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Theorem poimirlem17 37624
Description: Lemma for poimir 37640 establishing existence for poimirlem18 37625. (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 37623 . . . 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 13657 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℕ0)
98nn0red 12586 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℝ)
109adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ∈ ℝ)
111nnzd 12638 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ ℤ)
12 peano2zm 12658 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
1311, 12syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑁 − 1) ∈ ℤ)
1413zred 12720 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑁 − 1) ∈ ℝ)
1514adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) ∈ ℝ)
161nnred 12279 . . . . . . . . . . . . . . . 16 (𝜑𝑁 ∈ ℝ)
1716adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℝ)
18 elfzle2 13565 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ≤ (𝑁 − 1))
1918adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ≤ (𝑁 − 1))
2016ltm1d 12198 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑁 − 1) < 𝑁)
2120adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) < 𝑁)
2210, 15, 17, 19, 21lelttrd 11417 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 < 𝑁)
2322adantlr 715 . . . . . . . . . . . . 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 6907 . . . . . . . . . . . . . . 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 5475 . . . . . . . . . . . . . . . 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 8026 . . . . . . . . . . . . . . . 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 587 . . . . . . . . . . . . . . 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 2797 . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . 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 5176 . . . . . . . . . . . 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 4539 . . . . . . . . . . 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 3912 . . . . . . . . . 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 3482 . . . . . . . . . . . . 13 𝑦 ∈ V
34 oveq2 7439 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑦 → (1...𝑗) = (1...𝑦))
3534imaeq2d 6080 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑦 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑡)) “ (1...𝑦)))
3635xpeq1d 5718 . . . . . . . . . . . . . . 15 (𝑗 = 𝑦 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑡)) “ (1...𝑦)) × {1}))
37 oveq1 7438 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑦 → (𝑗 + 1) = (𝑦 + 1))
3837oveq1d 7446 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑦 → ((𝑗 + 1)...𝑁) = ((𝑦 + 1)...𝑁))
3938imaeq2d 6080 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑦 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑡)) “ ((𝑦 + 1)...𝑁)))
4039xpeq1d 5718 . . . . . . . . . . . . . . 15 (𝑗 = 𝑦 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑡)) “ ((𝑦 + 1)...𝑁)) × {0}))
4136, 40uneq12d 4179 . . . . . . . . . . . . . 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 7447 . . . . . . . . . . . . 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 3944 . . . . . . . . . . . 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 2fveq3 6912 . . . . . . . . . . . . . 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⟩)))
45 op1stg 8025 . . . . . . . . . . . . . . . . 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))))⟩)
4625, 1, 45sylancr 587 . . . . . . . . . . . . . . . 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))))⟩)
4746fveq2d 6911 . . . . . . . . . . . . . . 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))))⟩))
48 ovex 7464 . . . . . . . . . . . . . . . . 17 (1...𝑁) ∈ V
4948mptex 7243 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))) ∈ V
50 fvex 6920 . . . . . . . . . . . . . . . . 17 (2nd ‘(1st𝑇)) ∈ V
5148mptex 7243 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ∈ V
5250, 51coex 7953 . . . . . . . . . . . . . . . 16 ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) ∈ V
5349, 52op1st 8021 . . . . . . . . . . . . . . 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)))
5447, 53eqtrdi 2791 . . . . . . . . . . . . . 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))))
5544, 54sylan9eqr 2797 . . . . . . . . . . . . 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))))
56 fveq2 6907 . . . . . . . . . . . . . . . . . . 19 (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⟩))
5756, 46sylan9eqr 2797 . . . . . . . . . . . . . . . . . 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 6911 . . . . . . . . . . . . . . . . 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))))⟩))
5949, 52op2nd 8022 . . . . . . . . . . . . . . . . 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, 59eqtrdi 2791 . . . . . . . . . . . . . . . 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 6079 . . . . . . . . . . . . . . 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 5718 . . . . . . . . . . . . . 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 6079 . . . . . . . . . . . . . . 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 5718 . . . . . . . . . . . . . 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 4179 . . . . . . . . . . . . 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})))
