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Theorem poimirlem17 33397
Description: Lemma for poimir 33413 establishing existence for poimirlem18 33398. (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 33396 . . . 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 12417 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℕ0)
98nn0red 11337 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℝ)
109adantl 482 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ∈ ℝ)
111nnzd 11466 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ ℤ)
12 peano2zm 11405 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
1311, 12syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑁 − 1) ∈ ℤ)
1413zred 11467 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑁 − 1) ∈ ℝ)
1514adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) ∈ ℝ)
161nnred 11020 . . . . . . . . . . . . . . . 16 (𝜑𝑁 ∈ ℝ)
1716adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℝ)
18 elfzle2 12330 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ≤ (𝑁 − 1))
1918adantl 482 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ≤ (𝑁 − 1))
2016ltm1d 10941 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑁 − 1) < 𝑁)
2120adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) < 𝑁)
2210, 15, 17, 19, 21lelttrd 10180 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 < 𝑁)
2322adantlr 750 . . . . . . . . . . . . 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 6178 . . . . . . . . . . . . . . 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 4923 . . . . . . . . . . . . . . . 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 7166 . . . . . . . . . . . . . . . 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 694 . . . . . . . . . . . . . . 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 2676 . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . 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 4672 . . . . . . . . . . . 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 4085 . . . . . . . . . . 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 3533 . . . . . . . . . 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 3198 . . . . . . . . . . . . 13 𝑦 ∈ V
34 oveq2 6643 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑦 → (1...𝑗) = (1...𝑦))
3534imaeq2d 5454 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑦 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑡)) “ (1...𝑦)))
3635xpeq1d 5128 . . . . . . . . . . . . . . 15 (𝑗 = 𝑦 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑡)) “ (1...𝑦)) × {1}))
37 oveq1 6642 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑦 → (𝑗 + 1) = (𝑦 + 1))
3837oveq1d 6650 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑦 → ((𝑗 + 1)...𝑁) = ((𝑦 + 1)...𝑁))
3938imaeq2d 5454 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑦 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑡)) “ ((𝑦 + 1)...𝑁)))
4039xpeq1d 5128 . . . . . . . . . . . . . . 15 (𝑗 = 𝑦 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑡)) “ ((𝑦 + 1)...𝑁)) × {0}))
4136, 40uneq12d 3760 . . . . . . . . . . . . . 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 6651 . . . . . . . . . . . . 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 3552 . . . . . . . . . . . 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 6178 . . . . . . . . . . . . . . 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 6182 . . . . . . . . . . . . . 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 7165 . . . . . . . . . . . . . . . . 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 694 . . . . . . . . . . . . . . . 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 6182 . . . . . . . . . . . . . . 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 6663 . . . . . . . . . . . . . . . . 17 (1...𝑁) ∈ V
5049mptex 6471 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))) ∈ V
51 fvex 6188 . . . . . . . . . . . . . . . . 17 (2nd ‘(1st𝑇)) ∈ V
5249mptex 6471 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ∈ V
5351, 52coex 7103 . . . . . . . . . . . . . . . 16 ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) ∈ V
5450, 53op1st 7161 . . . . . . . . . . . . . . 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 2670 . . . . . . . . . . . . . 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 2676 . . . . . . . . . . . . 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 2676 . . . . . . . . . . . . . . . . . 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 6182 . . . . . . . . . . . . . . . . 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 7162 . . . . . . . . . . . . . . . . 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 2670 . . . . . . . . . . . . . . . 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 5453 . . . . . . . . . . . . . . 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 5128 . . . . . . . . . . . . . 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 5453 . . . . . . . . . . . . . . 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 5128 . . . . . . . . . . . . . 