9 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ 𝑧𝑇) → 𝑧𝑆)
18934adantr 474 . . . . . . . . . . . . . 14 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (1st𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
190 xpopth 7488 . . . . . . . . . . . . . 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 590 . . . . . . . . . . . . 13 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (((1st ‘(1st𝑧)) = (1st ‘(1st𝑇)) ∧ (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑇))) ↔ (1st𝑧) = (1st𝑇)))
19232adantr 474 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → 𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
193 xpopth 7488 . . . . . . . . . . . . . . . 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 221 . . . . . . . . . . . . . . 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 590 . . . . . . . . . . . . . 14 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (((1st𝑧) = (1st𝑇) ∧ (2nd𝑧) = (2nd𝑇)) → 𝑧 = 𝑇))
196101, 195mpan2d 684 . . . . . . . . . . . . 13 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → ((1st𝑧) = (1st𝑇) → 𝑧 = 𝑇))
197191, 196sylbid 232 . . . . . . . . . . . 12 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (((1st ‘(1st𝑧)) = (1st ‘(1st𝑇)) ∧ (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑇))) → 𝑧 = 𝑇))
198183, 197mpand 685 . . . . . . . . . . 11 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → ((2nd ‘(1st𝑧)) = (2nd ‘(1st𝑇)) → 𝑧 = 𝑇))
199198necon3d 2990 . . . . . . . . . 10 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (𝑧𝑇 → (2nd ‘(1st𝑧)) ≠ (2nd ‘(1st𝑇))))
200199imp 397 . . . . . . . . 9 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ 𝑧𝑇) → (2nd ‘(1st𝑧)) ≠ (2nd ‘(1st𝑇)))
201185, 3, 186, 187, 188, 200poimirlem9 34053 . . . . . . . 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 474 . . . . . . . 8 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ 𝑧𝑇) → (2nd𝑧) = (2nd𝑇))
203184, 201, 202jca31 510 . . . . . . 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 403 . . . . . 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 759 . . . . . . . 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 12692 . . . . . . . . . . . . . 14 ((2nd𝑇) ∈ (1...(𝑁 − 1)) → (2nd𝑇) ∈ ℕ)
207206nnred 11396 . . . . . . . . . . . . 13 ((2nd𝑇) ∈ (1...(𝑁 − 1)) → (2nd𝑇) ∈ ℝ)
208207ltp1d 11311 . . . . . . . . . . . . 13 ((2nd𝑇) ∈ (1...(𝑁 − 1)) → (2nd𝑇) < ((2nd𝑇) + 1))
209207, 208ltned 10514 . . . . . . . . . . . 12 ((2nd𝑇) ∈ (1...(𝑁 − 1)) → (2nd𝑇) ≠ ((2nd𝑇) + 1))
210209adantl 475 . . . . . . . . . . 11 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (2nd𝑇) ≠ ((2nd𝑇) + 1))
211 fveq1 6447 . . . . . . . . . . . . 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 11218 . . . . . . . . . . . . . . . . . . . . . 22 ((2nd𝑇) ∈ ℝ → (2nd𝑇) < ((2nd𝑇) + 1))
214212, 213ltned 10514 . . . . . . . . . . . . . . . . . . . . 21 ((2nd𝑇) ∈ ℝ → (2nd𝑇) ≠ ((2nd𝑇) + 1))
215 fvex 6461 . . . . . . . . . . . . . . . . . . . . . 22 (2nd𝑇) ∈ V
216 ovex 6956 . . . . . . . . . . . . . . . . . . . . . 22 ((2nd𝑇) + 1) ∈ V