6655, 65oveq12d 7449 . . . . . . . . . . . 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, 66eqtrid 2787 . . . . . . . . . . 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 480 . . . . . . . . . 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 2775 . . . . . . . . 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 5248 . . . . . . . 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 2746 . . . . . . 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 412 . . . . . 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 1925 . . . . 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 7439 . . . . . . . . . . 11 (1 = if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0) → (((1st ‘(1st𝑇))‘𝑛) + 1) = (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0)))
7574eleq1d 2824 . . . . . . . . . 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 7439 . . . . . . . . . . 11 (0 = if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0) → (((1st ‘(1st𝑇))‘𝑛) + 0) = (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0)))
7776eleq1d 2824 . . . . . . . . . 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 6907 . . . . . . . . . . . . . 14 (𝑛 = ((2nd ‘(1st𝑇))‘1) → ((1st ‘(1st𝑇))‘𝑛) = ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)))
7978oveq1d 7446 . . . . . . . . . . . . 13 (𝑛 = ((2nd ‘(1st𝑇))‘1) → (((1st ‘(1st𝑇))‘𝑛) + 1) = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1))
8079adantl 481 . . . . . . . . . . . 12 ((𝜑𝑛 = ((2nd ‘(1st𝑇))‘1)) → (((1st ‘(1st𝑇))‘𝑛) + 1) = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1))
81 elrabi 3690 . . . . . . . . . . . . . . . . . . 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 2857 . . . . . . . . . . . . . . . . . 18 (𝑇𝑆𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
83 xp1st 8045 . . . . . . . . . . . . . . . . . 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 8045 . . . . . . . . . . . . . . . . 17 ((1st𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
86 elmapi 8888 . . . . . . . . . . . . . . . . 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 8046 . . . . . . . . . . . . . . . . . . . 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 6837 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = (2nd ‘(1st𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
9250, 91elab 3681 . . . . . . . . . . . . . . . . . . 19 ((2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
9390, 92sylib 218 . . . . . . . . . . . . . . . . . 18 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
94 f1of 6849 . . . . . . . . . . . . . . . . . 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 12919 . . . . . . . . . . . . . . . . . . 19 ℕ = (ℤ‘1)
971, 96eleqtrdi 2849 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ (ℤ‘1))
98 eluzfz1 13568 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ (ℤ‘1) → 1 ∈ (1...𝑁))
9997, 98syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → 1 ∈ (1...𝑁))
10095, 99ffvelcdmd 7105 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd ‘(1st𝑇))‘1) ∈ (1...𝑁))
10187, 100ffvelcdmd 7105 . . . . . . . . . . . . . . 15 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) ∈ (0..^𝐾))
102 elfzonn0 13744 . . . . . . . . . . . . . . 15 (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) ∈ (0..^𝐾) → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) ∈ ℕ0)
103 peano2nn0 12564 . . . . . . . . . . . . . . 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 13737 . . . . . . . . . . . . . . . 16 (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) ∈ (0..^𝐾) ↔ (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) ∈ ℕ0𝐾 ∈ ℕ ∧ ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) < 𝐾))
106101, 105sylib 218 . . . . . . . . . . . . . . 15 (𝜑 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) ∈ ℕ0𝐾 ∈ ℕ ∧ ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) < 𝐾))
107106simp2d 1142 . . . . . . . . . . . . . 14 (𝜑𝐾 ∈ ℕ)
108104nn0red 12586 . . . . . . . . . . . . . . 15 (𝜑 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) ∈ ℝ)
109107nnred 12279 . . . . . . . . . . . . . . 15 (𝜑𝐾 ∈ ℝ)
110 elfzolt2 13705 . . . . . . . . . . . . . . . . 17 (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) ∈ (0..^𝐾) → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) < 𝐾)
111101, 110syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) < 𝐾)
112101, 102syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) ∈ ℕ0)
113112nn0zd 12637 . . . . . . . . . . . . . . . . 17 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) ∈ ℤ)
114107nnzd 12638 . . . . . . . . . . . . . . . . 17 (𝜑𝐾 ∈ ℤ)
115 zltp1le 12665 . . . . . . . . . . . . . . . . 17 ((((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) ∈ ℤ ∧ 𝐾 ∈ ℤ) → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) < 𝐾 ↔ (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) ≤ 𝐾))
116113, 114, 115syl2anc 584 . . . . . . . . . . . . . . . 16 (𝜑 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) < 𝐾 ↔ (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) ≤ 𝐾))
117111, 116mpbid 232 . . . . . . . . . . . . . . 15 (𝜑 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) ≤ 𝐾)
