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Theorem poimirlem17 32399
Description: Lemma for poimir 32415 establishing existence for poimirlem18 32400. (Contributed by Brendan Leahy, 21-Aug-2020.)
Hypotheses
Ref Expression
poimir.0 (𝜑𝑁 ∈ ℕ)
poimirlem22.s 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
poimirlem22.1 (𝜑𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)))
poimirlem22.2 (𝜑𝑇𝑆)
poimirlem18.3 ((𝜑𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 𝐾)
poimirlem18.4 (𝜑 → (2nd𝑇) = 0)
Assertion
Ref Expression
poimirlem17 (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 poimirlem17
StepHypRef Expression
1 poimir.0 . . . . 5 (𝜑𝑁 ∈ ℕ)
2 poimirlem22.s . . . . 5 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
3 poimirlem22.1 . . . . 5 (𝜑𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)))
4 poimirlem22.2 . . . . 5 (𝜑𝑇𝑆)
5 poimirlem18.3 . . . . 5 ((𝜑𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 𝐾)
6 poimirlem18.4 . . . . 5 (𝜑 → (2nd𝑇) = 0)
71, 2, 3, 4, 5, 6poimirlem16 32398 . . . 4 (𝜑𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))) ∘𝑓 + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})))))
8 elfznn0 12257 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℕ0)
98nn0red 11199 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℝ)
109adantl 480 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ∈ ℝ)
111nnzd 11313 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ ℤ)
12 peano2zm 11253 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
1311, 12syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑁 − 1) ∈ ℤ)
1413zred 11314 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑁 − 1) ∈ ℝ)
1514adantr 479 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) ∈ ℝ)
161nnred 10882 . . . . . . . . . . . . . . . 16 (𝜑𝑁 ∈ ℝ)
1716adantr 479 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℝ)
18 elfzle2 12171 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ≤ (𝑁 − 1))
1918adantl 480 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ≤ (𝑁 − 1))
2016ltm1d 10805 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑁 − 1) < 𝑁)
2120adantr 479 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) < 𝑁)
2210, 15, 17, 19, 21lelttrd 10046 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 < 𝑁)
2322adantlr 746 . . . . . . . . . . . . 13 (((𝜑𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 < 𝑁)
24 fveq2 6088 . . . . . . . . . . . . . . 15 (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ → (2nd𝑡) = (2nd ‘⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩))
25 opex 4853 . . . . . . . . . . . . . . . 16 ⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩ ∈ V
26 op2ndg 7049 . . . . . . . . . . . . . . . 16 ((⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩ ∈ V ∧ 𝑁 ∈ ℕ) → (2nd ‘⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) = 𝑁)
2725, 1, 26sylancr 693 . . . . . . . . . . . . . . 15 (𝜑 → (2nd ‘⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) = 𝑁)
2824, 27sylan9eqr 2665 . . . . . . . . . . . . . 14 ((𝜑𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → (2nd𝑡) = 𝑁)
2928adantr 479 . . . . . . . . . . . . 13 (((𝜑𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (2nd𝑡) = 𝑁)
3023, 29breqtrrd 4605 . . . . . . . . . . . 12 (((𝜑𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 < (2nd𝑡))
3130iftrued 4043 . . . . . . . . . . 11 (((𝜑𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) ∧ 𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = 𝑦)
3231csbeq1d 3505 . . . . . . . . . 10 (((𝜑𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) ∧ 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 / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))
33 vex 3175 . . . . . . . . . . . . 13 𝑦 ∈ V
34 oveq2 6535 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑦 → (1...𝑗) = (1...𝑦))
3534imaeq2d 5372 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑦 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑡)) “ (1...𝑦)))
3635xpeq1d 5052 . . . . . . . . . . . . . . 15 (𝑗 = 𝑦 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑡)) “ (1...𝑦)) × {1}))
37 oveq1 6534 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑦 → (𝑗 + 1) = (𝑦 + 1))
3837oveq1d 6542 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑦 → ((𝑗 + 1)...𝑁) = ((𝑦 + 1)...𝑁))
3938imaeq2d 5372 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑦 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑡)) “ ((𝑦 + 1)...𝑁)))
4039xpeq1d 5052 . . . . . . . . . . . . . . 15 (𝑗 = 𝑦 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑡)) “ ((𝑦 + 1)...𝑁)) × {0}))
4136, 40uneq12d 3729 . . . . . . . . . . . . . 14 (𝑗 = 𝑦 → ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑡)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑦 + 1)...𝑁)) × {0})))
4241oveq2d 6543 . . . . . . . . . . . . 13 (𝑗 = 𝑦 → ((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑦 + 1)...𝑁)) × {0}))))
4333, 42csbie 3524 . . . . . . . . . . . 12 𝑦 / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑦 + 1)...𝑁)) × {0})))
44 fveq2 6088 . . . . . . . . . . . . . . 15 (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ → (1st𝑡) = (1st ‘⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩))
4544fveq2d 6092 . . . . . . . . . . . . . 14 (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ → (1st ‘(1st𝑡)) = (1st ‘(1st ‘⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩)))
46 op1stg 7048 . . . . . . . . . . . . . . . . 17 ((⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩ ∈ V ∧ 𝑁 ∈ ℕ) → (1st ‘⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) = ⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩)
4725, 1, 46sylancr 693 . . . . . . . . . . . . . . . 16 (𝜑 → (1st ‘⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) = ⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩)
4847fveq2d 6092 . . . . . . . . . . . . . . 15 (𝜑 → (1st ‘(1st ‘⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩)) = (1st ‘⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩))
49 ovex 6555 . . . . . . . . . . . . . . . . 17 (1...𝑁) ∈ V
5049mptex 6368 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))) ∈ V
51 fvex 6098 . . . . . . . . . . . . . . . . 17 (2nd ‘(1st𝑇)) ∈ V
5249mptex 6368 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ∈ V
5351, 52coex 6988 . . . . . . . . . . . . . . . 16 ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) ∈ V
5450, 53op1st 7044 . . . . . . . . . . . . . . 15 (1st ‘⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩) = (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0)))
5548, 54syl6eq 2659 . . . . . . . . . . . . . 14 (𝜑 → (1st ‘(1st ‘⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩)) = (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))))
