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Theorem poimirlem22 33561
Description: Lemma for poimir 33572, 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 480 . . . 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 480 . . . 4 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)))
6 poimirlem22.2 . . . . 5 (𝜑𝑇𝑆)
76adantr 480 . . . 4 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → 𝑇𝑆)
8 simpr 476 . . . 4 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (2nd𝑇) ∈ (1...(𝑁 − 1)))
92, 3, 5, 7, 8poimirlem15 33554 . . 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 6229 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑇 → (2nd𝑡) = (2nd𝑇))
1110breq2d 4697 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑇 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑇)))
1211ifbid 4141 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = 𝑇 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)))
1312csbeq1d 3573 . . . . . . . . . . . . . . . . . . . . 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 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑇 → (1st𝑡) = (1st𝑇))
1514fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑇 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
1614fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = 𝑇 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑇)))
1716imaeq1d 5500 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑗)))
1817xpeq1d 5172 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}))
1916imaeq1d 5500 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)))
2019xpeq1d 5172 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
2118, 20uneq12d 3801 . . . . . . . . . . . . . . . . . . . . . . 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})))
2215, 21oveq12d 6708 . . . . . . . . . . . . . . . . . . . . . 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}))))
2322csbeq2dv 4025 . . . . . . . . . . . . . . . . . . . . 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}))))
2413, 23eqtrd 2685 . . . . . . . . . . . . . . . . . . . 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}))))
2524mpteq2dv 4778 . . . . . . . . . . . . . . . . . . 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})))))
2625eqeq2d 2661 . . . . . . . . . . . . . . . . . 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}))))))
2726, 3elrab2 3399 . . . . . . . . . . . . . . . . 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}))))))
2827simprbi 479 . . . . . . . . . . . . . . . 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})))))
296, 28syl 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})))))
3029adantr 480 . . . . . . . . . . . . . 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})))))
31 elrabi 3391 . . . . . . . . . . . . . . . . . . . . 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)))
3231, 3eleq2s 2748 . . . . . . . . . . . . . . . . . . . 20 (𝑇𝑆𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
336, 32syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
34 xp1st 7242 . . . . . . . . . . . . . . . . . . 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)}))
3533, 34syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1st𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
36 xp1st 7242 . . . . . . . . . . . . . . . . . 18 ((1st𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
3735, 36syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
38 elmapi 7921 . . . . . . . . . . . . . . . . 17 ((1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
3937, 38syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
40 elfzoelz 12509 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (0..^𝐾) → 𝑛 ∈ ℤ)
4140ssriv 3640 . . . . . . . . . . . . . . . 16 (0..^𝐾) ⊆ ℤ
42 fss 6094 . . . . . . . . . . . . . . . 16 (((1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾) ∧ (0..^𝐾) ⊆ ℤ) → (1st ‘(1st𝑇)):(1...𝑁)⟶ℤ)
4339, 41, 42sylancl 695 . . . . . . . . . . . . . . 15 (𝜑 → (1st ‘(1st𝑇)):(1...𝑁)⟶ℤ)
4443adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (1st ‘(1st𝑇)):(1...𝑁)⟶ℤ)
45 xp2nd 7243 . . . . . . . . . . . . . . . . 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)})
4635, 45syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
47 fvex 6239 . . . . . . . . . . . . . . . . 17 (2nd ‘(1st𝑇)) ∈ V
48 f1oeq1 6165 . . . . . . . . . . . . . . . . 17 (𝑓 = (2nd ‘(1st𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
4947, 48elab 3382 . . . . . . . . . . . . . . . 16 ((2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
5046, 49sylib 208 . . . . . . . . . . . . . . 15 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
5150adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
522, 30, 44, 51, 8poimirlem1 33540 . . . . . . . . . . . . 13 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑇))‘𝑛))
5352adantr 480 . . . . . . . . . . . 12 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑇))‘𝑛))
541ad3antrrr 766 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ (2nd𝑧) ≠ (2nd𝑇)) → 𝑁 ∈ ℕ)
55 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑧 → (2nd𝑡) = (2nd𝑧))
5655breq2d 4697 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑧 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑧)))