118 fvex 6920 . . . . . . . . . . . . . . . . . 18 ((2nd ‘(1st𝑇))‘1) ∈ V
119 eleq1 2827 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = ((2nd ‘(1st𝑇))‘1) → (𝑛 ∈ (1...𝑁) ↔ ((2nd ‘(1st𝑇))‘1) ∈ (1...𝑁)))
120119anbi2d 630 . . . . . . . . . . . . . . . . . . 19 (𝑛 = ((2nd ‘(1st𝑇))‘1) → ((𝜑𝑛 ∈ (1...𝑁)) ↔ (𝜑 ∧ ((2nd ‘(1st𝑇))‘1) ∈ (1...𝑁))))
121 fveq2 6907 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = ((2nd ‘(1st𝑇))‘1) → (𝑝𝑛) = (𝑝‘((2nd ‘(1st𝑇))‘1)))
122121neeq1d 2998 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = ((2nd ‘(1st𝑇))‘1) → ((𝑝𝑛) ≠ 𝐾 ↔ (𝑝‘((2nd ‘(1st𝑇))‘1)) ≠ 𝐾))
123122rexbidv 3177 . . . . . . . . . . . . . . . . . . 19 (𝑛 = ((2nd ‘(1st𝑇))‘1) → (∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 𝐾 ↔ ∃𝑝 ∈ ran 𝐹(𝑝‘((2nd ‘(1st𝑇))‘1)) ≠ 𝐾))
124120, 123imbi12d 344 . . . . . . . . . . . . . . . . . 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)))
125118, 124, 5vtocl 3558 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((2nd ‘(1st𝑇))‘1) ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘((2nd ‘(1st𝑇))‘1)) ≠ 𝐾)
126100, 125mpdan 687 . . . . . . . . . . . . . . . 16 (𝜑 → ∃𝑝 ∈ ran 𝐹(𝑝‘((2nd ‘(1st𝑇))‘1)) ≠ 𝐾)
127 fveq1 6906 . . . . . . . . . . . . . . . . . . . . . 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)))
12887ffnd 6738 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (1st ‘(1st𝑇)) Fn (1...𝑁))
129128adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1st ‘(1st𝑇)) Fn (1...𝑁))
130 1ex 11255 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1 ∈ V
131 fnconstg 6797 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (1 ∈ V → (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))))
132130, 131ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1)))
133 c0ex 11253 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 0 ∈ V
134 fnconstg 6797 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (0 ∈ V → (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
135133, 134ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))
136132, 135pm3.2i 470 . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
137 dff1o3 6855 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁) ∧ Fun (2nd ‘(1st𝑇))))
138137simprbi 496 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun (2nd ‘(1st𝑇)))
139 imain 6653 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (Fun (2nd ‘(1st𝑇)) → ((2nd ‘(1st𝑇)) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) ∩ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))))
14093, 138, 1393syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) ∩ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))))
141 nn0p1nn 12563 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℕ)
1428, 141syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℕ)
143142nnred 12279 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℝ)
144143ltp1d 12196 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) < ((𝑦 + 1) + 1))
145 fzdisj 13588 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑦 + 1) < ((𝑦 + 1) + 1) → ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅)
146145imaeq2d 6080 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑦 + 1) < ((𝑦 + 1) + 1) → ((2nd ‘(1st𝑇)) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ((2nd ‘(1st𝑇)) “ ∅))
147 ima0 6097 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((2nd ‘(1st𝑇)) “ ∅) = ∅
148146, 147eqtrdi 2791 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2796 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) ∩ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ∅)
151 fnun 6683 . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
152136, 150, 151sylancr 587 . . . . . . . . . . . . . . . . . . . . . . . . 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 6172 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((2nd ‘(1st𝑇)) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) ∪ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
154142peano2nnd 12281 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ ℕ)
155154, 96eleqtrdi 2849 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ (ℤ‘1))
1561nncnd 12280 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝜑𝑁 ∈ ℂ)
157 npcan1 11686 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
158156, 157syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
159158adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) = 𝑁)
160 elfzuz3 13558 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈ (ℤ𝑦))
161 eluzp1p1 12904 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑁 − 1) ∈ (ℤ𝑦) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑦 + 1)))
162160, 161syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑦 + 1)))
163162adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑦 + 1)))
164159, 163eqeltrrd 2840 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ‘(𝑦 + 1)))
165 fzsplit2 13586 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑦 + 1) + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ‘(𝑦 + 1))) → (1...𝑁) = ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))
166155, 164, 165syl2an2 686 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) = ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))
167166imaeq2d 6080 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = ((2nd ‘(1st𝑇)) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))))