5645, 55sylan9eqr 2665 . . . . . . . . . . . . 13 ((𝜑𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → (1st ‘(1st𝑡)) = (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))))
5744, 47sylan9eqr 2665 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → (1st𝑡) = ⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩)
5857fveq2d 6092 . . . . . . . . . . . . . . . . 17 ((𝜑𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → (2nd ‘(1st𝑡)) = (2nd ‘⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩))
5950, 53op2nd 7045 . . . . . . . . . . . . . . . . 17 (2nd ‘⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩) = ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))
6058, 59syl6eq 2659 . . . . . . . . . . . . . . . 16 ((𝜑𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → (2nd ‘(1st𝑡)) = ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))))
6160imaeq1d 5371 . . . . . . . . . . . . . . 15 ((𝜑𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → ((2nd ‘(1st𝑡)) “ (1...𝑦)) = (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)))
6261xpeq1d 5052 . . . . . . . . . . . . . 14 ((𝜑𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → (((2nd ‘(1st𝑡)) “ (1...𝑦)) × {1}) = ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}))
6360imaeq1d 5371 . . . . . . . . . . . . . . 15 ((𝜑𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → ((2nd ‘(1st𝑡)) “ ((𝑦 + 1)...𝑁)) = (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)))
6463xpeq1d 5052 . . . . . . . . . . . . . 14 ((𝜑𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → (((2nd ‘(1st𝑡)) “ ((𝑦 + 1)...𝑁)) × {0}) = ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))
6562, 64uneq12d 3729 . . . . . . . . . . . . 13 ((𝜑𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → ((((2nd ‘(1st𝑡)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑦 + 1)...𝑁)) × {0})) = (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})))
6656, 65oveq12d 6545 . . . . . . . . . . . 12 ((𝜑𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → ((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑦 + 1)...𝑁)) × {0}))) = ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))) ∘𝑓 + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))))
6743, 66syl5eq 2655 . . . . . . . . . . 11 ((𝜑𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → 𝑦 / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))) ∘𝑓 + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))))
6867adantr 479 . . . . . . . . . 10 (((𝜑𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))) ∘𝑓 + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))))
6932, 68eqtrd 2643 . . . . . . . . 9 (((𝜑𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) ∧ 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}))) = ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))) ∘𝑓 + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))))
7069mpteq2dva 4666 . . . . . . . 8 ((𝜑𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → (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)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))) ∘𝑓 + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})))))
7170eqeq2d 2619 . . . . . . 7 ((𝜑𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 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})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))) ∘𝑓 + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))))))
7271ex 448 . . . . . 6 (𝜑 → (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 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})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))) ∘𝑓 + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})))))))
7372alrimiv 1841 . . . . 5 (𝜑 → ∀𝑡(𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 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})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))) ∘𝑓 + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})))))))
74 oveq2 6535 . . . . . . . . . . 11 (1 = if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0) → (((1st ‘(1st𝑇))‘𝑛) + 1) = (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0)))
7574eleq1d 2671 . . . . . . . . . 10 (1 = if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0) → ((((1st ‘(1st𝑇))‘𝑛) + 1) ∈ (0..^𝐾) ↔ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0)) ∈ (0..^𝐾)))
76 oveq2 6535 . . . . . . . . . . 11 (0 = if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0) → (((1st ‘(1st𝑇))‘𝑛) + 0) = (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0)))
7776eleq1d 2671 . . . . . . . . . 10 (0 = if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0) → ((((1st ‘(1st𝑇))‘𝑛) + 0) ∈ (0..^𝐾) ↔ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0)) ∈ (0..^𝐾)))
78 fveq2 6088 . . . . . . . . . . . . . 14 (𝑛 = ((2nd ‘(1st𝑇))‘1) → ((1st ‘(1st𝑇))‘𝑛) = ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)))
7978oveq1d 6542 . . . . . . . . . . . . 13 (𝑛 = ((2nd ‘(1st𝑇))‘1) → (((1st ‘(1st𝑇))‘𝑛) + 1) = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1))
8079adantl 480 . . . . . . . . . . . 12 ((𝜑𝑛 = ((2nd ‘(1st𝑇))‘1)) → (((1st ‘(1st𝑇))‘𝑛) + 1) = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1))
81 elrabi 3327 . . . . . . . . . . . . . . . . . . 19 (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)))
8281, 2eleq2s 2705 . . . . . . . . . . . . . . . . . 18 (𝑇𝑆𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
83 xp1st 7066 . . . . . . . . . . . . . . . . . 18 (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)}))
844, 82, 833syl 18 . . . . . . . . . . . . . . . . 17 (𝜑 → (1st𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
85 xp1st 7066 . . . . . . . . . . . . . . . . 17 ((1st𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
86 elmapi 7742 . . . . . . . . . . . . . . . . 17 ((1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
8784, 85, 863syl 18 . . . . . . . . . . . . . . . 16 (𝜑 → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
884, 82syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
89 xp2nd 7067 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
9088, 83, 893syl 18 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
91 f1oeq1 6025 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = (2nd ‘(1st𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
9251, 91elab 3318 . . . . . . . . . . . . . . . . . . 19 ((2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
9390, 92sylib 206 . . . . . . . . . . . . . . . . . 18 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
94 f1of 6035 . . . . . . . . . . . . . . . . . 18 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)):(1...𝑁)⟶(1...𝑁))
9593, 94syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)⟶(1...𝑁))
96 nnuz 11555 . . . . . . . . . . . . . . . . . . 19 ℕ = (ℤ‘1)