5756ifbid 4141 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = 𝑧 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑧), 𝑦, (𝑦 + 1)))
5857csbeq1d 3573 . . . . . . . . . . . . . . . . . . . . 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}))))
59 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑧 → (1st𝑡) = (1st𝑧))
6059fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑧 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑧)))
6159fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = 𝑧 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑧)))
6261imaeq1d 5500 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑧 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑧)) “ (1...𝑗)))
6362xpeq1d 5172 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑧 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑧)) “ (1...𝑗)) × {1}))
6461imaeq1d 5500 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑧 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)))
6564xpeq1d 5172 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑧 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))
6663, 65uneq12d 3801 . . . . . . . . . . . . . . . . . . . . . . 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})))
6760, 66oveq12d 6708 . . . . . . . . . . . . . . . . . . . . . 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}))))
6867csbeq2dv 4025 . . . . . . . . . . . . . . . . . . . . 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}))))
6958, 68eqtrd 2685 . . . . . . . . . . . . . . . . . . . 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}))))
7069mpteq2dv 4778 . . . . . . . . . . . . . . . . . . 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})))))
7170eqeq2d 2661 . . . . . . . . . . . . . . . . . 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}))))))
7271, 3elrab2 3399 . . . . . . . . . . . . . . . . 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}))))))
7372simprbi 479 . . . . . . . . . . . . . . . 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})))))
7473ad2antlr 763 . . . . . . . . . . . . . . 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})))))
75 elrabi 3391 . . . . . . . . . . . . . . . . . . . . 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)))
7675, 3eleq2s 2748 . . . . . . . . . . . . . . . . . . . 20 (𝑧𝑆𝑧 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
77 xp1st 7242 . . . . . . . . . . . . . . . . . . . 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)}))
7876, 77syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑧𝑆 → (1st𝑧) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
79 xp1st 7242 . . . . . . . . . . . . . . . . . . 19 ((1st𝑧) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st𝑧)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
8078, 79syl 17 . . . . . . . . . . . . . . . . . 18 (𝑧𝑆 → (1st ‘(1st𝑧)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
81 elmapi 7921 . . . . . . . . . . . . . . . . . 18 ((1st ‘(1st𝑧)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st ‘(1st𝑧)):(1...𝑁)⟶(0..^𝐾))
8280, 81syl 17 . . . . . . . . . . . . . . . . 17 (𝑧𝑆 → (1st ‘(1st𝑧)):(1...𝑁)⟶(0..^𝐾))
83 fss 6094 . . . . . . . . . . . . . . . . 17 (((1st ‘(1st𝑧)):(1...𝑁)⟶(0..^𝐾) ∧ (0..^𝐾) ⊆ ℤ) → (1st ‘(1st𝑧)):(1...𝑁)⟶ℤ)
8482, 41, 83sylancl 695 . . . . . . . . . . . . . . . 16 (𝑧𝑆 → (1st ‘(1st𝑧)):(1...𝑁)⟶ℤ)
8584ad2antlr 763 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ (2nd𝑧) ≠ (2nd𝑇)) → (1st ‘(1st𝑧)):(1...𝑁)⟶ℤ)
86 xp2nd 7243 . . . . . . . . . . . . . . . . . 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)})
8778, 86syl 17 . . . . . . . . . . . . . . . . 17 (𝑧𝑆 → (2nd ‘(1st𝑧)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
88 fvex 6239 . . . . . . . . . . . . . . . . . 18 (2nd ‘(1st𝑧)) ∈ V
89 f1oeq1 6165 . . . . . . . . . . . . . . . . . 18 (𝑓 = (2nd ‘(1st𝑧)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁)))
9088, 89elab 3382 . . . . . . . . . . . . . . . . 17 ((2nd ‘(1st𝑧)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
9187, 90sylib 208 . . . . . . . . . . . . . . . 16 (𝑧𝑆 → (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
9291ad2antlr 763 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ (2nd𝑧) ≠ (2nd𝑇)) → (2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
93 simpllr 815 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ (2nd𝑧) ≠ (2nd𝑇)) → (2nd𝑇) ∈ (1...(𝑁 − 1)))
94 xp2nd 7243 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd𝑧) ∈ (0...𝑁))
9576, 94syl 17 . . . . . . . . . . . . . . . . 17 (𝑧𝑆 → (2nd𝑧) ∈ (0...𝑁))
9695adantl 481 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (2nd𝑧) ∈ (0...𝑁))
97 eldifsn 4350 . . . . . . . . . . . . . . . . 17 ((2nd𝑧) ∈ ((0...𝑁) ∖ {(2nd𝑇)}) ↔ ((2nd𝑧) ∈ (0...𝑁) ∧ (2nd𝑧) ≠ (2nd𝑇)))
9897biimpri 218 . . . . . . . . . . . . . . . 16 (((2nd𝑧) ∈ (0...𝑁) ∧ (2nd𝑧) ≠ (2nd𝑇)) → (2nd𝑧) ∈ ((0...𝑁) ∖ {(2nd𝑇)}))
9996, 98sylan 487 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ (2nd𝑧) ≠ (2nd𝑇)) → (2nd𝑧) ∈ ((0...𝑁) ∖ {(2nd𝑇)}))
10054, 74, 85, 92, 93, 99poimirlem2 33541 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ (2nd𝑧) ≠ (2nd𝑇)) → ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑇))‘𝑛))
101100ex 449 . . . . . . . . . . . . 13 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → ((2nd𝑧) ≠ (2nd𝑇) → ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑇))‘𝑛)))