168 f1ofo 6856 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁))
169 foima 6826 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
17093, 168, 1693syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
171170adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
172167, 171eqtr3d 2777 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) = (1...𝑁))
173153, 172eqtr3id 2789 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) ∪ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = (1...𝑁))
174173fneq2d 6663 . . . . . . . . . . . . . . . . . . . . . . . . 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)))
175152, 174mpbid 232 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁))
17648a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) ∈ V)
177 inidm 4235 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
178 eqidd 2736 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ ((2nd ‘(1st𝑇))‘1) ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) = ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)))
179 f1ofn 6850 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)) Fn (1...𝑁))
18093, 179syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑 → (2nd ‘(1st𝑇)) Fn (1...𝑁))
181180adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (2nd ‘(1st𝑇)) Fn (1...𝑁))
182 fzss2 13601 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑁 ∈ (ℤ‘(𝑦 + 1)) → (1...(𝑦 + 1)) ⊆ (1...𝑁))
183164, 182syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1...(𝑦 + 1)) ⊆ (1...𝑁))
184142, 96eleqtrdi 2849 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ (ℤ‘1))
185 eluzfz1 13568 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑦 + 1) ∈ (ℤ‘1) → 1 ∈ (1...(𝑦 + 1)))
186184, 185syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ (0...(𝑁 − 1)) → 1 ∈ (1...(𝑦 + 1)))
187186adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 1 ∈ (1...(𝑦 + 1)))
188 fnfvima 7253 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((2nd ‘(1st𝑇)) Fn (1...𝑁) ∧ (1...(𝑦 + 1)) ⊆ (1...𝑁) ∧ 1 ∈ (1...(𝑦 + 1))) → ((2nd ‘(1st𝑇))‘1) ∈ ((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))))
189181, 183, 187, 188syl3anc 1370 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇))‘1) ∈ ((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))))
190 fvun1 7000 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
191132, 135, 190mp3an12 1450 . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
192150, 189, 191syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
193130fvconst2 7224 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((2nd ‘(1st𝑇))‘1) ∈ ((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) → ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1})‘((2nd ‘(1st𝑇))‘1)) = 1)
194189, 193syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1})‘((2nd ‘(1st𝑇))‘1)) = 1)
195192, 194eqtrd 2775 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘1)) = 1)
196195adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 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)
197129, 175, 176, 176, 177, 178, 196ofval 7708 . . . . . . . . . . . . . . . . . . . . . . 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))
198100, 197mpidan 689 . . . . . . . . . . . . . . . . . . . . . 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))
199127, 198sylan9eqr 2797 . . . . . . . . . . . . . . . . . . . . 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))
200199adantllr 719 . . . . . . . . . . . . . . . . . . . 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))
201 fveq2 6907 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑡 = 𝑇 → (2nd𝑡) = (2nd𝑇))
202201breq2d 5160 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑡 = 𝑇 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑇)))
203202ifbid 4554 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑡 = 𝑇 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)))
204 2fveq3 6912 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑡 = 𝑇 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
205 2fveq3 6912 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑡 = 𝑇 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑇)))
206205imaeq1d 6079 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑗)))
207206xpeq1d 5718 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}))
208205imaeq1d 6079 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)))
209208xpeq1d 5718 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
210207, 209uneq12d 4179 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑡 = 𝑇 → ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))
211204, 210oveq12d 7449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (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}))))
212203, 211csbeq12dv 3917 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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}))))
213212mpteq2dv 5250 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})))))
214213eqeq2d 2746 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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}))))))
215214, 2elrab2 3698 . . . . . . . . . . . . . . . . . . . . . . . . . . 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}))))))