971, 96syl6eleq 2697 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ (ℤ‘1))
98 eluzfz1 12174 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ (ℤ‘1) → 1 ∈ (1...𝑁))
9997, 98syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → 1 ∈ (1...𝑁))
10095, 99ffvelrnd 6253 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd ‘(1st𝑇))‘1) ∈ (1...𝑁))
10187, 100ffvelrnd 6253 . . . . . . . . . . . . . . 15 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) ∈ (0..^𝐾))
102 elfzonn0 12335 . . . . . . . . . . . . . . 15 (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) ∈ (0..^𝐾) → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) ∈ ℕ0)
103 peano2nn0 11180 . . . . . . . . . . . . . . 15 (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) ∈ ℕ0 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) ∈ ℕ0)
104101, 102, 1033syl 18 . . . . . . . . . . . . . 14 (𝜑 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) ∈ ℕ0)
105 elfzo0 12331 . . . . . . . . . . . . . . . 16 (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) ∈ (0..^𝐾) ↔ (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) ∈ ℕ0𝐾 ∈ ℕ ∧ ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) < 𝐾))
106101, 105sylib 206 . . . . . . . . . . . . . . 15 (𝜑 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) ∈ ℕ0𝐾 ∈ ℕ ∧ ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) < 𝐾))
107106simp2d 1066 . . . . . . . . . . . . . 14 (𝜑𝐾 ∈ ℕ)
108 elfzolt2 12303 . . . . . . . . . . . . . . . . 17 (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) ∈ (0..^𝐾) → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) < 𝐾)
109101, 108syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) < 𝐾)
110101, 102syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) ∈ ℕ0)
111110nn0zd 11312 . . . . . . . . . . . . . . . . 17 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) ∈ ℤ)
112107nnzd 11313 . . . . . . . . . . . . . . . . 17 (𝜑𝐾 ∈ ℤ)
113 zltp1le 11260 . . . . . . . . . . . . . . . . 17 ((((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) ∈ ℤ ∧ 𝐾 ∈ ℤ) → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) < 𝐾 ↔ (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) ≤ 𝐾))
114111, 112, 113syl2anc 690 . . . . . . . . . . . . . . . 16 (𝜑 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) < 𝐾 ↔ (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) ≤ 𝐾))
115109, 114mpbid 220 . . . . . . . . . . . . . . 15 (𝜑 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) ≤ 𝐾)
116 fvex 6098 . . . . . . . . . . . . . . . . . 18 ((2nd ‘(1st𝑇))‘1) ∈ V
117 eleq1 2675 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = ((2nd ‘(1st𝑇))‘1) → (𝑛 ∈ (1...𝑁) ↔ ((2nd ‘(1st𝑇))‘1) ∈ (1...𝑁)))
118117anbi2d 735 . . . . . . . . . . . . . . . . . . 19 (𝑛 = ((2nd ‘(1st𝑇))‘1) → ((𝜑𝑛 ∈ (1...𝑁)) ↔ (𝜑 ∧ ((2nd ‘(1st𝑇))‘1) ∈ (1...𝑁))))
119 fveq2 6088 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = ((2nd ‘(1st𝑇))‘1) → (𝑝𝑛) = (𝑝‘((2nd ‘(1st𝑇))‘1)))
120119neeq1d 2840 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = ((2nd ‘(1st𝑇))‘1) → ((𝑝𝑛) ≠ 𝐾 ↔ (𝑝‘((2nd ‘(1st𝑇))‘1)) ≠ 𝐾))
121120rexbidv 3033 . . . . . . . . . . . . . . . . . . 19 (𝑛 = ((2nd ‘(1st𝑇))‘1) → (∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 𝐾 ↔ ∃𝑝 ∈ ran 𝐹(𝑝‘((2nd ‘(1st𝑇))‘1)) ≠ 𝐾))
122118, 121imbi12d 332 . . . . . . . . . . . . . . . . . 18 (𝑛 = ((2nd ‘(1st𝑇))‘1) → (((𝜑𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 𝐾) ↔ ((𝜑 ∧ ((2nd ‘(1st𝑇))‘1) ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘((2nd ‘(1st𝑇))‘1)) ≠ 𝐾)))
123116, 122, 5vtocl 3231 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((2nd ‘(1st𝑇))‘1) ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘((2nd ‘(1st𝑇))‘1)) ≠ 𝐾)
124100, 123mpdan 698 . . . . . . . . . . . . . . . 16 (𝜑 → ∃𝑝 ∈ ran 𝐹(𝑝‘((2nd ‘(1st𝑇))‘1)) ≠ 𝐾)
125 fveq1 6087 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 = ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))) → (𝑝‘((2nd ‘(1st𝑇))‘1)) = (((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))‘((2nd ‘(1st𝑇))‘1)))
126 ffn 5944 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾) → (1st ‘(1st𝑇)) Fn (1...𝑁))
12787, 126syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (1st ‘(1st𝑇)) Fn (1...𝑁))
128127adantr 479 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1st ‘(1st𝑇)) Fn (1...𝑁))
129 1ex 9891 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1 ∈ V
130 fnconstg 5991 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (1 ∈ V → (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))))
131129, 130ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1)))
132 c0ex 9890 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 0 ∈ V
133 fnconstg 5991 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (0 ∈ V → (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
134132, 133ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))
135131, 134pm3.2i 469 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) ∧ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
136 dff1o3 6041 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁) ∧ Fun (2nd ‘(1st𝑇))))
137136simprbi 478 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun (2nd ‘(1st𝑇)))
13893, 137syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → Fun (2nd ‘(1st𝑇)))
139 imain 5874 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (Fun (2nd ‘(1st𝑇)) → ((2nd ‘(1st𝑇)) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) ∩ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))))
140138, 139syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) ∩ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))))
141 nn0p1nn 11179 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℕ)
1428, 141syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℕ)
143142nnred 10882 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℝ)
144143ltp1d 10803 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) < ((𝑦 + 1) + 1))
145 fzdisj 12194 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑦 + 1) < ((𝑦 + 1) + 1) → ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅)
146145imaeq2d 5372 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑦 + 1) < ((𝑦 + 1) + 1) → ((2nd ‘(1st𝑇)) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ((2nd ‘(1st𝑇)) “ ∅))
147 ima0 5387 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((2nd ‘(1st𝑇)) “ ∅) = ∅
148146, 147syl6eq 2659 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑦 + 1) < ((𝑦 + 1) + 1) → ((2nd ‘(1st𝑇)) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ∅)
149144, 148syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd ‘(1st𝑇)) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ∅)
150140, 149sylan9req 2664 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) ∩ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ∅)