102101necon1bd 2841 . . . . . . . . . . . 12 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd𝑇) − 1))‘𝑛) ≠ ((𝐹‘(2nd𝑇))‘𝑛) → (2nd𝑧) = (2nd𝑇)))
10353, 102mpd 15 . . . . . . . . . . 11 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (2nd𝑧) = (2nd𝑇))
104 eleq1 2718 . . . . . . . . . . . . . . . 16 ((2nd𝑧) = (2nd𝑇) → ((2nd𝑧) ∈ (1...(𝑁 − 1)) ↔ (2nd𝑇) ∈ (1...(𝑁 − 1))))
105104biimparc 503 . . . . . . . . . . . . . . 15 (((2nd𝑇) ∈ (1...(𝑁 − 1)) ∧ (2nd𝑧) = (2nd𝑇)) → (2nd𝑧) ∈ (1...(𝑁 − 1)))
106105anim2i 592 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((2nd𝑇) ∈ (1...(𝑁 − 1)) ∧ (2nd𝑧) = (2nd𝑇))) → (𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))))
107106anassrs 681 . . . . . . . . . . . . 13 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ (2nd𝑧) = (2nd𝑇)) → (𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))))
10873adantl 481 . . . . . . . . . . . . . 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})))))
109 breq1 4688 . . . . . . . . . . . . . . . . . 18 (𝑦 = 0 → (𝑦 < (2nd𝑧) ↔ 0 < (2nd𝑧)))
110 id 22 . . . . . . . . . . . . . . . . . 18 (𝑦 = 0 → 𝑦 = 0)
111109, 110ifbieq1d 4142 . . . . . . . . . . . . . . . . 17 (𝑦 = 0 → if(𝑦 < (2nd𝑧), 𝑦, (𝑦 + 1)) = if(0 < (2nd𝑧), 0, (𝑦 + 1)))
112 elfznn 12408 . . . . . . . . . . . . . . . . . . . 20 ((2nd𝑧) ∈ (1...(𝑁 − 1)) → (2nd𝑧) ∈ ℕ)
113112nngt0d 11102 . . . . . . . . . . . . . . . . . . 19 ((2nd𝑧) ∈ (1...(𝑁 − 1)) → 0 < (2nd𝑧))
114113iftrued 4127 . . . . . . . . . . . . . . . . . 18 ((2nd𝑧) ∈ (1...(𝑁 − 1)) → if(0 < (2nd𝑧), 0, (𝑦 + 1)) = 0)
115114ad2antlr 763 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → if(0 < (2nd𝑧), 0, (𝑦 + 1)) = 0)
116111, 115sylan9eqr 2707 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ 𝑦 = 0) → if(𝑦 < (2nd𝑧), 𝑦, (𝑦 + 1)) = 0)
117116csbeq1d 3573 . . . . . . . . . . . . . . 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}))))
118 c0ex 10072 . . . . . . . . . . . . . . . . . 18 0 ∈ V
119 oveq2 6698 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 = 0 → (1...𝑗) = (1...0))
120 fz10 12400 . . . . . . . . . . . . . . . . . . . . . . . 24 (1...0) = ∅
121119, 120syl6eq 2701 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = 0 → (1...𝑗) = ∅)
122121imaeq2d 5501 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 0 → ((2nd ‘(1st𝑧)) “ (1...𝑗)) = ((2nd ‘(1st𝑧)) “ ∅))
123122xpeq1d 5172 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 0 → (((2nd ‘(1st𝑧)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑧)) “ ∅) × {1}))
124 oveq1 6697 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 = 0 → (𝑗 + 1) = (0 + 1))
125 0p1e1 11170 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0 + 1) = 1
126124, 125syl6eq 2701 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 = 0 → (𝑗 + 1) = 1)
127126oveq1d 6705 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = 0 → ((𝑗 + 1)...𝑁) = (1...𝑁))
128127imaeq2d 5501 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 0 → ((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑧)) “ (1...𝑁)))
129128xpeq1d 5172 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 0 → (((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0}))
130123, 129uneq12d 3801 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 0 → ((((2nd ‘(1st𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑧)) “ ∅) × {1}) ∪ (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0})))
131 ima0 5516 . . . . . . . . . . . . . . . . . . . . . . . 24 ((2nd ‘(1st𝑧)) “ ∅) = ∅
132131xpeq1i 5169 . . . . . . . . . . . . . . . . . . . . . . 23 (((2nd ‘(1st𝑧)) “ ∅) × {1}) = (∅ × {1})
133 0xp 5233 . . . . . . . . . . . . . . . . . . . . . . 23 (∅ × {1}) = ∅
134132, 133eqtri 2673 . . . . . . . . . . . . . . . . . . . . . 22 (((2nd ‘(1st𝑧)) “ ∅) × {1}) = ∅
135134uneq1i 3796 . . . . . . . . . . . . . . . . . . . . 21 ((((2nd ‘(1st𝑧)) “ ∅) × {1}) ∪ (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0})) = (∅ ∪ (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0}))
136 uncom 3790 . . . . . . . . . . . . . . . . . . . . 21 (∅ ∪ (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0})) = ((((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0}) ∪ ∅)
137 un0 4000 . . . . . . . . . . . . . . . . . . . . 21 ((((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0}) ∪ ∅) = (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0})
138135, 136, 1373eqtri 2677 . . . . . . . . . . . . . . . . . . . 20 ((((2nd ‘(1st𝑧)) “ ∅) × {1}) ∪ (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0})) = (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0})
139130, 138syl6eq 2701 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 0 → ((((2nd ‘(1st𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) × {0})) = (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0}))
140139oveq2d 6706 . . . . . . . . . . . . . . . . . 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})))
141118, 140csbie 3592 . . . . . . . . . . . . . . . . 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}))
142 f1ofo 6182 . . . . . . . . . . . . . . . . . . . . . 22 ((2nd ‘(1st𝑧)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑧)):(1...𝑁)–onto→(1...𝑁))
14391, 142syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑧𝑆 → (2nd ‘(1st𝑧)):(1...𝑁)–onto→(1...𝑁))