216215simprbi 496 . . . . . . . . . . . . . . . . . . . . . . . . . 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})))))
2174, 216syl 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})))))
218217rneqd 5952 . . . . . . . . . . . . . . . . . . . . . . . 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})))))
219218eleq2d 2825 . . . . . . . . . . . . . . . . . . . . . . 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}))))))
220 eqid 2735 . . . . . . . . . . . . . . . . . . . . . . . 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}))))
221 ovex 7464 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V
222221csbex 5317 . . . . . . . . . . . . . . . . . . . . . . . 24 if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V
223220, 222elrnmpti 5976 . . . . . . . . . . . . . . . . . . . . . . 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}))))
224219, 223bitrdi 287 . . . . . . . . . . . . . . . . . . . . . 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})))))
2256adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (2nd𝑇) = 0)
226 elfzle1 13564 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 ∈ (0...(𝑁 − 1)) → 0 ≤ 𝑦)
227226adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 0 ≤ 𝑦)
228225, 227eqbrtrd 5170 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (2nd𝑇) ≤ 𝑦)
229 0re 11261 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 0 ∈ ℝ
2306, 229eqeltrdi 2847 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑 → (2nd𝑇) ∈ ℝ)
231 lenlt 11337 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((2nd𝑇) ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((2nd𝑇) ≤ 𝑦 ↔ ¬ 𝑦 < (2nd𝑇)))
232230, 9, 231syl2an 596 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd𝑇) ≤ 𝑦 ↔ ¬ 𝑦 < (2nd𝑇)))
233228, 232mpbid 232 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ¬ 𝑦 < (2nd𝑇))
234233iffalsed 4542 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = (𝑦 + 1))
235234csbeq1d 3912 . . . . . . . . . . . . . . . . . . . . . . . . 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}))))
236 ovex 7464 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 + 1) ∈ V
237 oveq2 7439 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 = (𝑦 + 1) → (1...𝑗) = (1...(𝑦 + 1)))
238237imaeq2d 6080 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 = (𝑦 + 1) → ((2nd ‘(1st𝑇)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))))
239238xpeq1d 5718 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 = (𝑦 + 1) → (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}))
240 oveq1 7438 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑗 = (𝑦 + 1) → (𝑗 + 1) = ((𝑦 + 1) + 1))
241240oveq1d 7446 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 = (𝑦 + 1) → ((𝑗 + 1)...𝑁) = (((𝑦 + 1) + 1)...𝑁))
242241imaeq2d 6080 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 = (𝑦 + 1) → ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
243242xpeq1d 5718 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 = (𝑦 + 1) → (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))
244239, 243uneq12d 4179 . . . . . . . . . . . . . . . . . . . . . . . . . . 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})))
245244oveq2d 7447 . . . . . . . . . . . . . . . . . . . . . . . . . 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}))))
246236, 245csbie 3944 . . . . . . . . . . . . . . . . . . . . . . . . 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})))
247235, 246eqtrdi 2791 . . . . . . . . . . . . . . . . . . . . . . . 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}))))
248247eqeq2d 2746 . . . . . . . . . . . . . . . . . . . . . . 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})))))
249248rexbidva 3175 . . . . . . . . . . . . . . . . . . . . . 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})))))
250224, 249bitrd 279 . . . . . . . . . . . . . . . . . . . . 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})))))
251250biimpa 476 . . . . . . . . . . . . . . . . . . . 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}))))
252200, 251r19.29a 3160 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑝 ∈ ran 𝐹) → (𝑝‘((2nd ‘(1st𝑇))‘1)) = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1))
253 eqtr3 2761 . . . . . . . . . . . . . . . . . . . 20 (((𝑝‘((2nd ‘(1st𝑇))‘1)) = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) ∧ 𝐾 = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1)) → (𝑝‘((2nd ‘(1st𝑇))‘1)) = 𝐾)
254253ex 412 . . . . . . . . . . . . . . . . . . 19 ((𝑝‘((2nd ‘(1st𝑇))‘1)) = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) → (𝐾 = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) → (𝑝‘((2nd ‘(1st𝑇))‘1)) = 𝐾))
255252, 254syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑝 ∈ ran 𝐹) → (𝐾 = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) → (𝑝‘((2nd ‘(1st𝑇))‘1)) = 𝐾))
256255necon3d 2959 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ ran 𝐹) → ((𝑝‘((2nd ‘(1st𝑇))‘1)) ≠ 𝐾𝐾 ≠ (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1)))
257256rexlimdva 3153 . . . . . . . . . . . . . . . 16 (𝜑 → (∃𝑝 ∈ ran 𝐹(𝑝‘((2nd ‘(1st𝑇))‘1)) ≠ 𝐾𝐾 ≠ (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1)))
258126, 257mpd 15 . . . . . . . . . . . . . . 15 (𝜑𝐾 ≠ (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1))