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 3760 . . . . . . . . . . . . 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 6653 . . . . . . . . . . . 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 2666 . . . . . . . . . . 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 481 . . . . . . . . . 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 2654 . . . . . . . . 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 4735 . . . . . . . 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 2630 . . . . . . 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 450 . . . . . 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 1853 . . . . 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 6643 . . . . . . . . . . 11 (1 = if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0) → (((1st ‘(1st𝑇))‘𝑛) + 1) = (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0)))
7574eleq1d 2684 . . . . . . . . . 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 6643 . . . . . . . . . . 11 (0 = if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0) → (((1st ‘(1st𝑇))‘𝑛) + 0) = (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0)))
7776eleq1d 2684 . . . . . . . . . 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 6178 . . . . . . . . . . . . . 14 (𝑛 = ((2nd ‘(1st𝑇))‘1) → ((1st ‘(1st𝑇))‘𝑛) = ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)))
7978oveq1d 6650 . . . . . . . . . . . . 13 (𝑛 = ((2nd ‘(1st𝑇))‘1) → (((1st ‘(1st𝑇))‘𝑛) + 1) = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1))
8079adantl 482 . . . . . . . . . . . 12 ((𝜑𝑛 = ((2nd ‘(1st𝑇))‘1)) → (((1st ‘(1st𝑇))‘𝑛) + 1) = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1))
81 elrabi 3353 . . . . . . . . . . . . . . . . . . 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 2717 . . . . . . . . . . . . . . . . . 18 (𝑇𝑆𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
83 xp1st 7183 . . . . . . . . . . . . . . . . . 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 7183 . . . . . . . . . . . . . . . . 17 ((1st𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
86 elmapi 7864 . . . . . . . . . . . . . . . . 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 7184 . . . . . . . . . . . . . . . . . . . 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 6114 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = (2nd ‘(1st𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
9251, 91elab 3344 . . . . . . . . . . . . . . . . . . 19 ((2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
9390, 92sylib 208 . . . . . . . . . . . . . . . . . 18 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
94 f1of 6124 . . . . . . . . . . . . . . . . . 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 11708 . . . . . . . . . . . . . . . . . . 19 ℕ = (ℤ‘1)
971, 96syl6eleq 2709 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ (ℤ‘1))
98 eluzfz1 12333 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ (ℤ‘1) → 1 ∈ (1...𝑁))
9997, 98syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → 1 ∈ (1...𝑁))
10095, 99ffvelrnd 6346 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd ‘(1st𝑇))‘1) ∈ (1...𝑁))
10187, 100ffvelrnd 6346 . . . . . . . . . . . . . . 15 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) ∈ (0..^𝐾))
102 elfzonn0 12496 . . . . . . . . . . . . . . 15 (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) ∈ (0..^𝐾) → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) ∈ ℕ0)
103 peano2nn0 11318 . . . . . . . . . . . . . . 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 12492 . . . . . . . . . . . . . . . 16 (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) ∈ (0..^𝐾) ↔ (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) ∈ ℕ0𝐾 ∈ ℕ ∧ ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) < 𝐾))
106101, 105sylib 208 . . . . . . . . . . . . . . 15 (𝜑 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) ∈ ℕ0𝐾 ∈ ℕ ∧ ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) < 𝐾))
107106simp2d 1072 . . . . . . . . . . . . . 14 (𝜑𝐾 ∈ ℕ)
108 elfzolt2 12463 . . . . . . . . . . . . . . . . 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 11465 . . . . . . . . . . . . . . . . 17 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) ∈ ℤ)
112107nnzd 11466 . . . . . . . . . . . . . . . . 17 (𝜑𝐾 ∈ ℤ)
113 zltp1le 11412 . . . . . . . . . . . . . . . . 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 692 . . . . . . . . . . . . . . . 16 (𝜑 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) < 𝐾 ↔ (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) ≤ 𝐾))
115109, 114mpbid 222 . . . . . . . . . . . . . . 15 (𝜑 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) ≤ 𝐾)
116 fvex 6188 . . . . . . . . . . . . . . . . . 18 ((2nd ‘(1st𝑇))‘1) ∈ V
117 eleq1 2687 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = ((2nd ‘(1st𝑇))‘1) → (𝑛 ∈ (1...𝑁) ↔ ((2nd ‘(1st𝑇))‘1) ∈ (1...𝑁)))
118117anbi2d 739 . . . . . . . . . . . . . . . . . . 19 (𝑛 = ((2nd ‘(1st𝑇))‘1) → ((𝜑𝑛 ∈ (1...𝑁)) ↔ (𝜑 ∧ ((2nd ‘(1st𝑇))‘1) ∈ (1...𝑁))))