151 fnun 5897 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) ∧ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) ∧ (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) ∩ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ∅) → ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) ∪ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))))
152135, 150, 151sylancr 693 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) ∪ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))))
153 imaundi 5450 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((2nd ‘(1st𝑇)) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) ∪ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
154142peano2nnd 10884 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ ℕ)
155154, 96syl6eleq 2697 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ (ℤ‘1))
156155adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1) + 1) ∈ (ℤ‘1))
1571nncnd 10883 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝜑𝑁 ∈ ℂ)
158 npcan1 10306 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
159157, 158syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
160159adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) = 𝑁)
161 elfzuz3 12165 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈ (ℤ𝑦))
162 eluzp1p1 11545 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑁 − 1) ∈ (ℤ𝑦) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑦 + 1)))
163161, 162syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑦 + 1)))
164163adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑦 + 1)))
165160, 164eqeltrrd 2688 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ‘(𝑦 + 1)))
166 fzsplit2 12192 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑦 + 1) + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ‘(𝑦 + 1))) → (1...𝑁) = ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))
167156, 165, 166syl2anc 690 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) = ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))
168167imaeq2d 5372 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = ((2nd ‘(1st𝑇)) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))))
169 f1ofo 6042 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁))
170 foima 6018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
17193, 169, 1703syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
172171adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
173168, 172eqtr3d 2645 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) = (1...𝑁))
174153, 173syl5eqr 2657 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) ∪ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = (1...𝑁))
175174fneq2d 5882 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) ∪ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) ↔ ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁)))
176152, 175mpbid 220 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁))
17749a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) ∈ V)
178 inidm 3783 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
179 eqidd 2610 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ ((2nd ‘(1st𝑇))‘1) ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) = ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)))
180 f1ofn 6036 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)) Fn (1...𝑁))
18193, 180syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑 → (2nd ‘(1st𝑇)) Fn (1...𝑁))
182181adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (2nd ‘(1st𝑇)) Fn (1...𝑁))
183 fzss2 12207 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑁 ∈ (ℤ‘(𝑦 + 1)) → (1...(𝑦 + 1)) ⊆ (1...𝑁))
184165, 183syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1...(𝑦 + 1)) ⊆ (1...𝑁))
185142, 96syl6eleq 2697 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ (ℤ‘1))
186 eluzfz1 12174 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑦 + 1) ∈ (ℤ‘1) → 1 ∈ (1...(𝑦 + 1)))
187185, 186syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ (0...(𝑁 − 1)) → 1 ∈ (1...(𝑦 + 1)))
188187adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 1 ∈ (1...(𝑦 + 1)))
189 fnfvima 6378 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((2nd ‘(1st𝑇)) Fn (1...𝑁) ∧ (1...(𝑦 + 1)) ⊆ (1...𝑁) ∧ 1 ∈ (1...(𝑦 + 1))) → ((2nd ‘(1st𝑇))‘1) ∈ ((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))))
190182, 184, 188, 189syl3anc 1317 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇))‘1) ∈ ((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))))
191 fvun1 6164 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) ∧ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) ∧ ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) ∩ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ∅ ∧ ((2nd ‘(1st𝑇))‘1) ∈ ((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))))) → (((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘1)) = ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1})‘((2nd ‘(1st𝑇))‘1)))
192131, 134, 191mp3an12 1405 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) ∩ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ∅ ∧ ((2nd ‘(1st𝑇))‘1) ∈ ((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1)))) → (((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘1)) = ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1})‘((2nd ‘(1st𝑇))‘1)))
193150, 190, 192syl2anc 690 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘1)) = ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1})‘((2nd ‘(1st𝑇))‘1)))
194129fvconst2 6352 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((2nd ‘(1st𝑇))‘1) ∈ ((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) → ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1})‘((2nd ‘(1st𝑇))‘1)) = 1)
195190, 194syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1})‘((2nd ‘(1st𝑇))‘1)) = 1)
196193, 195eqtrd 2643 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘1)) = 1)
197196adantr 479 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ ((2nd ‘(1st𝑇))‘1) ∈ (1...𝑁)) → (((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘1)) = 1)
198128, 176, 177, 177, 178, 179, 197ofval 6781 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ ((2nd ‘(1st𝑇))‘1) ∈ (1...𝑁)) → (((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))‘((2nd ‘(1st𝑇))‘1)) = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1))
199100, 198mpidan 700 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))‘((2nd ‘(1st𝑇))‘1)) = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1))
200125, 199sylan9eqr 2665 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑝 = ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))) → (𝑝‘((2nd ‘(1st𝑇))‘1)) = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1))