259108, 109, 117, 258leneltd 11413 . . . . . . . . . . . . . 14 (𝜑 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) < 𝐾)
260 elfzo0 13737 . . . . . . . . . . . . . 14 ((((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) ∈ (0..^𝐾) ↔ ((((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) ∈ ℕ0𝐾 ∈ ℕ ∧ (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) < 𝐾))
261104, 107, 259, 260syl3anbrc 1342 . . . . . . . . . . . . 13 (𝜑 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) ∈ (0..^𝐾))
262261adantr 480 . . . . . . . . . . . 12 ((𝜑𝑛 = ((2nd ‘(1st𝑇))‘1)) → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) ∈ (0..^𝐾))
26380, 262eqeltrd 2839 . . . . . . . . . . 11 ((𝜑𝑛 = ((2nd ‘(1st𝑇))‘1)) → (((1st ‘(1st𝑇))‘𝑛) + 1) ∈ (0..^𝐾))
264263adantlr 715 . . . . . . . . . 10 (((𝜑𝑛 ∈ (1...𝑁)) ∧ 𝑛 = ((2nd ‘(1st𝑇))‘1)) → (((1st ‘(1st𝑇))‘𝑛) + 1) ∈ (0..^𝐾))
26587ffvelcdmda 7104 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ (0..^𝐾))
266 elfzonn0 13744 . . . . . . . . . . . . . . 15 (((1st ‘(1st𝑇))‘𝑛) ∈ (0..^𝐾) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℕ0)
267265, 266syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℕ0)
268267nn0cnd 12587 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℂ)
269268addridd 11459 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑇))‘𝑛) + 0) = ((1st ‘(1st𝑇))‘𝑛))
270269, 265eqeltrd 2839 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑇))‘𝑛) + 0) ∈ (0..^𝐾))
271270adantr 480 . . . . . . . . . 10 (((𝜑𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st𝑇))‘1)) → (((1st ‘(1st𝑇))‘𝑛) + 0) ∈ (0..^𝐾))
27275, 77, 264, 271ifbothda 4569 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0)) ∈ (0..^𝐾))
273272fmpttd 7135 . . . . . . . 8 (𝜑 → (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))):(1...𝑁)⟶(0..^𝐾))
274 ovex 7464 . . . . . . . . 9 (0..^𝐾) ∈ V
275274, 48elmap 8910 . . . . . . . 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))
276273, 275sylibr 234 . . . . . . 7 (𝜑 → (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))) ∈ ((0..^𝐾) ↑m (1...𝑁)))
277 simpr 484 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (1...(𝑁 − 1))) → 𝑛 ∈ (1...(𝑁 − 1)))
278 1z 12645 . . . . . . . . . . . . . . . . 17 1 ∈ ℤ
27913, 278jctil 519 . . . . . . . . . . . . . . . 16 (𝜑 → (1 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ))
280 elfzelz 13561 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ ℤ)
281280, 278jctir 520 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (1...(𝑁 − 1)) → (𝑛 ∈ ℤ ∧ 1 ∈ ℤ))
282 fzaddel 13595 . . . . . . . . . . . . . . . 16 (((1 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) ∧ (𝑛 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑛 ∈ (1...(𝑁 − 1)) ↔ (𝑛 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1))))
283279, 281, 282syl2an 596 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (1...(𝑁 − 1))) → (𝑛 ∈ (1...(𝑁 − 1)) ↔ (𝑛 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1))))
284277, 283mpbid 232 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...(𝑁 − 1))) → (𝑛 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1)))
285158oveq2d 7447 . . . . . . . . . . . . . . 15 (𝜑 → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 + 1)...𝑁))
286285adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...(𝑁 − 1))) → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 + 1)...𝑁))
287284, 286eleqtrd 2841 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...(𝑁 − 1))) → (𝑛 + 1) ∈ ((1 + 1)...𝑁))
288287ralrimiva 3144 . . . . . . . . . . . 12 (𝜑 → ∀𝑛 ∈ (1...(𝑁 − 1))(𝑛 + 1) ∈ ((1 + 1)...𝑁))
289 simpr 484 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ ((1 + 1)...𝑁)) → 𝑦 ∈ ((1 + 1)...𝑁))
290 peano2z 12656 . . . . . . . . . . . . . . . . . . 19 (1 ∈ ℤ → (1 + 1) ∈ ℤ)
291278, 290ax-mp 5 . . . . . . . . . . . . . . . . . 18 (1 + 1) ∈ ℤ
29211, 291jctil 519 . . . . . . . . . . . . . . . . 17 (𝜑 → ((1 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ))
293 elfzelz 13561 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ((1 + 1)...𝑁) → 𝑦 ∈ ℤ)
294293, 278jctir 520 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ((1 + 1)...𝑁) → (𝑦 ∈ ℤ ∧ 1 ∈ ℤ))
295 fzsubel 13597 . . . . . . . . . . . . . . . . 17 ((((1 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑦 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑦 ∈ ((1 + 1)...𝑁) ↔ (𝑦 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1))))
296292, 294, 295syl2an 596 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ ((1 + 1)...𝑁)) → (𝑦 ∈ ((1 + 1)...𝑁) ↔ (𝑦 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1))))
297289, 296mpbid 232 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ ((1 + 1)...𝑁)) → (𝑦 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1)))
298 ax-1cn 11211 . . . . . . . . . . . . . . . . 17 1 ∈ ℂ
299298, 298pncan3oi 11522 . . . . . . . . . . . . . . . 16 ((1 + 1) − 1) = 1
300299oveq1i 7441 . . . . . . . . . . . . . . 15 (((1 + 1) − 1)...(𝑁 − 1)) = (1...(𝑁 − 1))
301297, 300eleqtrdi 2849 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ ((1 + 1)...𝑁)) → (𝑦 − 1) ∈ (1...(𝑁 − 1)))
302293zcnd 12721 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ((1 + 1)...𝑁) → 𝑦 ∈ ℂ)
303 elfznn 13590 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ ℕ)