119 fveq2 6178 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = ((2nd ‘(1st𝑇))‘1) → (𝑝𝑛) = (𝑝‘((2nd ‘(1st𝑇))‘1)))
120119neeq1d 2850 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = ((2nd ‘(1st𝑇))‘1) → ((𝑝𝑛) ≠ 𝐾 ↔ (𝑝‘((2nd ‘(1st𝑇))‘1)) ≠ 𝐾))
121120rexbidv 3048 . . . . . . . . . . . . . . . . . . 19 (𝑛 = ((2nd ‘(1st𝑇))‘1) → (∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 𝐾 ↔ ∃𝑝 ∈ ran 𝐹(𝑝‘((2nd ‘(1st𝑇))‘1)) ≠ 𝐾))
122118, 121imbi12d 334 . . . . . . . . . . . . . . . . . 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 3254 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((2nd ‘(1st𝑇))‘1) ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘((2nd ‘(1st𝑇))‘1)) ≠ 𝐾)
124100, 123mpdan 701 . . . . . . . . . . . . . . . 16 (𝜑 → ∃𝑝 ∈ ran 𝐹(𝑝‘((2nd ‘(1st𝑇))‘1)) ≠ 𝐾)
125 fveq1 6177 . . . . . . . . . . . . . . . . . . . . . 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 6032 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾) → (1st ‘(1st𝑇)) Fn (1...𝑁))
12787, 126syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (1st ‘(1st𝑇)) Fn (1...𝑁))
128127adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1st ‘(1st𝑇)) Fn (1...𝑁))
129 1ex 10020 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1 ∈ V
130 fnconstg 6080 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 10019 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 0 ∈ V
133 fnconstg 6080 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 471 . . . . . . . . . . . . . . . . . . . . . . . . . 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 6130 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁) ∧ Fun (2nd ‘(1st𝑇))))
137136simprbi 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun (2nd ‘(1st𝑇)))
13893, 137syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → Fun (2nd ‘(1st𝑇)))
139 imain 5962 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11317 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℕ)
1428, 141syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℕ)
143142nnred 11020 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℝ)
144143ltp1d 10939 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) < ((𝑦 + 1) + 1))
145 fzdisj 12353 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑦 + 1) < ((𝑦 + 1) + 1) → ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅)
146145imaeq2d 5454 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑦 + 1) < ((𝑦 + 1) + 1) → ((2nd ‘(1st𝑇)) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ((2nd ‘(1st𝑇)) “ ∅))
147 ima0 5469 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((2nd ‘(1st𝑇)) “ ∅) = ∅
148146, 147syl6eq 2670 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2675 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) ∩ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ∅)
151 fnun 5985 . . . . . . . . . . . . . . . . . . . . . . . . . 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 694 . . . . . . . . . . . . . . . . . . . . . . . . 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 5533 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((2nd ‘(1st𝑇)) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) ∪ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
154142peano2nnd 11022 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ ℕ)
155154, 96syl6eleq 2709 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ (ℤ‘1))
156155adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1) + 1) ∈ (ℤ‘1))
1571nncnd 11021 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝜑𝑁 ∈ ℂ)
158 npcan1 10440 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
159157, 158syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
160159adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) = 𝑁)
161 elfzuz3 12324 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈ (ℤ𝑦))
162 eluzp1p1 11698 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑦 + 1)))
165160, 164eqeltrrd 2700 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ‘(𝑦 + 1)))
166 fzsplit2 12351 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑦 + 1) + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ‘(𝑦 + 1))) → (1...𝑁) = ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))
167156, 165, 166syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) = ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))
168167imaeq2d 5454 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = ((2nd ‘(1st𝑇)) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))))
169 f1ofo 6131 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁))
170 foima 6107 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
173168, 172eqtr3d 2656 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) = (1...𝑁))
174153, 173syl5eqr 2668 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) ∪ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = (1...𝑁))
175174fneq2d 5970 . . . . . . . . . . . . . . . . . . . . . . . . 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 222 . . . . . . . . . . . . . . . . . . . . . . . 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 3814 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