201200adantllr 750 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑝 ∈ ran 𝐹) ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑝 = ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))) → (𝑝‘((2nd ‘(1st𝑇))‘1)) = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1))
202 fveq2 6088 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑡 = 𝑇 → (2nd𝑡) = (2nd𝑇))
203202breq2d 4589 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑡 = 𝑇 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑇)))
204203ifbid 4057 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑡 = 𝑇 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)))
205204csbeq1d 3505 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (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}))))
206 fveq2 6088 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑡 = 𝑇 → (1st𝑡) = (1st𝑇))
207206fveq2d 6092 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑡 = 𝑇 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
208206fveq2d 6092 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑡 = 𝑇 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑇)))
209208imaeq1d 5371 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑗)))
210209xpeq1d 5052 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}))
211208imaeq1d 5371 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)))
212211xpeq1d 5052 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
213210, 212uneq12d 3729 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑡 = 𝑇 → ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))
214207, 213oveq12d 6545 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (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}))))
215214csbeq2dv 3943 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (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}))))
216205, 215eqtrd 2643 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (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}))))
217216mpteq2dv 4667 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (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})))))
218217eqeq2d 2619 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (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}))))))
219218, 2elrab2 3332 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (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}))))))
220219simprbi 478 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (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})))))
2214, 220syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (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})))))
222221rneqd 5261 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ran 𝐹 = ran (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})))))
223222eleq2d 2672 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝑝 ∈ ran 𝐹𝑝 ∈ ran (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}))))))
224 eqid 2609 . . . . . . . . . . . . . . . . . . . . . . . 24 (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}))))
225 ovex 6555 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V
226225csbex 4716 . . . . . . . . . . . . . . . . . . . . . . . 24 if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V
227224, 226elrnmpti 5284 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑝 ∈ ran (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))𝑝 = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
228223, 227syl6bb 274 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑝 ∈ ran 𝐹 ↔ ∃𝑦 ∈ (0...(𝑁 − 1))𝑝 = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
2296adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (2nd𝑇) = 0)
230 elfzle1 12170 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 ∈ (0...(𝑁 − 1)) → 0 ≤ 𝑦)
231230adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 0 ≤ 𝑦)
232229, 231eqbrtrd 4599 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (2nd𝑇) ≤ 𝑦)
233 0re 9896 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 0 ∈ ℝ
2346, 233syl6eqel 2695 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑 → (2nd𝑇) ∈ ℝ)
235 lenlt 9967 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((2nd𝑇) ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((2nd𝑇) ≤ 𝑦 ↔ ¬ 𝑦 < (2nd𝑇)))
236234, 9, 235syl2an 492 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd𝑇) ≤ 𝑦 ↔ ¬ 𝑦 < (2nd𝑇)))
237232, 236mpbid 220 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ¬ 𝑦 < (2nd𝑇))
238237iffalsed 4046 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = (𝑦 + 1))
239238csbeq1d 3505 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑦 ∈ (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 + 1) / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
240 ovex 6555 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 + 1) ∈ V
241 oveq2 6535 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 = (𝑦 + 1) → (1...𝑗) = (1...(𝑦 + 1)))
242241imaeq2d 5372 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 = (𝑦 + 1) → ((2nd ‘(1st𝑇)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))))
243242xpeq1d 5052 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 = (𝑦 + 1) → (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}))
244 oveq1 6534 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑗 = (𝑦 + 1) → (𝑗 + 1) = ((𝑦 + 1) + 1))
245244oveq1d 6542 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 = (𝑦 + 1) → ((𝑗 + 1)...𝑁) = (((𝑦 + 1) + 1)...𝑁))
246245imaeq2d 5372 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 = (𝑦 + 1) → ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
247246xpeq1d 5052 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 = (𝑦 + 1) → (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))
248243, 247uneq12d 3729 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 = (𝑦 + 1) → ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))
249248oveq2d 6543 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 = (𝑦 + 1) → ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))
250240, 249csbie 3524 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 + 1) / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))
251239, 250syl6eq 2659 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))
252251eqeq2d 2619 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑝 = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) ↔ 𝑝 = ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))))
253252rexbidva 3030 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (∃𝑦 ∈ (0...(𝑁 − 1))𝑝 = 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))𝑝 = ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))))
254228, 253bitrd 266 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑝 ∈ ran 𝐹 ↔ ∃𝑦 ∈ (0...(𝑁 − 1))𝑝 = ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))))
255254biimpa 499 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑝 ∈ ran 𝐹) → ∃𝑦 ∈ (0...(𝑁 − 1))𝑝 = ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))