304303nncnd 12280 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ ℂ)
305 subadd2 11510 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑛 ∈ ℂ) → ((𝑦 − 1) = 𝑛 ↔ (𝑛 + 1) = 𝑦))
306298, 305mp3an2 1448 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℂ) → ((𝑦 − 1) = 𝑛 ↔ (𝑛 + 1) = 𝑦))
307306bicomd 223 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℂ) → ((𝑛 + 1) = 𝑦 ↔ (𝑦 − 1) = 𝑛))
308 eqcom 2742 . . . . . . . . . . . . . . . . . 18 (𝑦 = (𝑛 + 1) ↔ (𝑛 + 1) = 𝑦)
309 eqcom 2742 . . . . . . . . . . . . . . . . . 18 (𝑛 = (𝑦 − 1) ↔ (𝑦 − 1) = 𝑛)
310307, 308, 3093bitr4g 314 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℂ) → (𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1)))
311302, 304, 310syl2an 596 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ((1 + 1)...𝑁) ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1)))
312311ralrimiva 3144 . . . . . . . . . . . . . . 15 (𝑦 ∈ ((1 + 1)...𝑁) → ∀𝑛 ∈ (1...(𝑁 − 1))(𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1)))
313312adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ ((1 + 1)...𝑁)) → ∀𝑛 ∈ (1...(𝑁 − 1))(𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1)))
314 reu6i 3737 . . . . . . . . . . . . . 14 (((𝑦 − 1) ∈ (1...(𝑁 − 1)) ∧ ∀𝑛 ∈ (1...(𝑁 − 1))(𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1))) → ∃!𝑛 ∈ (1...(𝑁 − 1))𝑦 = (𝑛 + 1))
315301, 313, 314syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ ((1 + 1)...𝑁)) → ∃!𝑛 ∈ (1...(𝑁 − 1))𝑦 = (𝑛 + 1))
316315ralrimiva 3144 . . . . . . . . . . . 12 (𝜑 → ∀𝑦 ∈ ((1 + 1)...𝑁)∃!𝑛 ∈ (1...(𝑁 − 1))𝑦 = (𝑛 + 1))
317 eqid 2735 . . . . . . . . . . . . 13 (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) = (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1))
318317f1ompt 7131 . . . . . . . . . . . 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)))
319288, 316, 318sylanbrc 583 . . . . . . . . . . 11 (𝜑 → (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)):(1...(𝑁 − 1))–1-1-onto→((1 + 1)...𝑁))
320 f1osng 6890 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ ∧ 1 ∈ V) → {⟨𝑁, 1⟩}:{𝑁}–1-1-onto→{1})
3211, 130, 320sylancl 586 . . . . . . . . . . 11 (𝜑 → {⟨𝑁, 1⟩}:{𝑁}–1-1-onto→{1})
32214, 16ltnled 11406 . . . . . . . . . . . . . 14 (𝜑 → ((𝑁 − 1) < 𝑁 ↔ ¬ 𝑁 ≤ (𝑁 − 1)))
32320, 322mpbid 232 . . . . . . . . . . . . 13 (𝜑 → ¬ 𝑁 ≤ (𝑁 − 1))
324 elfzle2 13565 . . . . . . . . . . . . 13 (𝑁 ∈ (1...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1))
325323, 324nsyl 140 . . . . . . . . . . . 12 (𝜑 → ¬ 𝑁 ∈ (1...(𝑁 − 1)))
326 disjsn 4716 . . . . . . . . . . . 12 (((1...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (1...(𝑁 − 1)))
327325, 326sylibr 234 . . . . . . . . . . 11 (𝜑 → ((1...(𝑁 − 1)) ∩ {𝑁}) = ∅)
328 1re 11259 . . . . . . . . . . . . . . . 16 1 ∈ ℝ
329328ltp1i 12170 . . . . . . . . . . . . . . 15 1 < (1 + 1)
330291zrei 12617 . . . . . . . . . . . . . . . 16 (1 + 1) ∈ ℝ
331328, 330ltnlei 11380 . . . . . . . . . . . . . . 15 (1 < (1 + 1) ↔ ¬ (1 + 1) ≤ 1)
332329, 331mpbi 230 . . . . . . . . . . . . . 14 ¬ (1 + 1) ≤ 1
333 elfzle1 13564 . . . . . . . . . . . . . 14 (1 ∈ ((1 + 1)...𝑁) → (1 + 1) ≤ 1)
334332, 333mto 197 . . . . . . . . . . . . 13 ¬ 1 ∈ ((1 + 1)...𝑁)
335 disjsn 4716 . . . . . . . . . . . . 13 ((((1 + 1)...𝑁) ∩ {1}) = ∅ ↔ ¬ 1 ∈ ((1 + 1)...𝑁))
336334, 335mpbir 231 . . . . . . . . . . . 12 (((1 + 1)...𝑁) ∩ {1}) = ∅
337336a1i 11 . . . . . . . . . . 11 (𝜑 → (((1 + 1)...𝑁) ∩ {1}) = ∅)
338 f1oun 6868 . . . . . . . . . . 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}))
339319, 321, 327, 337, 338syl22anc 839 . . . . . . . . . 10 (𝜑 → ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {⟨𝑁, 1⟩}):((1...(𝑁 − 1)) ∪ {𝑁})–1-1-onto→(((1 + 1)...𝑁) ∪ {1}))
340130a1i 11 . . . . . . . . . . . . 13 (𝜑 → 1 ∈ V)
341158, 97eqeltrd 2839 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘1))
342 uzid 12891 . . . . . . . . . . . . . . . . 17 ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)))
343 peano2uz 12941 . . . . . . . . . . . . . . . . 17 ((𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
34413, 342, 3433syl 18 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
345158, 344eqeltrrd 2840 . . . . . . . . . . . . . . 15 (𝜑𝑁 ∈ (ℤ‘(𝑁 − 1)))
346 fzsplit2 13586 . . . . . . . . . . . . . . 15 ((((𝑁 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ‘(𝑁 − 1))) → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
347341, 345, 346syl2anc 584 . . . . . . . . . . . . . 14 (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
348158oveq1d 7446 . . . . . . . . . . . . . . . 16 (𝜑 → (((𝑁 − 1) + 1)...𝑁) = (𝑁...𝑁))
349 fzsn 13603 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
35011, 349syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑁...𝑁) = {𝑁})
351348, 350eqtrd 2775 . . . . . . . . . . . . . . 15 (𝜑 → (((𝑁 − 1) + 1)...𝑁) = {𝑁})
352351uneq2d 4178 . . . . . . . . . . . . . 14 (𝜑 → ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = ((1...(𝑁 − 1)) ∪ {𝑁}))
353347, 352eqtr2d 2776 . . . . . . . . . . . . 13 (𝜑 → ((1...(𝑁 − 1)) ∪ {𝑁}) = (1...𝑁))
354 iftrue 4537 . . . . . . . . . . . . . 14 (𝑛 = 𝑁 → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = 1)
355354adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑛 = 𝑁) → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = 1)