179 eqidd 2621 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ ((2nd ‘(1st𝑇))‘1) ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) = ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)))
180 f1ofn 6125 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (2nd ‘(1st𝑇)) Fn (1...𝑁))
183 fzss2 12366 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2709 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ (ℤ‘1))
186 eluzfz1 12333 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑦 + 1) ∈ (ℤ‘1) → 1 ∈ (1...(𝑦 + 1)))
187185, 186syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ (0...(𝑁 − 1)) → 1 ∈ (1...(𝑦 + 1)))
188187adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 1 ∈ (1...(𝑦 + 1)))
189 fnfvima 6481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1324 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇))‘1) ∈ ((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))))
191 fvun1 6256 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1412 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 692 . . . . . . . . . . . . . . . . . . . . . . . . . 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 6454 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2654 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘1)) = 1)
197196adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 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 6891 . . . . . . . . . . . . . . . . . . . . . . 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 703 . . . . . . . . . . . . . . . . . . . . . 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 2676 . . . . . . . . . . . . . . . . . . . . 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 754 . . . . . . . . . . . . . . . . . . . 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 6178 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑡 = 𝑇 → (2nd𝑡) = (2nd𝑇))
203202breq2d 4656 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑡 = 𝑇 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑇)))
204203ifbid 4099 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑡 = 𝑇 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)))
205204csbeq1d 3533 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6178 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑡 = 𝑇 → (1st𝑡) = (1st𝑇))
207206fveq2d 6182 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑡 = 𝑇 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
208206fveq2d 6182 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑡 = 𝑇 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑇)))
209208imaeq1d 5453 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑗)))
210209xpeq1d 5128 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}))
211208imaeq1d 5453 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)))
212211xpeq1d 5128 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
213210, 212uneq12d 3760 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6653 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3983 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2654 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4736 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2630 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3360 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . . . . . . . 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 5342 . . . . . . . . . . . . . . . . . . . . . . . 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 2685 . . . . . . . . . . . . . . . . . . . . . . 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 2620 . . . . . . . . . . . . . . . . . . . . . . . 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 6663 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V
226225csbex 4784 . . . . . . . . . . . . . . . . . . . . . . . 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 5365 . . . . . . . . . . . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (2nd𝑇) = 0)
230 elfzle1 12329 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 ∈ (0...(𝑁 − 1)) → 0 ≤ 𝑦)
231230adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 0 ≤ 𝑦)
232229, 231eqbrtrd 4666 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (2nd𝑇) ≤ 𝑦)
233 0re 10025 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 0 ∈ ℝ
2346, 233syl6eqel 2707 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑 → (2nd𝑇) ∈ ℝ)
235 lenlt 10101 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((2nd𝑇) ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((2nd𝑇) ≤ 𝑦 ↔ ¬ 𝑦 < (2nd𝑇)))
236234, 9, 235syl2an 494 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd𝑇) ≤ 𝑦 ↔ ¬ 𝑦 < (2nd𝑇)))
237232, 236mpbid 222 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ¬ 𝑦 < (2nd𝑇))
238237iffalsed 4088 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = (𝑦 + 1))
239238csbeq1d 3533 . . . . . . . . . . . . . . . . . . . . . . . . 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 6663 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 + 1) ∈ V