144 foima 6158 . . . . . . . . . . . . . . . . . . . . 21 ((2nd ‘(1st𝑧)):(1...𝑁)–onto→(1...𝑁) → ((2nd ‘(1st𝑧)) “ (1...𝑁)) = (1...𝑁))
145143, 144syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑧𝑆 → ((2nd ‘(1st𝑧)) “ (1...𝑁)) = (1...𝑁))
146145xpeq1d 5172 . . . . . . . . . . . . . . . . . . 19 (𝑧𝑆 → (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0}) = ((1...𝑁) × {0}))
147146oveq2d 6706 . . . . . . . . . . . . . . . . . 18 (𝑧𝑆 → ((1st ‘(1st𝑧)) ∘𝑓 + (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0})) = ((1st ‘(1st𝑧)) ∘𝑓 + ((1...𝑁) × {0})))
148 ovexd 6720 . . . . . . . . . . . . . . . . . . 19 (𝑧𝑆 → (1...𝑁) ∈ V)
149 ffn 6083 . . . . . . . . . . . . . . . . . . . 20 ((1st ‘(1st𝑧)):(1...𝑁)⟶(0..^𝐾) → (1st ‘(1st𝑧)) Fn (1...𝑁))
15082, 149syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑧𝑆 → (1st ‘(1st𝑧)) Fn (1...𝑁))
151 fnconstg 6131 . . . . . . . . . . . . . . . . . . . 20 (0 ∈ V → ((1...𝑁) × {0}) Fn (1...𝑁))
152118, 151mp1i 13 . . . . . . . . . . . . . . . . . . 19 (𝑧𝑆 → ((1...𝑁) × {0}) Fn (1...𝑁))
153 eqidd 2652 . . . . . . . . . . . . . . . . . . 19 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑧))‘𝑛) = ((1st ‘(1st𝑧))‘𝑛))
154118fvconst2 6510 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {0})‘𝑛) = 0)
155154adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → (((1...𝑁) × {0})‘𝑛) = 0)
15682ffvelrnda 6399 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑧))‘𝑛) ∈ (0..^𝐾))
157 elfzonn0 12552 . . . . . . . . . . . . . . . . . . . . . 22 (((1st ‘(1st𝑧))‘𝑛) ∈ (0..^𝐾) → ((1st ‘(1st𝑧))‘𝑛) ∈ ℕ0)
158156, 157syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑧))‘𝑛) ∈ ℕ0)
159158nn0cnd 11391 . . . . . . . . . . . . . . . . . . . 20 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑧))‘𝑛) ∈ ℂ)
160159addid1d 10274 . . . . . . . . . . . . . . . . . . 19 ((𝑧𝑆𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑧))‘𝑛) + 0) = ((1st ‘(1st𝑧))‘𝑛))
161148, 150, 152, 150, 153, 155, 160offveq 6960 . . . . . . . . . . . . . . . . . 18 (𝑧𝑆 → ((1st ‘(1st𝑧)) ∘𝑓 + ((1...𝑁) × {0})) = (1st ‘(1st𝑧)))
162147, 161eqtrd 2685 . . . . . . . . . . . . . . . . 17 (𝑧𝑆 → ((1st ‘(1st𝑧)) ∘𝑓 + (((2nd ‘(1st𝑧)) “ (1...𝑁)) × {0})) = (1st ‘(1st𝑧)))
163141, 162syl5eq 2697 . . . . . . . . . . . . . . . 16 (𝑧𝑆0 / 𝑗((1st ‘(1st𝑧)) ∘𝑓 + ((((2nd ‘(1st𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))) = (1st ‘(1st𝑧)))
164163ad2antlr 763 . . . . . . . . . . . . . . 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)))
165117, 164eqtrd 2685 . . . . . . . . . . . . . 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)))
166 nnm1nn0 11372 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
1671, 166syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑁 − 1) ∈ ℕ0)
168 0elfz 12475 . . . . . . . . . . . . . . . 16 ((𝑁 − 1) ∈ ℕ0 → 0 ∈ (0...(𝑁 − 1)))
169167, 168syl 17 . . . . . . . . . . . . . . 15 (𝜑 → 0 ∈ (0...(𝑁 − 1)))
170169ad2antrr 762 . . . . . . . . . . . . . 14 (((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → 0 ∈ (0...(𝑁 − 1)))
171 fvexd 6241 . . . . . . . . . . . . . 14 (((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (1st ‘(1st𝑧)) ∈ V)
172108, 165, 170, 171fvmptd 6327 . . . . . . . . . . . . 13 (((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (𝐹‘0) = (1st ‘(1st𝑧)))
173107, 172sylan 487 . . . . . . . . . . . 12 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ (2nd𝑧) = (2nd𝑇)) ∧ 𝑧𝑆) → (𝐹‘0) = (1st ‘(1st𝑧)))
174173an32s 863 . . . . . . . . . . 11 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ (2nd𝑧) = (2nd𝑇)) → (𝐹‘0) = (1st ‘(1st𝑧)))
175103, 174mpdan 703 . . . . . . . . . 10 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (𝐹‘0) = (1st ‘(1st𝑧)))
176 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑇 → (2nd𝑧) = (2nd𝑇))
177176eleq1d 2715 . . . . . . . . . . . . . . 15 (𝑧 = 𝑇 → ((2nd𝑧) ∈ (1...(𝑁 − 1)) ↔ (2nd𝑇) ∈ (1...(𝑁 − 1))))
178177anbi2d 740 . . . . . . . . . . . . . 14 (𝑧 = 𝑇 → ((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) ↔ (𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1)))))
179 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑇 → (1st𝑧) = (1st𝑇))
180179fveq2d 6233 . . . . . . . . . . . . . . 15 (𝑧 = 𝑇 → (1st ‘(1st𝑧)) = (1st ‘(1st𝑇)))
181180eqeq2d 2661 . . . . . . . . . . . . . 14 (𝑧 = 𝑇 → ((𝐹‘0) = (1st ‘(1st𝑧)) ↔ (𝐹‘0) = (1st ‘(1st𝑇))))
182178, 181imbi12d 333 . . . . . . . . . . . . 13 (𝑧 = 𝑇 → (((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → (𝐹‘0) = (1st ‘(1st𝑧))) ↔ ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (𝐹‘0) = (1st ‘(1st𝑇)))))
183172expcom 450 . . . . . . . . . . . . 13 (𝑧𝑆 → ((𝜑 ∧ (2nd𝑧) ∈ (1...(𝑁 − 1))) → (𝐹‘0) = (1st ‘(1st𝑧))))
184182, 183vtoclga 3303 . . . . . . . . . . . 12 (𝑇𝑆 → ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (𝐹‘0) = (1st ‘(1st𝑇))))
1857, 184mpcom 38 . . . . . . . . . . 11 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (𝐹‘0) = (1st ‘(1st𝑇)))
186185adantr 480 . . . . . . . . . 10 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (𝐹‘0) = (1st ‘(1st𝑇)))
187175, 186eqtr3d 2687 . . . . . . . . 9 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (1st ‘(1st𝑧)) = (1st ‘(1st𝑇)))
188187adantr 480 . . . . . . . 8 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ 𝑧𝑇) → (1st ‘(1st𝑧)) = (1st ‘(1st𝑇)))