217215, 216, 216, 215fpr 6689 . . . . . . . . . . . . . . . . . . . . 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 6430 . . . . . . . . . . . . . . . . . . . . 21 ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})):((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})–1-1-onto→((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})
220 f1of 6393 . . . . . . . . . . . . . . . . . . . . 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 4264 . . . . . . . . . . . . . . . . . . . . 21 ({(2nd𝑇), ((2nd𝑇) + 1)} ∩ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = ∅
223 fun 6318 . . . . . . . . . . . . . . . . . . . . 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 681 . . . . . . . . . . . . . . . . . . . 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 580 . . . . . . . . . . . . . . . . . . 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 4529 . . . . . . . . . . . . . . . . . . . 20 (2nd𝑇) ∈ {(2nd𝑇), ((2nd𝑇) + 1)}
227 elun1 4003 . . . . . . . . . . . . . . . . . . . 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 6537 . . . . . . . . . . . . . . . . . . 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 580 . . . . . . . . . . . . . . . . . 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 6294 . . . . . . . . . . . . . . . . . . . . 21 ((2nd𝑇) ∈ ℝ → {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} Fn {(2nd𝑇), ((2nd𝑇) + 1)})
232 fnresi 6256 . . . . . . . . . . . . . . . . . . . . . 22 ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) Fn ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})
233222, 226pm3.2i 464 . . . . . . . . . . . . . . . . . . . . . 22 (({(2nd𝑇), ((2nd𝑇) + 1)} ∩ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = ∅ ∧ (2nd𝑇) ∈ {(2nd𝑇), ((2nd𝑇) + 1)})
234 fvun1 6531 . . . . . . . . . . . . . . . . . . . . . 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 1526 . . . . . . . . . . . . . . . . . . . . 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 6729 . . . . . . . . . . . . . . . . . . . . 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 2814 . . . . . . . . . . . . . . . . . . 19 ((2nd𝑇) ∈ ℝ → (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))‘(2nd𝑇)) = ((2nd𝑇) + 1))
240239fveq2d 6452 . . . . . . . . . . . . . . . . . 18 ((2nd𝑇) ∈ ℝ → ((2nd ‘(1st𝑇))‘(({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))‘(2nd𝑇))) = ((2nd ‘(1st𝑇))‘((2nd𝑇) + 1)))
241230, 240eqtrd 2814 . . . . . . . . . . . . . . . . 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 2788 . . . . . . . . . . . . . . 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 475 . . . . . . . . . . . . . 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 6392 . . . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (2nd ‘(1st𝑇)):(1...𝑁)–1-1→(1...𝑁))
2481nncnd 11397 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑁 ∈ ℂ)
249 npcan1 10803 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
250248, 249syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
251164nn0zd 11837 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑁 − 1) ∈ ℤ)
252 uzid 12012 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)))