256201, 255r19.29a 3059 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑝 ∈ ran 𝐹) → (𝑝‘((2nd ‘(1st𝑇))‘1)) = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1))
257 eqtr3 2630 . . . . . . . . . . . . . . . . . . . 20 (((𝑝‘((2nd ‘(1st𝑇))‘1)) = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) ∧ 𝐾 = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1)) → (𝑝‘((2nd ‘(1st𝑇))‘1)) = 𝐾)
258257ex 448 . . . . . . . . . . . . . . . . . . 19 ((𝑝‘((2nd ‘(1st𝑇))‘1)) = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) → (𝐾 = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) → (𝑝‘((2nd ‘(1st𝑇))‘1)) = 𝐾))
259256, 258syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑝 ∈ ran 𝐹) → (𝐾 = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) → (𝑝‘((2nd ‘(1st𝑇))‘1)) = 𝐾))
260259necon3d 2802 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ ran 𝐹) → ((𝑝‘((2nd ‘(1st𝑇))‘1)) ≠ 𝐾𝐾 ≠ (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1)))
261260rexlimdva 3012 . . . . . . . . . . . . . . . 16 (𝜑 → (∃𝑝 ∈ ran 𝐹(𝑝‘((2nd ‘(1st𝑇))‘1)) ≠ 𝐾𝐾 ≠ (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1)))
262124, 261mpd 15 . . . . . . . . . . . . . . 15 (𝜑𝐾 ≠ (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1))
263104nn0red 11199 . . . . . . . . . . . . . . . 16 (𝜑 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) ∈ ℝ)
264107nnred 10882 . . . . . . . . . . . . . . . 16 (𝜑𝐾 ∈ ℝ)
265263, 264ltlend 10033 . . . . . . . . . . . . . . 15 (𝜑 → ((((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) < 𝐾 ↔ ((((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) ≤ 𝐾𝐾 ≠ (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1))))
266115, 262, 265mpbir2and 958 . . . . . . . . . . . . . 14 (𝜑 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) < 𝐾)
267 elfzo0 12331 . . . . . . . . . . . . . 14 ((((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) ∈ (0..^𝐾) ↔ ((((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) ∈ ℕ0𝐾 ∈ ℕ ∧ (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) < 𝐾))
268104, 107, 266, 267syl3anbrc 1238 . . . . . . . . . . . . 13 (𝜑 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) ∈ (0..^𝐾))
269268adantr 479 . . . . . . . . . . . 12 ((𝜑𝑛 = ((2nd ‘(1st𝑇))‘1)) → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) ∈ (0..^𝐾))
27080, 269eqeltrd 2687 . . . . . . . . . . 11 ((𝜑𝑛 = ((2nd ‘(1st𝑇))‘1)) → (((1st ‘(1st𝑇))‘𝑛) + 1) ∈ (0..^𝐾))
271270adantlr 746 . . . . . . . . . 10 (((𝜑𝑛 ∈ (1...𝑁)) ∧ 𝑛 = ((2nd ‘(1st𝑇))‘1)) → (((1st ‘(1st𝑇))‘𝑛) + 1) ∈ (0..^𝐾))
27287ffvelrnda 6252 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ (0..^𝐾))
273 elfzonn0 12335 . . . . . . . . . . . . . . 15 (((1st ‘(1st𝑇))‘𝑛) ∈ (0..^𝐾) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℕ0)
274272, 273syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℕ0)
275274nn0cnd 11200 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℂ)
276275addid1d 10087 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑇))‘𝑛) + 0) = ((1st ‘(1st𝑇))‘𝑛))
277276, 272eqeltrd 2687 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑇))‘𝑛) + 0) ∈ (0..^𝐾))
278277adantr 479 . . . . . . . . . 10 (((𝜑𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st𝑇))‘1)) → (((1st ‘(1st𝑇))‘𝑛) + 0) ∈ (0..^𝐾))
27975, 77, 271, 278ifbothda 4072 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0)) ∈ (0..^𝐾))
280 eqid 2609 . . . . . . . . 9 (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))) = (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0)))
281279, 280fmptd 6277 . . . . . . . 8 (𝜑 → (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))):(1...𝑁)⟶(0..^𝐾))
282 ovex 6555 . . . . . . . . 9 (0..^𝐾) ∈ V
283282, 49elmap 7749 . . . . . . . 8 ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) ↔ (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))):(1...𝑁)⟶(0..^𝐾))
284281, 283sylibr 222 . . . . . . 7 (𝜑 → (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
285 simpr 475 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (1...(𝑁 − 1))) → 𝑛 ∈ (1...(𝑁 − 1)))
286 1z 11240 . . . . . . . . . . . . . . . . 17 1 ∈ ℤ
28713, 286jctil 557 . . . . . . . . . . . . . . . 16 (𝜑 → (1 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ))
288 elfzelz 12168 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ ℤ)
289288, 286jctir 558 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (1...(𝑁 − 1)) → (𝑛 ∈ ℤ ∧ 1 ∈ ℤ))
290 fzaddel 12201 . . . . . . . . . . . . . . . 16 (((1 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) ∧ (𝑛 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑛 ∈ (1...(𝑁 − 1)) ↔ (𝑛 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1))))
291287, 289, 290syl2an 492 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (1...(𝑁 − 1))) → (𝑛 ∈ (1...(𝑁 − 1)) ↔ (𝑛 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1))))
292285, 291mpbid 220 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...(𝑁 − 1))) → (𝑛 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1)))
293159oveq2d 6543 . . . . . . . . . . . . . . 15 (𝜑 → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 + 1)...𝑁))
294293adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...(𝑁 − 1))) → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 + 1)...𝑁))
295292, 294eleqtrd 2689 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...(𝑁 − 1))) → (𝑛 + 1) ∈ ((1 + 1)...𝑁))
296295ralrimiva 2948 . . . . . . . . . . . 12 (𝜑 → ∀𝑛 ∈ (1...(𝑁 − 1))(𝑛 + 1) ∈ ((1 + 1)...𝑁))
297 simpr 475 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ ((1 + 1)...𝑁)) → 𝑦 ∈ ((1 + 1)...𝑁))
298 peano2z 11251 . . . . . . . . . . . . . . . . . . 19 (1 ∈ ℤ → (1 + 1) ∈ ℤ)
299286, 298ax-mp 5 . . . . . . . . . . . . . . . . . 18 (1 + 1) ∈ ℤ
30011, 299jctil 557 . . . . . . . . . . . . . . . . 17 (𝜑 → ((1 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ))
301 elfzelz 12168 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ((1 + 1)...𝑁) → 𝑦 ∈ ℤ)
302301, 286jctir 558 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ((1 + 1)...𝑁) → (𝑦 ∈ ℤ ∧ 1 ∈ ℤ))
303 fzsubel 12203 . . . . . . . . . . . . . . . . 17 ((((1 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑦 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑦 ∈ ((1 + 1)...𝑁) ↔ (𝑦 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1))))
304300, 302, 303syl2an 492 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ ((1 + 1)...𝑁)) → (𝑦 ∈ ((1 + 1)...𝑁) ↔ (𝑦 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1))))
305297, 304mpbid 220 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ ((1 + 1)...𝑁)) → (𝑦 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1)))
306 ax-1cn 9850 . . . . . . . . . . . . . . . . 17 1 ∈ ℂ
307306, 306pncan3oi 10148 . . . . . . . . . . . . . . . 16 ((1 + 1) − 1) = 1