1891ad3antrrr 766 . . . . . . . . 9 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ 𝑧𝑇) → 𝑁 ∈ ℕ)
1906ad3antrrr 766 . . . . . . . . 9 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ 𝑧𝑇) → 𝑇𝑆)
191 simpllr 815 . . . . . . . . 9 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ 𝑧𝑇) → (2nd𝑇) ∈ (1...(𝑁 − 1)))
192 simplr 807 . . . . . . . . 9 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ 𝑧𝑇) → 𝑧𝑆)
19335adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (1st𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
194 xpopth 7251 . . . . . . . . . . . . . 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)))
19578, 193, 194syl2anr 494 . . . . . . . . . . . . 13 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (((1st ‘(1st𝑧)) = (1st ‘(1st𝑇)) ∧ (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑇))) ↔ (1st𝑧) = (1st𝑇)))
19633adantr 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → 𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
197 xpopth 7251 . . . . . . . . . . . . . . . 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))
198197biimpd 219 . . . . . . . . . . . . . . 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))
19976, 196, 198syl2anr 494 . . . . . . . . . . . . . 14 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (((1st𝑧) = (1st𝑇) ∧ (2nd𝑧) = (2nd𝑇)) → 𝑧 = 𝑇))
200103, 199mpan2d 710 . . . . . . . . . . . . 13 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → ((1st𝑧) = (1st𝑇) → 𝑧 = 𝑇))
201195, 200sylbid 230 . . . . . . . . . . . 12 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (((1st ‘(1st𝑧)) = (1st ‘(1st𝑇)) ∧ (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑇))) → 𝑧 = 𝑇))
202187, 201mpand 711 . . . . . . . . . . 11 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → ((2nd ‘(1st𝑧)) = (2nd ‘(1st𝑇)) → 𝑧 = 𝑇))
203202necon3d 2844 . . . . . . . . . 10 (((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) → (𝑧𝑇 → (2nd ‘(1st𝑧)) ≠ (2nd ‘(1st𝑇))))
204203imp 444 . . . . . . . . 9 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ 𝑧𝑇) → (2nd ‘(1st𝑧)) ≠ (2nd ‘(1st𝑇)))
205189, 3, 190, 191, 192, 204poimirlem9 33548 . . . . . . . 8 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ 𝑧𝑇) → (2nd ‘(1st𝑧)) = ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))))
206103adantr 480 . . . . . . . 8 ((((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) ∧ 𝑧𝑆) ∧ 𝑧𝑇) → (2nd𝑧) = (2nd𝑇))
207188, 205, 206jca31 556 . . . . . . 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)))
208207ex 449 . . . . . 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))))
209 simplr 807 . . . . . . . 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)})))))
210 elfznn 12408 . . . . . . . . . . . . . 14 ((2nd𝑇) ∈ (1...(𝑁 − 1)) → (2nd𝑇) ∈ ℕ)
211210nnred 11073 . . . . . . . . . . . . 13 ((2nd𝑇) ∈ (1...(𝑁 − 1)) → (2nd𝑇) ∈ ℝ)
212211ltp1d 10992 . . . . . . . . . . . . 13 ((2nd𝑇) ∈ (1...(𝑁 − 1)) → (2nd𝑇) < ((2nd𝑇) + 1))
213211, 212ltned 10211 . . . . . . . . . . . 12 ((2nd𝑇) ∈ (1...(𝑁 − 1)) → (2nd𝑇) ≠ ((2nd𝑇) + 1))
214213adantl 481 . . . . . . . . . . 11 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (2nd𝑇) ≠ ((2nd𝑇) + 1))
215 fveq1 6228 . . . . . . . . . . . . 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)))
216 id 22 . . . . . . . . . . . . . . . . . . . . . 22 ((2nd𝑇) ∈ ℝ → (2nd𝑇) ∈ ℝ)
217 ltp1 10899 . . . . . . . . . . . . . . . . . . . . . 22 ((2nd𝑇) ∈ ℝ → (2nd𝑇) < ((2nd𝑇) + 1))
218216, 217ltned 10211 . . . . . . . . . . . . . . . . . . . . 21 ((2nd𝑇) ∈ ℝ → (2nd𝑇) ≠ ((2nd𝑇) + 1))
219 fvex 6239 . . . . . . . . . . . . . . . . . . . . . 22 (2nd𝑇) ∈ V
220 ovex 6718 . . . . . . . . . . . . . . . . . . . . . 22 ((2nd𝑇) + 1) ∈ V
221219, 220, 220, 219fpr 6461 . . . . . . . . . . . . . . . . . . . . 21 ((2nd𝑇) ≠ ((2nd𝑇) + 1) → {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩}:{(2nd𝑇), ((2nd𝑇) + 1)}⟶{((2nd𝑇) + 1), (2nd𝑇)})
222218, 221syl 17 . . . . . . . . . . . . . . . . . . . 20 ((2nd𝑇) ∈ ℝ → {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩}:{(2nd𝑇), ((2nd𝑇) + 1)}⟶{((2nd𝑇) + 1), (2nd𝑇)})
223 f1oi 6212 . . . . . . . . . . . . . . . . . . . . 21 ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})):((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})–1-1-onto→((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})
224 f1of 6175 . . . . . . . . . . . . . . . . . . . . 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)}))
225223, 224ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})):((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})⟶((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})
226 disjdif 4073 . . . . . . . . . . . . . . . . . . . . 21 ({(2nd𝑇), ((2nd𝑇) + 1)} ∩ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = ∅
227 fun 6104 . . . . . . . . . . . . . . . . . . . . 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)})))
228226, 227mpan2 707 . . . . . . . . . . . . . . . . . . . 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)})))
229222, 225, 228sylancl 695 . . . . . . . . . . . . . . . . . . 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)})))
230219prid1 4329 . . . . . . . . . . . . . . . . . . . 20 (2nd𝑇) ∈ {(2nd𝑇), ((2nd𝑇) + 1)}
231 elun1 3813 . . . . . . . . . . . . . . . . . . . 20 ((2nd𝑇) ∈ {(2nd𝑇), ((2nd𝑇) + 1)} → (2nd𝑇) ∈ ({(2nd𝑇), ((2nd𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))