253251, 252syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)))
254 peano2uz 12052 . . . . . . . . . . . . . . . . . . 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 2860 . . . . . . . . . . . . . . . . 17 (𝜑𝑁 ∈ (ℤ‘(𝑁 − 1)))
257 fzss2 12703 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ (ℤ‘(𝑁 − 1)) → (1...(𝑁 − 1)) ⊆ (1...𝑁))
258256, 257syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
259258sselda 3821 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (2nd𝑇) ∈ (1...𝑁))
260 fzp1elp1 12716 . . . . . . . . . . . . . . . . 17 ((2nd𝑇) ∈ (1...(𝑁 − 1)) → ((2nd𝑇) + 1) ∈ (1...((𝑁 − 1) + 1)))
261260adantl 475 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → ((2nd𝑇) + 1) ∈ (1...((𝑁 − 1) + 1)))
262250oveq2d 6940 . . . . . . . . . . . . . . . . 17 (𝜑 → (1...((𝑁 − 1) + 1)) = (1...𝑁))
263262adantr 474 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (1...((𝑁 − 1) + 1)) = (1...𝑁))
264261, 263eleqtrd 2861 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → ((2nd𝑇) + 1) ∈ (1...𝑁))
265 f1veqaeq 6788 . . . . . . . . . . . . . . 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 827 . . . . . . . . . . . . . 14 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (((2nd ‘(1st𝑇))‘(2nd𝑇)) = ((2nd ‘(1st𝑇))‘((2nd𝑇) + 1)) → (2nd𝑇) = ((2nd𝑇) + 1)))
267244, 266sylbid 232 . . . . . . . . . . . . 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 2990 . . . . . . . . . . 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 6453 . . . . . . . . . . 11 (𝑧 = 𝑇 → (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑇)))
272271neeq1d 3028 . . . . . . . . . 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 239 . . . . . . . . 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 2992 . . . . . . . 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 474 . . . . . 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 204 . . . . 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 7489 . . . . . . . 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 7489 . . . . . . . . . 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 623 . . . . . . . 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 271 . . . . . . 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 475 . . . . 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 274 . . . 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 3148 . . 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 3609 . . 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 579 . 2 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → ∃!𝑧𝑆 𝑧𝑇)
289 xp2nd 7480 . . . . . . 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 528 . . . . 5 (𝜑 → (¬ (2nd𝑇) ∈ (1...(𝑁 − 1)) ↔ ((2nd𝑇) ∈ (0...𝑁) ∧ ¬ (2nd𝑇) ∈ (1...(𝑁 − 1)))))
2921nnnn0d 11707 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℕ0)
293 nn0uz 12033 . . . . . . . . . . . 12 0 = (ℤ‘0)
294292, 293syl6eleq 2869 . . . . . . . . . . 11 (𝜑𝑁 ∈ (ℤ‘0))
295 fzpred 12711 . . . . . . . . . . 11 (𝑁 ∈ (ℤ‘0) → (0...𝑁) = ({0} ∪ ((0 + 1)...𝑁)))
296294, 295syl 17 . . . . . . . . . 10 (𝜑 → (0...𝑁) = ({0} ∪ ((0 + 1)...𝑁)))