241 oveq2 6643 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 = (𝑦 + 1) → (1...𝑗) = (1...(𝑦 + 1)))
242241imaeq2d 5454 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 = (𝑦 + 1) → ((2nd ‘(1st𝑇)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))))
243242xpeq1d 5128 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 = (𝑦 + 1) → (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}))
244 oveq1 6642 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑗 = (𝑦 + 1) → (𝑗 + 1) = ((𝑦 + 1) + 1))
245244oveq1d 6650 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 = (𝑦 + 1) → ((𝑗 + 1)...𝑁) = (((𝑦 + 1) + 1)...𝑁))
246245imaeq2d 5454 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 = (𝑦 + 1) → ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
247246xpeq1d 5128 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 = (𝑦 + 1) → (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))
248243, 247uneq12d 3760 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6651 . . . . . . . . . . . . . . . . . . . . . . . . . 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 3552 . . . . . . . . . . . . . . . . . . . . . . . . 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 2670 . . . . . . . . . . . . . . . . . . . . . . . 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 2630 . . . . . . . . . . . . . . . . . . . . . . 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 3045 . . . . . . . . . . . . . . . . . . . . . 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 268 . . . . . . . . . . . . . . . . . . . . 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 501 . . . . . . . . . . . . . . . . . . . 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 3074 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑝 ∈ ran 𝐹) → (𝑝‘((2nd ‘(1st𝑇))‘1)) = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1))
257 eqtr3 2641 . . . . . . . . . . . . . . . . . . . 20 (((𝑝‘((2nd ‘(1st𝑇))‘1)) = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) ∧ 𝐾 = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1)) → (𝑝‘((2nd ‘(1st𝑇))‘1)) = 𝐾)
258257ex 450 . . . . . . . . . . . . . . . . . . 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 2812 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ ran 𝐹) → ((𝑝‘((2nd ‘(1st𝑇))‘1)) ≠ 𝐾𝐾 ≠ (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1)))
261260rexlimdva 3027 . . . . . . . . . . . . . . . 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 11337 . . . . . . . . . . . . . . . 16 (𝜑 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) ∈ ℝ)
264107nnred 11020 . . . . . . . . . . . . . . . 16 (𝜑𝐾 ∈ ℝ)
265263, 264ltlend 10167 . . . . . . . . . . . . . . 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 956 . . . . . . . . . . . . . 14 (𝜑 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) < 𝐾)
267 elfzo0 12492 . . . . . . . . . . . . . 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 1244 . . . . . . . . . . . . 13 (𝜑 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) ∈ (0..^𝐾))
269268adantr 481 . . . . . . . . . . . 12 ((𝜑𝑛 = ((2nd ‘(1st𝑇))‘1)) → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘1)) + 1) ∈ (0..^𝐾))
27080, 269eqeltrd 2699 . . . . . . . . . . 11 ((𝜑𝑛 = ((2nd ‘(1st𝑇))‘1)) → (((1st ‘(1st𝑇))‘𝑛) + 1) ∈ (0..^𝐾))
271270adantlr 750 . . . . . . . . . 10 (((𝜑𝑛 ∈ (1...𝑁)) ∧ 𝑛 = ((2nd ‘(1st𝑇))‘1)) → (((1st ‘(1st𝑇))‘𝑛) + 1) ∈ (0..^𝐾))
27287ffvelrnda 6345 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ (0..^𝐾))
273 elfzonn0 12496 . . . . . . . . . . . . . . 15 (((1st ‘(1st𝑇))‘𝑛) ∈ (0..^𝐾) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℕ0)
274272, 273syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℕ0)
275274nn0cnd 11338 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℂ)
276275addid1d 10221 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑇))‘𝑛) + 0) = ((1st ‘(1st𝑇))‘𝑛))
277276, 272eqeltrd 2699 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑇))‘𝑛) + 0) ∈ (0..^𝐾))
278277adantr 481 . . . . . . . . . 10 (((𝜑𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st𝑇))‘1)) → (((1st ‘(1st𝑇))‘𝑛) + 0) ∈ (0..^𝐾))
27975, 77, 271, 278ifbothda 4114 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0)) ∈ (0..^𝐾))
280 eqid 2620 . . . . . . . . 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 6371 . . . . . . . 8 (𝜑 → (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))):(1...𝑁)⟶(0..^𝐾))
282 ovex 6663 . . . . . . . . 9 (0..^𝐾) ∈ V
283282, 49elmap 7871 . . . . . . . 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 224 . . . . . . 7 (𝜑 → (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
285 simpr 477 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (1...(𝑁 − 1))) → 𝑛 ∈ (1...(𝑁 − 1)))
286 1z 11392 . . . . . . . . . . . . . . . . 17 1 ∈ ℤ
28713, 286jctil 559 . . . . . . . . . . . . . . . 16 (𝜑 → (1 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ))
288 elfzelz 12327 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ ℤ)