232230, 231ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (2nd𝑇) ∈ ({(2nd𝑇), ((2nd𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
233 fvco3 6314 . . . . . . . . . . . . . . . . . . 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))))
234229, 232, 233sylancl 695 . . . . . . . . . . . . . . . . . 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))))
235 ffn 6083 . . . . . . . . . . . . . . . . . . . . . 22 ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩}:{(2nd𝑇), ((2nd𝑇) + 1)}⟶{((2nd𝑇) + 1), (2nd𝑇)} → {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} Fn {(2nd𝑇), ((2nd𝑇) + 1)})
236222, 235syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((2nd𝑇) ∈ ℝ → {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} Fn {(2nd𝑇), ((2nd𝑇) + 1)})
237 fnresi 6046 . . . . . . . . . . . . . . . . . . . . . 22 ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) Fn ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})
238226, 230pm3.2i 470 . . . . . . . . . . . . . . . . . . . . . 22 (({(2nd𝑇), ((2nd𝑇) + 1)} ∩ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = ∅ ∧ (2nd𝑇) ∈ {(2nd𝑇), ((2nd𝑇) + 1)})
239 fvun1 6308 . . . . . . . . . . . . . . . . . . . . . 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)))
240237, 238, 239mp3an23 1456 . . . . . . . . . . . . . . . . . . . . 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)))
241236, 240syl 17 . . . . . . . . . . . . . . . . . . . 20 ((2nd𝑇) ∈ ℝ → (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))‘(2nd𝑇)) = ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩}‘(2nd𝑇)))
242219, 220fvpr1 6497 . . . . . . . . . . . . . . . . . . . . 21 ((2nd𝑇) ≠ ((2nd𝑇) + 1) → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩}‘(2nd𝑇)) = ((2nd𝑇) + 1))
243218, 242syl 17 . . . . . . . . . . . . . . . . . . . 20 ((2nd𝑇) ∈ ℝ → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩}‘(2nd𝑇)) = ((2nd𝑇) + 1))
244241, 243eqtrd 2685 . . . . . . . . . . . . . . . . . . 19 ((2nd𝑇) ∈ ℝ → (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))‘(2nd𝑇)) = ((2nd𝑇) + 1))
245244fveq2d 6233 . . . . . . . . . . . . . . . . . 18 ((2nd𝑇) ∈ ℝ → ((2nd ‘(1st𝑇))‘(({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))‘(2nd𝑇))) = ((2nd ‘(1st𝑇))‘((2nd𝑇) + 1)))
246234, 245eqtrd 2685 . . . . . . . . . . . . . . . . 17 ((2nd𝑇) ∈ ℝ → (((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))‘(2nd𝑇)) = ((2nd ‘(1st𝑇))‘((2nd𝑇) + 1)))
247211, 246syl 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)))
248247eqeq2d 2661 . . . . . . . . . . . . . . 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))))
249248adantl 481 . . . . . . . . . . . . . 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))))
250 f1of1 6174 . . . . . . . . . . . . . . . . 17 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)):(1...𝑁)–1-1→(1...𝑁))
25150, 250syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–1-1→(1...𝑁))
252251adantr 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (2nd ‘(1st𝑇)):(1...𝑁)–1-1→(1...𝑁))
2531nncnd 11074 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑁 ∈ ℂ)
254 npcan1 10493 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
255253, 254syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
256167nn0zd 11518 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑁 − 1) ∈ ℤ)
257 uzid 11740 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)))
258256, 257syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)))
259 peano2uz 11779 . . . . . . . . . . . . . . . . . . 19 ((𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
260258, 259syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
261255, 260eqeltrrd 2731 . . . . . . . . . . . . . . . . 17 (𝜑𝑁 ∈ (ℤ‘(𝑁 − 1)))
262 fzss2 12419 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ (ℤ‘(𝑁 − 1)) → (1...(𝑁 − 1)) ⊆ (1...𝑁))
263261, 262syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
264263sselda 3636 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (2nd𝑇) ∈ (1...𝑁))
265 fzp1elp1 12432 . . . . . . . . . . . . . . . . 17 ((2nd𝑇) ∈ (1...(𝑁 − 1)) → ((2nd𝑇) + 1) ∈ (1...((𝑁 − 1) + 1)))
266265adantl 481 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → ((2nd𝑇) + 1) ∈ (1...((𝑁 − 1) + 1)))
267255oveq2d 6706 . . . . . . . . . . . . . . . . 17 (𝜑 → (1...((𝑁 − 1) + 1)) = (1...𝑁))
268267adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (1...((𝑁 − 1) + 1)) = (1...𝑁))
269266, 268eleqtrd 2732 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → ((2nd𝑇) + 1) ∈ (1...𝑁))
270 f1veqaeq 6554 . . . . . . . . . . . . . . 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)))
271252, 264, 269, 270syl12anc 1364 . . . . . . . . . . . . . 14 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (((2nd ‘(1st𝑇))‘(2nd𝑇)) = ((2nd ‘(1st𝑇))‘((2nd𝑇) + 1)) → (2nd𝑇) = ((2nd𝑇) + 1)))
272249, 271sylbid 230 . . . . . . . . . . . . 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)))
273215, 272syl5 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)))
274273necon3d 2844 . . . . . . . . . . 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)}))))))
275214, 274mpd 15 . . . . . . . . . 10 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (2nd ‘(1st𝑇)) ≠ ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))))