297123oveq1i 6934 . . . . . . . . . . 11 ((0 + 1)...𝑁) = (1...𝑁)
298297uneq2i 3987 . . . . . . . . . 10 ({0} ∪ ((0 + 1)...𝑁)) = ({0} ∪ (1...𝑁))
299296, 298syl6eq 2830 . . . . . . . . 9 (𝜑 → (0...𝑁) = ({0} ∪ (1...𝑁)))
300299difeq1d 3950 . . . . . . . 8 (𝜑 → ((0...𝑁) ∖ (1...(𝑁 − 1))) = (({0} ∪ (1...𝑁)) ∖ (1...(𝑁 − 1))))
301 difundir 4107 . . . . . . . . . 10 (({0} ∪ (1...𝑁)) ∖ (1...(𝑁 − 1))) = (({0} ∖ (1...(𝑁 − 1))) ∪ ((1...𝑁) ∖ (1...(𝑁 − 1))))
302 0lt1 10900 . . . . . . . . . . . . . 14 0 < 1
303 0re 10380 . . . . . . . . . . . . . . 15 0 ∈ ℝ
304 1re 10378 . . . . . . . . . . . . . . 15 1 ∈ ℝ
305303, 304ltnlei 10499 . . . . . . . . . . . . . 14 (0 < 1 ↔ ¬ 1 ≤ 0)
306302, 305mpbi 222 . . . . . . . . . . . . 13 ¬ 1 ≤ 0
307 elfzle1 12666 . . . . . . . . . . . . 13 (0 ∈ (1...(𝑁 − 1)) → 1 ≤ 0)
308306, 307mto 189 . . . . . . . . . . . 12 ¬ 0 ∈ (1...(𝑁 − 1))
309 incom 4028 . . . . . . . . . . . . . 14 ((1...(𝑁 − 1)) ∩ {0}) = ({0} ∩ (1...(𝑁 − 1)))
310309eqeq1i 2783 . . . . . . . . . . . . 13 (((1...(𝑁 − 1)) ∩ {0}) = ∅ ↔ ({0} ∩ (1...(𝑁 − 1))) = ∅)
311 disjsn 4478 . . . . . . . . . . . . 13 (((1...(𝑁 − 1)) ∩ {0}) = ∅ ↔ ¬ 0 ∈ (1...(𝑁 − 1)))
312 disj3 4246 . . . . . . . . . . . . 13 (({0} ∩ (1...(𝑁 − 1))) = ∅ ↔ {0} = ({0} ∖ (1...(𝑁 − 1))))
313310, 311, 3123bitr3i 293 . . . . . . . . . . . 12 (¬ 0 ∈ (1...(𝑁 − 1)) ↔ {0} = ({0} ∖ (1...(𝑁 − 1))))
314308, 313mpbi 222 . . . . . . . . . . 11 {0} = ({0} ∖ (1...(𝑁 − 1)))
315314uneq1i 3986 . . . . . . . . . 10 ({0} ∪ ((1...𝑁) ∖ (1...(𝑁 − 1)))) = (({0} ∖ (1...(𝑁 − 1))) ∪ ((1...𝑁) ∖ (1...(𝑁 − 1))))
316301, 315eqtr4i 2805 . . . . . . . . 9 (({0} ∪ (1...𝑁)) ∖ (1...(𝑁 − 1))) = ({0} ∪ ((1...𝑁) ∖ (1...(𝑁 − 1))))
317 difundir 4107 . . . . . . . . . . . 12 (((1...(𝑁 − 1)) ∪ {𝑁}) ∖ (1...(𝑁 − 1))) = (((1...(𝑁 − 1)) ∖ (1...(𝑁 − 1))) ∪ ({𝑁} ∖ (1...(𝑁 − 1))))
318 difid 4179 . . . . . . . . . . . . 13 ((1...(𝑁 − 1)) ∖ (1...(𝑁 − 1))) = ∅
319318uneq1i 3986 . . . . . . . . . . . 12 (((1...(𝑁 − 1)) ∖ (1...(𝑁 − 1))) ∪ ({𝑁} ∖ (1...(𝑁 − 1)))) = (∅ ∪ ({𝑁} ∖ (1...(𝑁 − 1))))
320 uncom 3980 . . . . . . . . . . . . 13 (∅ ∪ ({𝑁} ∖ (1...(𝑁 − 1)))) = (({𝑁} ∖ (1...(𝑁 − 1))) ∪ ∅)
321 un0 4193 . . . . . . . . . . . . 13 (({𝑁} ∖ (1...(𝑁 − 1))) ∪ ∅) = ({𝑁} ∖ (1...(𝑁 − 1)))
322320, 321eqtri 2802 . . . . . . . . . . . 12 (∅ ∪ ({𝑁} ∖ (1...(𝑁 − 1)))) = ({𝑁} ∖ (1...(𝑁 − 1)))
323317, 319, 3223eqtri 2806 . . . . . . . . . . 11 (((1...(𝑁 − 1)) ∪ {𝑁}) ∖ (1...(𝑁 − 1))) = ({𝑁} ∖ (1...(𝑁 − 1)))
324 nnuz 12034 . . . . . . . . . . . . . . . 16 ℕ = (ℤ‘1)
3251, 324syl6eleq 2869 . . . . . . . . . . . . . . 15 (𝜑𝑁 ∈ (ℤ‘1))
326250, 325eqeltrd 2859 . . . . . . . . . . . . . 14 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘1))
327 fzsplit2 12688 . . . . . . . . . . . . . 14 ((((𝑁 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ‘(𝑁 − 1))) → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
328326, 256, 327syl2anc 579 . . . . . . . . . . . . 13 (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
329250oveq1d 6939 . . . . . . . . . . . . . . 15 (𝜑 → (((𝑁 − 1) + 1)...𝑁) = (𝑁...𝑁))
3301nnzd 11838 . . . . . . . . . . . . . . . 16 (𝜑𝑁 ∈ ℤ)