3561, 340, 353, 355fmptapd 7191 . . . . . . . . . . . 12 (𝜑 → ((𝑛 ∈ (1...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ∪ {⟨𝑁, 1⟩}) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))
357 eleq1 2827 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑁 → (𝑛 ∈ (1...(𝑁 − 1)) ↔ 𝑁 ∈ (1...(𝑁 − 1))))
358357notbid 318 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑁 → (¬ 𝑛 ∈ (1...(𝑁 − 1)) ↔ ¬ 𝑁 ∈ (1...(𝑁 − 1))))
359325, 358syl5ibrcom 247 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑛 = 𝑁 → ¬ 𝑛 ∈ (1...(𝑁 − 1))))
360359necon2ad 2953 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛𝑁))
361360imp 406 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (1...(𝑁 − 1))) → 𝑛𝑁)
362 ifnefalse 4543 . . . . . . . . . . . . . . 15 (𝑛𝑁 → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = (𝑛 + 1))
363361, 362syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...(𝑁 − 1))) → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = (𝑛 + 1))
364363mpteq2dva 5248 . . . . . . . . . . . . 13 (𝜑 → (𝑛 ∈ (1...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)))
365364uneq1d 4177 . . . . . . . . . . . 12 (𝜑 → ((𝑛 ∈ (1...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ∪ {⟨𝑁, 1⟩}) = ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {⟨𝑁, 1⟩}))
366356, 365eqtr3d 2777 . . . . . . . . . . 11 (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {⟨𝑁, 1⟩}))
367347, 352eqtrd 2775 . . . . . . . . . . 11 (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁}))
368 uzid 12891 . . . . . . . . . . . . . 14 (1 ∈ ℤ → 1 ∈ (ℤ‘1))
369 peano2uz 12941 . . . . . . . . . . . . . 14 (1 ∈ (ℤ‘1) → (1 + 1) ∈ (ℤ‘1))
370278, 368, 369mp2b 10 . . . . . . . . . . . . 13 (1 + 1) ∈ (ℤ‘1)
371 fzsplit2 13586 . . . . . . . . . . . . 13 (((1 + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ‘1)) → (1...𝑁) = ((1...1) ∪ ((1 + 1)...𝑁)))
372370, 97, 371sylancr 587 . . . . . . . . . . . 12 (𝜑 → (1...𝑁) = ((1...1) ∪ ((1 + 1)...𝑁)))
373 fzsn 13603 . . . . . . . . . . . . . . 15 (1 ∈ ℤ → (1...1) = {1})
374278, 373ax-mp 5 . . . . . . . . . . . . . 14 (1...1) = {1}
375374uneq1i 4174 . . . . . . . . . . . . 13 ((1...1) ∪ ((1 + 1)...𝑁)) = ({1} ∪ ((1 + 1)...𝑁))
376375equncomi 4170 . . . . . . . . . . . 12 ((1...1) ∪ ((1 + 1)...𝑁)) = (((1 + 1)...𝑁) ∪ {1})
377372, 376eqtrdi 2791 . . . . . . . . . . 11 (𝜑 → (1...𝑁) = (((1 + 1)...𝑁) ∪ {1}))
378366, 367, 377f1oeq123d 6843 . . . . . . . . . 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})))
379339, 378mpbird 257 . . . . . . . . 9 (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)–1-1-onto→(1...𝑁))
380 f1oco 6872 . . . . . . . . 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))
38193, 379, 380syl2anc 584 . . . . . . . 8 (𝜑 → ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–1-1-onto→(1...𝑁))
382 f1oeq1 6837 . . . . . . . . 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)))
38352, 382elab 3681 . . . . . . . 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))
384381, 383sylibr 234 . . . . . . 7 (𝜑 → ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
385276, 384opelxpd 5728 . . . . . 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)}))
3861nnnn0d 12585 . . . . . . 7 (𝜑𝑁 ∈ ℕ0)
387 nn0fz0 13662 . . . . . . 7 (𝑁 ∈ ℕ0𝑁 ∈ (0...𝑁))
388386, 387sylib 218 . . . . . 6 (𝜑𝑁 ∈ (0...𝑁))
389385, 388opelxpd 5728 . . . . 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)))
390 elrab3t 3694 . . . . 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}))))))
39173, 389, 390syl2anc 584 . . . 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}))))))
3927, 391mpbird 257 . . 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}))))})
393392, 2eleqtrrdi 2850 . 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)
394 fveqeq2 6916 . . . . . 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))
39527, 394syl5ibcom 245 . . . . 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))
3961nnne0d 12314 . . . . . 6 (𝜑𝑁 ≠ 0)
397 neeq1 3001 . . . . . 6 ((2nd𝑇) = 𝑁 → ((2nd𝑇) ≠ 0 ↔ 𝑁 ≠ 0))
398396, 397syl5ibrcom 247 . . . . 5 (𝜑 → ((2nd𝑇) = 𝑁 → (2nd𝑇) ≠ 0))
399395, 398syld 47 . . . 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))
400399necon2d 2961 . . 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))
4016, 400mpd 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)
402 neeq1 3001 . . 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))
403402rspcev 3622 . 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)
404393, 401, 403syl2anc 584 1 (𝜑 → ∃𝑧𝑆 𝑧𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086  wal 1535   = wceq 1537  wcel 2106  {cab 2712  wne 2938  wral 3059  wrex 3068  ∃!wreu 3376  {crab 3433  Vcvv 3478  csb 3908  cun 3961  cin 3962  wss 3963  c0 4339  ifcif 4531  {csn 4631  cop 4637   class class class wbr 5148  cmpt 5231   × cxp 5687  ccnv 5688  ran crn 5690  cima 5692  ccom 5693  Fun wfun 6557   Fn wfn 6558  wf 6559  ontowfo 6561  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  f cof 7695  1st c1st 8011  2nd c2nd 8012  m cmap 8865  cc 11151  cr 11152  0cc0 11153  1c1 11154   + caddc 11156   < clt 11293  cle 11294  cmin 11490  cn 12264  0cn0 12524  cz 12611  cuz 12876  ...cfz 13544  ..^cfzo 13691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692
This theorem is referenced by:  poimirlem18  37625
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