276179fveq2d 6233 . . . . . . . . . . 11 (𝑧 = 𝑇 → (2nd ‘(1st𝑧)) = (2nd ‘(1st𝑇)))
277276neeq1d 2882 . . . . . . . . . 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)}))))))
278275, 277syl5ibrcom 237 . . . . . . . . 9 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → (𝑧 = 𝑇 → (2nd ‘(1st𝑧)) ≠ ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))))
279278necon2d 2846 . . . . . . . 8 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → ((2nd ‘(1st𝑧)) = ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) → 𝑧𝑇))
280209, 279syl5 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))
281280adantr 480 . . . . . 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))
282208, 281impbid 202 . . . . 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))))
283 eqop 7252 . . . . . . . 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))))
284 eqop 7252 . . . . . . . . . 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)})))))))
28577, 284syl 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)})))))))
286285anbi1d 741 . . . . . . . 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))))
287283, 286bitrd 268 . . . . . . 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))))
28876, 287syl 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))))
289288adantl 481 . . . . 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))))
290282, 289bitr4d 271 . . . 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)⟩))
291290ralrimiva 2995 . . 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)⟩))
292 reu6i 3430 . . 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)
2939, 291, 292syl2anc 694 . 2 ((𝜑 ∧ (2nd𝑇) ∈ (1...(𝑁 − 1))) → ∃!𝑧𝑆 𝑧𝑇)
294 xp2nd 7243 . . . . . . 7 (𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd𝑇) ∈ (0...𝑁))
29533, 294syl 17 . . . . . 6 (𝜑 → (2nd𝑇) ∈ (0...𝑁))
296295biantrurd 528 . . . . 5 (𝜑 → (¬ (2nd𝑇) ∈ (1...(𝑁 − 1)) ↔ ((2nd𝑇) ∈ (0...𝑁) ∧ ¬ (2nd𝑇) ∈ (1...(𝑁 − 1)))))
2971nnnn0d 11389 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℕ0)
298 nn0uz 11760 . . . . . . . . . . . 12 0 = (ℤ‘0)
299297, 298syl6eleq 2740 . . . . . . . . . . 11 (𝜑𝑁 ∈ (ℤ‘0))
300 fzpred 12427 . . . . . . . . . . 11 (𝑁 ∈ (ℤ‘0) → (0...𝑁) = ({0} ∪ ((0 + 1)...𝑁)))
301299, 300syl 17 . . . . . . . . . 10 (𝜑 → (0...𝑁) = ({0} ∪ ((0 + 1)...𝑁)))
302125oveq1i 6700 . . . . . . . . . . 11 ((0 + 1)...𝑁) = (1...𝑁)
303302uneq2i 3797 . . . . . . . . . 10 ({0} ∪ ((0 + 1)...𝑁)) = ({0} ∪ (1...𝑁))
304301, 303syl6eq 2701 . . . . . . . . 9 (𝜑 → (0...𝑁) = ({0} ∪ (1...𝑁)))
305304difeq1d 3760 . . . . . . . 8 (𝜑 → ((0...𝑁) ∖ (1...(𝑁 − 1))) = (({0} ∪ (1...𝑁)) ∖ (1...(𝑁 − 1))))
306 difundir 3913 . . . . . . . . . 10 (({0} ∪ (1...𝑁)) ∖ (1...(𝑁 − 1))) = (({0} ∖ (1...(𝑁 − 1))) ∪ ((1...𝑁) ∖ (1...(𝑁 − 1))))
307 0lt1 10588 . . . . . . . . . . . . . 14 0 < 1
308 0re 10078 . . . . . . . . . . . . . . 15 0 ∈ ℝ
309 1re 10077 . . . . . . . . . . . . . . 15 1 ∈ ℝ
310308, 309ltnlei 10196 . . . . . . . . . . . . . 14 (0 < 1 ↔ ¬ 1 ≤ 0)
311307, 310mpbi 220 . . . . . . . . . . . . 13 ¬ 1 ≤ 0
312 elfzle1 12382 . . . . . . . . . . . . 13 (0 ∈ (1...(𝑁 − 1)) → 1 ≤ 0)
313311, 312mto 188 . . . . . . . . . . . 12 ¬ 0 ∈ (1...(𝑁 − 1))
314 incom 3838 . . . . . . . . . . . . . 14 ((1...(𝑁 − 1)) ∩ {0}) = ({0} ∩ (1...(𝑁 − 1)))
315314eqeq1i 2656 . . . . . . . . . . . . 13 (((1...(𝑁 − 1)) ∩ {0}) = ∅ ↔ ({0} ∩ (1...(𝑁 − 1))) = ∅)
316 disjsn 4278 . . . . . . . . . . . . 13 (((1...(𝑁 − 1)) ∩ {0}) = ∅ ↔ ¬ 0 ∈ (1...(𝑁 − 1)))
317 disj3 4054 . . . . . . . . . . . . 13 (({0} ∩ (1...(𝑁 − 1))) = ∅ ↔ {0} = ({0} ∖ (1...(𝑁 − 1))))
318315, 316, 3173bitr3i 290 . . . . . . . . . . . 12 (¬ 0 ∈ (1...(𝑁 − 1)) ↔ {0} = ({0} ∖ (1...(𝑁 − 1))))
319313, 318mpbi 220 . . . . . . . . . . 11 {0} = ({0} ∖ (1...(𝑁 − 1)))
320319uneq1i 3796 . . . . . . . . . 10 ({0} ∪ ((1...𝑁) ∖ (1...(𝑁 − 1)))) = (({0} ∖ (1...(𝑁 − 1))) ∪ ((1...𝑁) ∖ (1...(𝑁 − 1))))
321306, 320eqtr4i 2676 . . . . . . . . 9 (({0} ∪ (1...𝑁)) ∖ (1...(𝑁 − 1))) = ({0} ∪ ((1...𝑁) ∖ (1...(𝑁 − 1))))
322 difundir 3913 . . . . . . . . . . . 12 (((1...(𝑁 − 1)) ∪ {𝑁}) ∖ (1...(𝑁 − 1))) = (((1...(𝑁 − 1)) ∖ (1...(𝑁 − 1))) ∪ ({𝑁} ∖ (1...(𝑁 − 1))))
323 difid 3981 . . . . . . . . . . . . 13 ((1...(𝑁 − 1)) ∖ (1...(𝑁 − 1))) = ∅
324323uneq1i 3796 . . . . . . . . . . . 12 (((1...(𝑁 − 1)) ∖ (1...(𝑁 − 1))) ∪ ({𝑁} ∖ (1...(𝑁 − 1)))) = (∅ ∪ ({𝑁} ∖ (1...(𝑁 − 1))))
325 uncom 3790 . . . . . . . . . . . . 13 (∅ ∪ ({𝑁} ∖ (1...(𝑁 − 1)))) = (({𝑁} ∖ (1...(𝑁 − 1))) ∪ ∅)
326 un0 4000 . . . . . . . . . . . . 13 (({𝑁} ∖ (1...(𝑁 − 1))) ∪ ∅) = ({𝑁} ∖ (1...(𝑁 − 1)))
327325, 326eqtri 2673 . . . . . . . . . . . 12 (∅ ∪ ({𝑁} ∖ (1...(𝑁 − 1)))) = ({𝑁} ∖ (1...(𝑁 − 1)))
328322, 324, 3273eqtri 2677 . . . . . . . . . . 11 (((1...(𝑁 − 1)) ∪ {𝑁}) ∖ (1...(𝑁 − 1))) = ({𝑁} ∖ (1...(𝑁 − 1)))
329 nnuz 11761 . . . . . . . . . . . . . . . 16 ℕ = (ℤ‘1)
3301, 329syl6eleq 2740 . . . . . . . . . . . . . . 15 (𝜑𝑁 ∈ (ℤ‘1))
331255, 330eqeltrd 2730 . . . . . . . . . . . . . 14 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘1))
332 fzsplit2 12404 . . . . . . . . . . . . . 14 ((((𝑁 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ‘(𝑁 − 1))) → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
333331, 261, 332syl2anc 694 . . . . . . . . . . . . 13 (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