331 fzsn 12705 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
332330, 331syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (𝑁...𝑁) = {𝑁})
333329, 332eqtrd 2814 . . . . . . . . . . . . . 14 (𝜑 → (((𝑁 − 1) + 1)...𝑁) = {𝑁})
334333uneq2d 3990 . . . . . . . . . . . . 13 (𝜑 → ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = ((1...(𝑁 − 1)) ∪ {𝑁}))
335328, 334eqtrd 2814 . . . . . . . . . . . 12 (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁}))
336335difeq1d 3950 . . . . . . . . . . 11 (𝜑 → ((1...𝑁) ∖ (1...(𝑁 − 1))) = (((1...(𝑁 − 1)) ∪ {𝑁}) ∖ (1...(𝑁 − 1))))
3371nnred 11396 . . . . . . . . . . . . . . 15 (𝜑𝑁 ∈ ℝ)
338337ltm1d 11313 . . . . . . . . . . . . . 14 (𝜑 → (𝑁 − 1) < 𝑁)
339164nn0red 11708 . . . . . . . . . . . . . . 15 (𝜑 → (𝑁 − 1) ∈ ℝ)
340339, 337ltnled 10525 . . . . . . . . . . . . . 14 (𝜑 → ((𝑁 − 1) < 𝑁 ↔ ¬ 𝑁 ≤ (𝑁 − 1)))
341338, 340mpbid 224 . . . . . . . . . . . . 13 (𝜑 → ¬ 𝑁 ≤ (𝑁 − 1))
342 elfzle2 12667 . . . . . . . . . . . . 13 (𝑁 ∈ (1...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1))
343341, 342nsyl 138 . . . . . . . . . . . 12 (𝜑 → ¬ 𝑁 ∈ (1...(𝑁 − 1)))
344 incom 4028 . . . . . . . . . . . . . 14 ((1...(𝑁 − 1)) ∩ {𝑁}) = ({𝑁} ∩ (1...(𝑁 − 1)))
345344eqeq1i 2783 . . . . . . . . . . . . 13 (((1...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ({𝑁} ∩ (1...(𝑁 − 1))) = ∅)
346 disjsn 4478 . . . . . . . . . . . . 13 (((1...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (1...(𝑁 − 1)))
347 disj3 4246 . . . . . . . . . . . . 13 (({𝑁} ∩ (1...(𝑁 − 1))) = ∅ ↔ {𝑁} = ({𝑁} ∖ (1...(𝑁 − 1))))
348345, 346, 3473bitr3i 293 . . . . . . . . . . . 12 𝑁 ∈ (1...(𝑁 − 1)) ↔ {𝑁} = ({𝑁} ∖ (1...(𝑁 − 1))))
349343, 348sylib 210 . . . . . . . . . . 11 (𝜑 → {𝑁} = ({𝑁} ∖ (1...(𝑁 − 1))))
350323, 336, 3493eqtr4a 2840 . . . . . . . . . 10 (𝜑 → ((1...𝑁) ∖ (1...(𝑁 − 1))) = {𝑁})
351350uneq2d 3990 . . . . . . . . 9 (𝜑 → ({0} ∪ ((1...𝑁) ∖ (1...(𝑁 − 1)))) = ({0} ∪ {𝑁}))
352316, 351syl5eq 2826 . . . . . . . 8 (𝜑 → (({0} ∪ (1...𝑁)) ∖ (1...(𝑁 − 1))) = ({0} ∪ {𝑁}))
353300, 352eqtrd 2814 . . . . . . 7 (𝜑 → ((0...𝑁) ∖ (1...(𝑁 − 1))) = ({0} ∪ {𝑁}))
354353eleq2d 2845 . . . . . 6 (𝜑 → ((2nd𝑇) ∈ ((0...𝑁) ∖ (1...(𝑁 − 1))) ↔ (2nd𝑇) ∈ ({0} ∪ {𝑁})))
355 eldif 3802 . . . . . 6 ((2nd𝑇) ∈ ((0...𝑁) ∖ (1...(𝑁 − 1))) ↔ ((2nd𝑇) ∈ (0...𝑁) ∧ ¬ (2nd𝑇) ∈ (1...(𝑁 − 1))))
356 elun 3976 . . . . . . 7 ((2nd𝑇) ∈ ({0} ∪ {𝑁}) ↔ ((2nd𝑇) ∈ {0} ∨ (2nd𝑇) ∈ {𝑁}))
357215elsn 4413 . . . . . . . 8 ((2nd𝑇) ∈ {0} ↔ (2nd𝑇) = 0)
358215elsn 4413 . . . . . . . 8 ((2nd𝑇) ∈ {𝑁} ↔ (2nd𝑇) = 𝑁)
359357, 358orbi12i 901 . . . . . . 7 (((2nd𝑇) ∈ {0} ∨ (2nd𝑇) ∈ {𝑁}) ↔ ((2nd𝑇) = 0 ∨ (2nd𝑇) = 𝑁))
360356, 359bitri 267 . . . . . 6 ((2nd𝑇) ∈ ({0} ∪ {𝑁}) ↔ ((2nd𝑇) = 0 ∨ (2nd𝑇) = 𝑁))
361354, 355, 3603bitr3g 305 . . . . 5 (𝜑 → (((2nd𝑇) ∈ (0...𝑁) ∧ ¬ (2nd𝑇) ∈ (1...(𝑁 − 1))) ↔ ((2nd𝑇) = 0 ∨ (2nd𝑇) = 𝑁)))
362291, 361bitrd 271 . . . 4 (𝜑 → (¬ (2nd𝑇) ∈ (1...(𝑁 − 1)) ↔ ((2nd𝑇) = 0 ∨ (2nd𝑇) = 𝑁)))
363362biimpa 470 . . 3 ((𝜑 ∧ ¬ (2nd𝑇) ∈ (1...(𝑁 − 1))) → ((2nd𝑇) = 0 ∨ (2nd𝑇) = 𝑁))
3641adantr 474 . . . . 5 ((𝜑 ∧ (2nd𝑇) = 0) → 𝑁 ∈ ℕ)
3654adantr 474 . . . . 5 ((𝜑 ∧ (2nd𝑇) = 0) → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)))