308307oveq1i 6537 . . . . . . . . . . . . . . 15 (((1 + 1) − 1)...(𝑁 − 1)) = (1...(𝑁 − 1))
309305, 308syl6eleq 2697 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ ((1 + 1)...𝑁)) → (𝑦 − 1) ∈ (1...(𝑁 − 1)))
310301zcnd 11315 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ((1 + 1)...𝑁) → 𝑦 ∈ ℂ)
311 elfznn 12196 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ ℕ)
312311nncnd 10883 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ ℂ)
313 subadd2 10136 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑛 ∈ ℂ) → ((𝑦 − 1) = 𝑛 ↔ (𝑛 + 1) = 𝑦))
314306, 313mp3an2 1403 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℂ) → ((𝑦 − 1) = 𝑛 ↔ (𝑛 + 1) = 𝑦))
315314bicomd 211 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℂ) → ((𝑛 + 1) = 𝑦 ↔ (𝑦 − 1) = 𝑛))
316 eqcom 2616 . . . . . . . . . . . . . . . . . 18 (𝑦 = (𝑛 + 1) ↔ (𝑛 + 1) = 𝑦)
317 eqcom 2616 . . . . . . . . . . . . . . . . . 18 (𝑛 = (𝑦 − 1) ↔ (𝑦 − 1) = 𝑛)
318315, 316, 3173bitr4g 301 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℂ) → (𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1)))
319310, 312, 318syl2an 492 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ((1 + 1)...𝑁) ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1)))
320319ralrimiva 2948 . . . . . . . . . . . . . . 15 (𝑦 ∈ ((1 + 1)...𝑁) → ∀𝑛 ∈ (1...(𝑁 − 1))(𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1)))
321320adantl 480 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ ((1 + 1)...𝑁)) → ∀𝑛 ∈ (1...(𝑁 − 1))(𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1)))
322 reu6i 3363 . . . . . . . . . . . . . 14 (((𝑦 − 1) ∈ (1...(𝑁 − 1)) ∧ ∀𝑛 ∈ (1...(𝑁 − 1))(𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1))) → ∃!𝑛 ∈ (1...(𝑁 − 1))𝑦 = (𝑛 + 1))
323309, 321, 322syl2anc 690 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ ((1 + 1)...𝑁)) → ∃!𝑛 ∈ (1...(𝑁 − 1))𝑦 = (𝑛 + 1))
324323ralrimiva 2948 . . . . . . . . . . . 12 (𝜑 → ∀𝑦 ∈ ((1 + 1)...𝑁)∃!𝑛 ∈ (1...(𝑁 − 1))𝑦 = (𝑛 + 1))
325 eqid 2609 . . . . . . . . . . . . 13 (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) = (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1))
326325f1ompt 6275 . . . . . . . . . . . 12 ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)):(1...(𝑁 − 1))–1-1-onto→((1 + 1)...𝑁) ↔ (∀𝑛 ∈ (1...(𝑁 − 1))(𝑛 + 1) ∈ ((1 + 1)...𝑁) ∧ ∀𝑦 ∈ ((1 + 1)...𝑁)∃!𝑛 ∈ (1...(𝑁 − 1))𝑦 = (𝑛 + 1)))
327296, 324, 326sylanbrc 694 . . . . . . . . . . 11 (𝜑 → (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)):(1...(𝑁 − 1))–1-1-onto→((1 + 1)...𝑁))
328 f1osng 6074 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ ∧ 1 ∈ V) → {⟨𝑁, 1⟩}:{𝑁}–1-1-onto→{1})
3291, 129, 328sylancl 692 . . . . . . . . . . 11 (𝜑 → {⟨𝑁, 1⟩}:{𝑁}–1-1-onto→{1})
33014, 16ltnled 10035 . . . . . . . . . . . . . 14 (𝜑 → ((𝑁 − 1) < 𝑁 ↔ ¬ 𝑁 ≤ (𝑁 − 1)))
33120, 330mpbid 220 . . . . . . . . . . . . 13 (𝜑 → ¬ 𝑁 ≤ (𝑁 − 1))
332 elfzle2 12171 . . . . . . . . . . . . 13 (𝑁 ∈ (1...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1))
333331, 332nsyl 133 . . . . . . . . . . . 12 (𝜑 → ¬ 𝑁 ∈ (1...(𝑁 − 1)))
334 disjsn 4191 . . . . . . . . . . . 12 (((1...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (1...(𝑁 − 1)))
335333, 334sylibr 222 . . . . . . . . . . 11 (𝜑 → ((1...(𝑁 − 1)) ∩ {𝑁}) = ∅)
336 1re 9895 . . . . . . . . . . . . . . . 16 1 ∈ ℝ
337336ltp1i 10776 . . . . . . . . . . . . . . 15 1 < (1 + 1)
338299zrei 11216 . . . . . . . . . . . . . . . 16 (1 + 1) ∈ ℝ
339336, 338ltnlei 10009 . . . . . . . . . . . . . . 15 (1 < (1 + 1) ↔ ¬ (1 + 1) ≤ 1)
340337, 339mpbi 218 . . . . . . . . . . . . . 14 ¬ (1 + 1) ≤ 1
341 elfzle1 12170 . . . . . . . . . . . . . 14 (1 ∈ ((1 + 1)...𝑁) → (1 + 1) ≤ 1)
342340, 341mto 186 . . . . . . . . . . . . 13 ¬ 1 ∈ ((1 + 1)...𝑁)
343 disjsn 4191 . . . . . . . . . . . . 13 ((((1 + 1)...𝑁) ∩ {1}) = ∅ ↔ ¬ 1 ∈ ((1 + 1)...𝑁))
344342, 343mpbir 219 . . . . . . . . . . . 12 (((1 + 1)...𝑁) ∩ {1}) = ∅
345344a1i 11 . . . . . . . . . . 11 (𝜑 → (((1 + 1)...𝑁) ∩ {1}) = ∅)
346 f1oun 6054 . . . . . . . . . . 11 ((((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)):(1...(𝑁 − 1))–1-1-onto→((1 + 1)...𝑁) ∧ {⟨𝑁, 1⟩}:{𝑁}–1-1-onto→{1}) ∧ (((1...(𝑁 − 1)) ∩ {𝑁}) = ∅ ∧ (((1 + 1)...𝑁) ∩ {1}) = ∅)) → ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {⟨𝑁, 1⟩}):((1...(𝑁 − 1)) ∪ {𝑁})–1-1-onto→(((1 + 1)...𝑁) ∪ {1}))
347327, 329, 335, 345, 346syl22anc 1318 . . . . . . . . . 10 (𝜑 → ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {⟨𝑁, 1⟩}):((1...(𝑁 − 1)) ∪ {𝑁})–1-1-onto→(((1 + 1)...𝑁) ∪ {1}))
348 elex 3184 . . . . . . . . . . . . . 14 (𝑁 ∈ ℕ → 𝑁 ∈ V)
3491, 348syl 17 . . . . . . . . . . . . 13 (𝜑𝑁 ∈ V)
350129a1i 11 . . . . . . . . . . . . 13 (𝜑 → 1 ∈ V)
351159, 97eqeltrd 2687 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘1))
352 uzid 11534 . . . . . . . . . . . . . . . . 17 ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)))
353 peano2uz 11573 . . . . . . . . . . . . . . . . 17 ((𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
35413, 352, 3533syl 18 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
355159, 354eqeltrrd 2688 . . . . . . . . . . . . . . 15 (𝜑𝑁 ∈ (ℤ‘(𝑁 − 1)))
356 fzsplit2 12192 . . . . . . . . . . . . . . 15 ((((𝑁 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ‘(𝑁 − 1))) → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
357351, 355, 356syl2anc 690 . . . . . . . . . . . . . 14 (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
358159oveq1d 6542 . . . . . . . . . . . . . . . 16 (𝜑 → (((𝑁 − 1) + 1)...𝑁) = (𝑁...𝑁))
359 fzsn 12209 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
36011, 359syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑁...𝑁) = {𝑁})
361358, 360eqtrd 2643 . . . . . . . . . . . . . . 15 (𝜑 → (((𝑁 − 1) + 1)...𝑁) = {𝑁})
362361uneq2d 3728 . . . . . . . . . . . . . 14 (𝜑 → ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = ((1...(𝑁 − 1)) ∪ {𝑁}))
363357, 362eqtr2d 2644 . . . . . . . . . . . . 13 (𝜑 → ((1...(𝑁 − 1)) ∪ {𝑁}) = (1...𝑁))
364 iftrue 4041 . . . . . . . . . . . . . 14 (𝑛 = 𝑁 → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = 1)
365364adantl 480 . . . . . . . . . . . . 13 ((𝜑𝑛 = 𝑁) → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = 1)
366349, 350, 363, 365fmptapd 6320 . . . . . . . . . . . 12 (𝜑 → ((𝑛 ∈ (1...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ∪ {⟨𝑁, 1⟩}) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))
367 eleq1 2675 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑁 → (𝑛 ∈ (1...(𝑁 − 1)) ↔ 𝑁 ∈ (1...(𝑁 − 1))))
368367notbid 306 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑁 → (¬ 𝑛 ∈ (1...(𝑁 − 1)) ↔ ¬ 𝑁 ∈ (1...(𝑁 − 1))))
369333, 368syl5ibrcom 235 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑛 = 𝑁 → ¬ 𝑛 ∈ (1...(𝑁 − 1))))
370369necon2ad 2796 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛𝑁))
371370imp 443 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (1...(𝑁 − 1))) → 𝑛𝑁)