334255oveq1d 6705 . . . . . . . . . . . . . . 15 (𝜑 → (((𝑁 − 1) + 1)...𝑁) = (𝑁...𝑁))
3351nnzd 11519 . . . . . . . . . . . . . . . 16 (𝜑𝑁 ∈ ℤ)
336 fzsn 12421 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
337335, 336syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (𝑁...𝑁) = {𝑁})
338334, 337eqtrd 2685 . . . . . . . . . . . . . 14 (𝜑 → (((𝑁 − 1) + 1)...𝑁) = {𝑁})
339338uneq2d 3800 . . . . . . . . . . . . 13 (𝜑 → ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = ((1...(𝑁 − 1)) ∪ {𝑁}))
340333, 339eqtrd 2685 . . . . . . . . . . . 12 (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁}))
341340difeq1d 3760 . . . . . . . . . . 11 (𝜑 → ((1...𝑁) ∖ (1...(𝑁 − 1))) = (((1...(𝑁 − 1)) ∪ {𝑁}) ∖ (1...(𝑁 − 1))))
3421nnred 11073 . . . . . . . . . . . . . . 15 (𝜑𝑁 ∈ ℝ)
343342ltm1d 10994 . . . . . . . . . . . . . 14 (𝜑 → (𝑁 − 1) < 𝑁)
344167nn0red 11390 . . . . . . . . . . . . . . 15 (𝜑 → (𝑁 − 1) ∈ ℝ)
345344, 342ltnled 10222 . . . . . . . . . . . . . 14 (𝜑 → ((𝑁 − 1) < 𝑁 ↔ ¬ 𝑁 ≤ (𝑁 − 1)))
346343, 345mpbid 222 . . . . . . . . . . . . 13 (𝜑 → ¬ 𝑁 ≤ (𝑁 − 1))
347 elfzle2 12383 . . . . . . . . . . . . 13 (𝑁 ∈ (1...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1))
348346, 347nsyl 135 . . . . . . . . . . . 12 (𝜑 → ¬ 𝑁 ∈ (1...(𝑁 − 1)))
349 incom 3838 . . . . . . . . . . . . . 14 ((1...(𝑁 − 1)) ∩ {𝑁}) = ({𝑁} ∩ (1...(𝑁 − 1)))
350349eqeq1i 2656 . . . . . . . . . . . . 13 (((1...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ({𝑁} ∩ (1...(𝑁 − 1))) = ∅)
351 disjsn 4278 . . . . . . . . . . . . 13 (((1...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (1...(𝑁 − 1)))
352 disj3 4054 . . . . . . . . . . . . 13 (({𝑁} ∩ (1...(𝑁 − 1))) = ∅ ↔ {𝑁} = ({𝑁} ∖ (1...(𝑁 − 1))))
353350, 351, 3523bitr3i 290 . . . . . . . . . . . 12 𝑁 ∈ (1...(𝑁 − 1)) ↔ {𝑁} = ({𝑁} ∖ (1...(𝑁 − 1))))
354348, 353sylib 208 . . . . . . . . . . 11 (𝜑 → {𝑁} = ({𝑁} ∖ (1...(𝑁 − 1))))
355328, 341, 3543eqtr4a 2711 . . . . . . . . . 10 (𝜑 → ((1...𝑁) ∖ (1...(𝑁 − 1))) = {𝑁})
356355uneq2d 3800 . . . . . . . . 9 (𝜑 → ({0} ∪ ((1...𝑁) ∖ (1...(𝑁 − 1)))) = ({0} ∪ {𝑁}))
357321, 356syl5eq 2697 . . . . . . . 8 (𝜑 → (({0} ∪ (1...𝑁)) ∖ (1...(𝑁 − 1))) = ({0} ∪ {𝑁}))
358305, 357eqtrd 2685 . . . . . . 7 (𝜑 → ((0...𝑁) ∖ (1...(𝑁 − 1))) = ({0} ∪ {𝑁}))
359358eleq2d 2716 . . . . . 6 (𝜑 → ((2nd𝑇) ∈ ((0...𝑁) ∖ (1...(𝑁 − 1))) ↔ (2nd𝑇) ∈ ({0} ∪ {𝑁})))
360 eldif 3617 . . . . . 6 ((2nd𝑇) ∈ ((0...𝑁) ∖ (1...(𝑁 − 1))) ↔ ((2nd𝑇) ∈ (0...𝑁) ∧ ¬ (2nd𝑇) ∈ (1...(𝑁 − 1))))
361 elun 3786 . . . . . . 7 ((2nd𝑇) ∈ ({0} ∪ {𝑁}) ↔ ((2nd𝑇) ∈ {0} ∨ (2nd𝑇) ∈ {𝑁}))
362219elsn 4225 . . . . . . . 8 ((2nd𝑇) ∈ {0} ↔ (2nd𝑇) = 0)
363219elsn 4225 . . . . . . . 8 ((2nd𝑇) ∈ {𝑁} ↔ (2nd𝑇) = 𝑁)
364362, 363orbi12i 542 . . . . . . 7 (((2nd𝑇) ∈ {0} ∨ (2nd𝑇) ∈ {𝑁}) ↔ ((2nd𝑇) = 0 ∨ (2nd𝑇) = 𝑁))
365361, 364bitri 264 . . . . . 6 ((2nd𝑇) ∈ ({0} ∪ {𝑁}) ↔ ((2nd𝑇) = 0 ∨ (2nd𝑇) = 𝑁))
366359, 360, 3653bitr3g 302 . . . . 5 (𝜑 → (((2nd𝑇) ∈ (0...𝑁) ∧ ¬ (2nd𝑇) ∈ (1...(𝑁 − 1))) ↔ ((2nd𝑇) = 0 ∨ (2nd𝑇) = 𝑁)))
367296, 366bitrd 268 . . . 4 (𝜑 → (¬ (2nd𝑇) ∈ (1...(𝑁 − 1)) ↔ ((2nd𝑇) = 0 ∨ (2nd𝑇) = 𝑁)))
368367biimpa 500 . . 3 ((𝜑 ∧ ¬ (2nd𝑇) ∈ (1...(𝑁 − 1))) → ((2nd𝑇) = 0 ∨ (2nd𝑇) = 𝑁))
3691adantr 480 . . . . 5 ((𝜑 ∧ (2nd𝑇) = 0) → 𝑁 ∈ ℕ)
3704adantr 480 . . . . 5 ((𝜑 ∧ (2nd𝑇) = 0) → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)))
3716adantr 480 . . . . 5 ((𝜑 ∧ (2nd𝑇) = 0) → 𝑇𝑆)
372 poimirlem22.4 . . . . . 6 ((𝜑𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 𝐾)
373372adantlr 751 . . . . 5 (((𝜑 ∧ (2nd𝑇) = 0) ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 𝐾)
374 simpr 476 . . . . 5 ((𝜑 ∧ (2nd𝑇) = 0) → (2nd𝑇) = 0)
375369, 3, 370, 371, 373, 374poimirlem18 33557 . . . 4 ((𝜑 ∧ (2nd𝑇) = 0) → ∃!𝑧𝑆 𝑧𝑇)
3761adantr 480 . . . . 5 ((𝜑 ∧ (2nd𝑇) = 𝑁) → 𝑁 ∈ ℕ)
3774adantr 480 . . . . 5 ((𝜑 ∧ (2nd𝑇) = 𝑁) → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)))
3786adantr 480 . . . . 5 ((𝜑 ∧ (2nd𝑇) = 𝑁) → 𝑇𝑆)
379 poimirlem22.3 . . . . . 6 ((𝜑𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 0)
380379adantlr 751 . . . . 5 (((𝜑 ∧ (2nd𝑇) = 𝑁) ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 0)
381 simpr 476 . . . . 5 ((𝜑 ∧ (2nd𝑇) = 𝑁) → (2nd𝑇) = 𝑁)
382376, 3, 377, 378, 380, 381poimirlem21 33560 . . . 4 ((𝜑 ∧ (2nd𝑇) = 𝑁) → ∃!𝑧𝑆 𝑧𝑇)
383375, 382jaodan 843 . . 3 ((𝜑 ∧ ((2nd𝑇) = 0 ∨ (2nd𝑇) = 𝑁)) → ∃!𝑧𝑆 𝑧𝑇)
384368, 383syldan 486 . 2 ((𝜑 ∧ ¬ (2nd𝑇) ∈ (1...(𝑁 − 1))) → ∃!𝑧𝑆 𝑧𝑇)
385293, 384pm2.61dan 849 1 (𝜑 → ∃!𝑧𝑆 𝑧𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1523  wcel 2030  {cab 2637  wne 2823  wral 2941  wrex 2942  ∃!wreu 2943  ∃*wrmo 2944  {crab 2945  Vcvv 3231  csb 3566  cdif 3604  cun 3605  cin 3606  wss 3607  c0 3948  ifcif 4119  {csn 4210  {cpr 4212  cop 4216   class class class wbr 4685  cmpt 4762   I cid 5052   × cxp 5141  ran crn 5144  cres 5145  cima 5146  ccom 5147   Fn wfn 5921  wf 5922  1-1wf1 5923  ontowfo 5924  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  𝑓 cof 6937  1st c1st 7208  2nd c2nd 7209  𝑚 cmap 7899  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   < clt 10112  cle 10113  cmin 10304  cn 11058  0cn0 11330  cz 11415  cuz 11725  ...cfz 12364  ..^cfzo 12504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-seq 12842  df-fac 13101  df-bc 13130  df-hash 13158
This theorem is referenced by:  poimirlem27  33566
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