3666adantr 474 . . . . 5 ((𝜑 ∧ (2nd𝑇) = 0) → 𝑇𝑆)
367 poimirlem22.4 . . . . . 6 ((𝜑𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 𝐾)
368367adantlr 705 . . . . 5 (((𝜑 ∧ (2nd𝑇) = 0) ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 𝐾)
369 simpr 479 . . . . 5 ((𝜑 ∧ (2nd𝑇) = 0) → (2nd𝑇) = 0)
370364, 3, 365, 366, 368, 369poimirlem18 34062 . . . 4 ((𝜑 ∧ (2nd𝑇) = 0) → ∃!𝑧𝑆 𝑧𝑇)
3711adantr 474 . . . . 5 ((𝜑 ∧ (2nd𝑇) = 𝑁) → 𝑁 ∈ ℕ)
3724adantr 474 . . . . 5 ((𝜑 ∧ (2nd𝑇) = 𝑁) → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)))
3736adantr 474 . . . . 5 ((𝜑 ∧ (2nd𝑇) = 𝑁) → 𝑇𝑆)
374 poimirlem22.3 . . . . . 6 ((𝜑𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 0)
375374adantlr 705 . . . . 5 (((𝜑 ∧ (2nd𝑇) = 𝑁) ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 0)
376 simpr 479 . . . . 5 ((𝜑 ∧ (2nd𝑇) = 𝑁) → (2nd𝑇) = 𝑁)
377371, 3, 372, 373, 375, 376poimirlem21 34065 . . . 4 ((𝜑 ∧ (2nd𝑇) = 𝑁) → ∃!𝑧𝑆 𝑧𝑇)
378370, 377jaodan 943 . . 3 ((𝜑 ∧ ((2nd𝑇) = 0 ∨ (2nd𝑇) = 𝑁)) → ∃!𝑧𝑆 𝑧𝑇)
379363, 378syldan 585 . 2 ((𝜑 ∧ ¬ (2nd𝑇) ∈ (1...(𝑁 − 1))) → ∃!𝑧𝑆 𝑧𝑇)
380288, 379pm2.61dan 803 1 (𝜑 → ∃!𝑧𝑆 𝑧𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  wo 836   = wceq 1601  wcel 2107  {cab 2763  wne 2969  wral 3090  wrex 3091  ∃!wreu 3092  ∃*wrmo 3093  {crab 3094  Vcvv 3398  csb 3751  cdif 3789  cun 3790  cin 3791  wss 3792  c0 4141  ifcif 4307  {csn 4398  {cpr 4400  cop 4404   class class class wbr 4888  cmpt 4967   I cid 5262   × cxp 5355  ran crn 5358  cres 5359  cima 5360  ccom 5361   Fn wfn 6132  wf 6133  1-1wf1 6134  ontowfo 6135  1-1-ontowf1o 6136  cfv 6137  (class class class)co 6924  𝑓 cof 7174  1st c1st 7445  2nd c2nd 7446  𝑚 cmap 8142  cc 10272  cr 10273  0cc0 10274  1c1 10275   + caddc 10277   < clt 10413  cle 10414  cmin 10608  cn 11379  0cn0 11647  cz 11733  cuz 11997  ...cfz 12648  ..^cfzo 12789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-cnex 10330  ax-resscn 10331  ax-1cn 10332  ax-icn 10333  ax-addcl 10334  ax-addrcl 10335  ax-mulcl 10336  ax-mulrcl 10337  ax-mulcom 10338  ax-addass 10339  ax-mulass 10340  ax-distr 10341  ax-i2m1 10342  ax-1ne0 10343  ax-1rid 10344  ax-rnegex 10345  ax-rrecex 10346  ax-cnre 10347  ax-pre-lttri 10348  ax-pre-lttrn 10349  ax-pre-ltadd 10350  ax-pre-mulgt0 10351
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-int 4713  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-riota 6885  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-of 7176  df-om 7346  df-1st 7447  df-2nd 7448  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-1o 7845  df-2o 7846  df-oadd 7849  df-er 8028  df-map 8144  df-pm 8145  df-en 8244  df-dom 8245  df-sdom 8246  df-fin 8247  df-card 9100  df-cda 9327  df-pnf 10415  df-mnf 10416  df-xr 10417  df-ltxr 10418  df-le 10419  df-sub 10610  df-neg 10611  df-div 11036  df-nn 11380  df-2 11443  df-3 11444  df-n0 11648  df-xnn0 11720  df-z 11734  df-uz 11998  df-fz 12649  df-fzo 12790  df-seq 13125  df-fac 13385  df-bc 13414  df-hash 13442
This theorem is referenced by:  poimirlem27  34071
  Copyright terms: Public domain W3C validator