372 ifnefalse 4047 . . . . . . . . . . . . . . 15 (𝑛𝑁 → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = (𝑛 + 1))
373371, 372syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...(𝑁 − 1))) → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = (𝑛 + 1))
374373mpteq2dva 4666 . . . . . . . . . . . . 13 (𝜑 → (𝑛 ∈ (1...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)))
375374uneq1d 3727 . . . . . . . . . . . 12 (𝜑 → ((𝑛 ∈ (1...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ∪ {⟨𝑁, 1⟩}) = ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {⟨𝑁, 1⟩}))
376366, 375eqtr3d 2645 . . . . . . . . . . 11 (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {⟨𝑁, 1⟩}))
377357, 362eqtrd 2643 . . . . . . . . . . 11 (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁}))
378 uzid 11534 . . . . . . . . . . . . . 14 (1 ∈ ℤ → 1 ∈ (ℤ‘1))
379 peano2uz 11573 . . . . . . . . . . . . . 14 (1 ∈ (ℤ‘1) → (1 + 1) ∈ (ℤ‘1))
380286, 378, 379mp2b 10 . . . . . . . . . . . . 13 (1 + 1) ∈ (ℤ‘1)
381 fzsplit2 12192 . . . . . . . . . . . . 13 (((1 + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ‘1)) → (1...𝑁) = ((1...1) ∪ ((1 + 1)...𝑁)))
382380, 97, 381sylancr 693 . . . . . . . . . . . 12 (𝜑 → (1...𝑁) = ((1...1) ∪ ((1 + 1)...𝑁)))
383 fzsn 12209 . . . . . . . . . . . . . . 15 (1 ∈ ℤ → (1...1) = {1})
384286, 383ax-mp 5 . . . . . . . . . . . . . 14 (1...1) = {1}
385384uneq1i 3724 . . . . . . . . . . . . 13 ((1...1) ∪ ((1 + 1)...𝑁)) = ({1} ∪ ((1 + 1)...𝑁))
386385equncomi 3720 . . . . . . . . . . . 12 ((1...1) ∪ ((1 + 1)...𝑁)) = (((1 + 1)...𝑁) ∪ {1})
387382, 386syl6eq 2659 . . . . . . . . . . 11 (𝜑 → (1...𝑁) = (((1 + 1)...𝑁) ∪ {1}))
388376, 377, 387f1oeq123d 6031 . . . . . . . . . 10 (𝜑 → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {⟨𝑁, 1⟩}):((1...(𝑁 − 1)) ∪ {𝑁})–1-1-onto→(((1 + 1)...𝑁) ∪ {1})))
389347, 388mpbird 245 . . . . . . . . 9 (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)–1-1-onto→(1...𝑁))
390 f1oco 6057 . . . . . . . . 9 (((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ∧ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)–1-1-onto→(1...𝑁)) → ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–1-1-onto→(1...𝑁))
39193, 389, 390syl2anc 690 . . . . . . . 8 (𝜑 → ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–1-1-onto→(1...𝑁))
392 f1oeq1 6025 . . . . . . . . 9 (𝑓 = ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–1-1-onto→(1...𝑁)))
39353, 392elab 3318 . . . . . . . 8 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–1-1-onto→(1...𝑁))
394391, 393sylibr 222 . . . . . . 7 (𝜑 → ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
395 opelxpi 5062 . . . . . . 7 (((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) ∧ ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → ⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩ ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
396284, 394, 395syl2anc 690 . . . . . 6 (𝜑 → ⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩ ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
3971nnnn0d 11198 . . . . . . 7 (𝜑𝑁 ∈ ℕ0)
398 nn0fz0 12261 . . . . . . 7 (𝑁 ∈ ℕ0𝑁 ∈ (0...𝑁))
399397, 398sylib 206 . . . . . 6 (𝜑𝑁 ∈ (0...𝑁))
400 opelxpi 5062 . . . . . 6 ((⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩ ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑁 ∈ (0...𝑁)) → ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
401396, 399, 400syl2anc 690 . . . . 5 (𝜑 → ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
402 elrab3t 3329 . . . . 5 ((∀𝑡(𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 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})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))) ∘𝑓 + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})))))) ∧ ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ ∈ {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}))))} ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))) ∘𝑓 + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))))))
40373, 401, 402syl2anc 690 . . . 4 (𝜑 → (⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ ∈ {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}))))} ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))) ∘𝑓 + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))))))
4047, 403mpbird 245 . . 3 (𝜑 → ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ ∈ {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}))))})
405404, 2syl6eleqr 2698 . 2 (𝜑 → ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ ∈ 𝑆)
406 fveq2 6088 . . . . . . 7 (⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ = 𝑇 → (2nd ‘⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) = (2nd𝑇))
407406eqeq1d 2611 . . . . . 6 (⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ = 𝑇 → ((2nd ‘⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) = 𝑁 ↔ (2nd𝑇) = 𝑁))
40827, 407syl5ibcom 233 . . . . 5 (𝜑 → (⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ = 𝑇 → (2nd𝑇) = 𝑁))
4091nnne0d 10912 . . . . . 6 (𝜑𝑁 ≠ 0)
410 neeq1 2843 . . . . . 6 ((2nd𝑇) = 𝑁 → ((2nd𝑇) ≠ 0 ↔ 𝑁 ≠ 0))
411409, 410syl5ibrcom 235 . . . . 5 (𝜑 → ((2nd𝑇) = 𝑁 → (2nd𝑇) ≠ 0))
412408, 411syld 45 . . . 4 (𝜑 → (⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ = 𝑇 → (2nd𝑇) ≠ 0))
413412necon2d 2804 . . 3 (𝜑 → ((2nd𝑇) = 0 → ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ ≠ 𝑇))
4146, 413mpd 15 . 2 (𝜑 → ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ ≠ 𝑇)
415 neeq1 2843 . . 3 (𝑧 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ → (𝑧𝑇 ↔ ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ ≠ 𝑇))
416415rspcev 3281 . 2 ((⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ ∈ 𝑆 ∧ ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ ≠ 𝑇) → ∃𝑧𝑆 𝑧𝑇)
417405, 414, 416syl2anc 690 1 (𝜑 → ∃𝑧𝑆 𝑧𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030  wal 1472   = wceq 1474  wcel 1976  {cab 2595  wne 2779  wral 2895  wrex 2896  ∃!wreu 2897  {crab 2899  Vcvv 3172  csb 3498  cun 3537  cin 3538  wss 3539  c0 3873  ifcif 4035  {csn 4124  cop 4130   class class class wbr 4577  cmpt 4637   × cxp 5026  ccnv 5027  ran crn 5029  cima 5031  ccom 5032  Fun wfun 5784   Fn wfn 5785  wf 5786  ontowfo 5788  1-1-ontowf1o 5789  cfv 5790  (class class class)co 6527  𝑓 cof 6770  1st c1st 7034  2nd c2nd 7035  𝑚 cmap 7721  cc 9790  cr 9791  0cc0 9792  1c1 9793   + caddc 9795   < clt 9930  cle 9931  cmin 10117  cn 10867  0cn0 11139  cz 11210  cuz 11519  ...cfz 12152  ..^cfzo 12289
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-of 6772  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-er 7606  df-map 7723  df-en 7819  df-dom 7820  df-sdom 7821  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-n0 11140  df-z 11211  df-uz 11520  df-fz 12153  df-fzo 12290
This theorem is referenced by:  poimirlem18  32400
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