289288, 286jctir 560 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (1...(𝑁 − 1)) → (𝑛 ∈ ℤ ∧ 1 ∈ ℤ))
290 fzaddel 12360 . . . . . . . . . . . . . . . 16 (((1 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) ∧ (𝑛 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑛 ∈ (1...(𝑁 − 1)) ↔ (𝑛 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1))))
291287, 289, 290syl2an 494 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (1...(𝑁 − 1))) → (𝑛 ∈ (1...(𝑁 − 1)) ↔ (𝑛 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1))))
292285, 291mpbid 222 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...(𝑁 − 1))) → (𝑛 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1)))
293159oveq2d 6651 . . . . . . . . . . . . . . 15 (𝜑 → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 + 1)...𝑁))
294293adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...(𝑁 − 1))) → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 + 1)...𝑁))
295292, 294eleqtrd 2701 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...(𝑁 − 1))) → (𝑛 + 1) ∈ ((1 + 1)...𝑁))
296295ralrimiva 2963 . . . . . . . . . . . 12 (𝜑 → ∀𝑛 ∈ (1...(𝑁 − 1))(𝑛 + 1) ∈ ((1 + 1)...𝑁))
297 simpr 477 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ ((1 + 1)...𝑁)) → 𝑦 ∈ ((1 + 1)...𝑁))
298 peano2z 11403 . . . . . . . . . . . . . . . . . . 19 (1 ∈ ℤ → (1 + 1) ∈ ℤ)
299286, 298ax-mp 5 . . . . . . . . . . . . . . . . . 18 (1 + 1) ∈ ℤ
30011, 299jctil 559 . . . . . . . . . . . . . . . . 17 (𝜑 → ((1 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ))
301 elfzelz 12327 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ((1 + 1)...𝑁) → 𝑦 ∈ ℤ)
302301, 286jctir 560 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ((1 + 1)...𝑁) → (𝑦 ∈ ℤ ∧ 1 ∈ ℤ))
303 fzsubel 12362 . . . . . . . . . . . . . . . . 17 ((((1 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑦 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑦 ∈ ((1 + 1)...𝑁) ↔ (𝑦 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1))))
304300, 302, 303syl2an 494 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ ((1 + 1)...𝑁)) → (𝑦 ∈ ((1 + 1)...𝑁) ↔ (𝑦 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1))))
305297, 304mpbid 222 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ ((1 + 1)...𝑁)) → (𝑦 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1)))
306 ax-1cn 9979 . . . . . . . . . . . . . . . . 17 1 ∈ ℂ
307306, 306pncan3oi 10282 . . . . . . . . . . . . . . . 16 ((1 + 1) − 1) = 1
308307oveq1i 6645 . . . . . . . . . . . . . . 15 (((1 + 1) − 1)...(𝑁 − 1)) = (1...(𝑁 − 1))
309305, 308syl6eleq 2709 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ ((1 + 1)...𝑁)) → (𝑦 − 1) ∈ (1...(𝑁 − 1)))
310301zcnd 11468 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ((1 + 1)...𝑁) → 𝑦 ∈ ℂ)
311 elfznn 12355 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ ℕ)
312311nncnd 11021 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ ℂ)
313 subadd2 10270 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑛 ∈ ℂ) → ((𝑦 − 1) = 𝑛 ↔ (𝑛 + 1) = 𝑦))
314306, 313mp3an2 1410 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℂ) → ((𝑦 − 1) = 𝑛 ↔ (𝑛 + 1) = 𝑦))
315314bicomd 213 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℂ) → ((𝑛 + 1) = 𝑦 ↔ (𝑦 − 1) = 𝑛))
316 eqcom 2627 . . . . . . . . . . . . . . . . . 18 (𝑦 = (𝑛 + 1) ↔ (𝑛 + 1) = 𝑦)
317 eqcom 2627 . . . . . . . . . . . . . . . . . 18 (𝑛 = (𝑦 − 1) ↔ (𝑦 − 1) = 𝑛)
318315, 316, 3173bitr4g 303 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℂ) → (𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1)))
319310, 312, 318syl2an 494 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ((1 + 1)...𝑁) ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1)))
320319ralrimiva 2963 . . . . . . . . . . . . . . 15 (𝑦 ∈ ((1 + 1)...𝑁) → ∀𝑛 ∈ (1...(𝑁 − 1))(𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1)))
321320adantl 482 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ ((1 + 1)...𝑁)) → ∀𝑛 ∈ (1...(𝑁 − 1))(𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1)))
322 reu6i 3391 . . . . . . . . . . . . . 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 692 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ ((1 + 1)...𝑁)) → ∃!𝑛 ∈ (1...(𝑁 − 1))𝑦 = (𝑛 + 1))
324323ralrimiva 2963 . . . . . . . . . . . 12 (𝜑 → ∀𝑦 ∈ ((1 + 1)...𝑁)∃!𝑛 ∈ (1...(𝑁 − 1))𝑦 = (𝑛 + 1))
325 eqid 2620 . . . . . . . . . . . . 13 (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) = (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1))
326325f1ompt 6368 . . . . . . . . . . . 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 697 . . . . . . . . . . 11 (𝜑 → (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)):(1...(𝑁 − 1))–1-1-onto→((1 + 1)...𝑁))
328 f1osng 6164 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ ∧ 1 ∈ V) → {⟨𝑁, 1⟩}:{𝑁}–1-1-onto→{1})
3291, 129, 328sylancl 693 . . . . . . . . . . 11 (𝜑 → {⟨𝑁, 1⟩}:{𝑁}–1-1-onto→{1})
33014, 16ltnled 10169 . . . . . . . . . . . . . 14 (𝜑 → ((𝑁 − 1) < 𝑁 ↔ ¬ 𝑁 ≤ (𝑁 − 1)))
33120, 330mpbid 222 . . . . . . . . . . . . 13 (𝜑 → ¬ 𝑁 ≤ (𝑁 − 1))
332 elfzle2 12330 . . . . . . . . . . . . 13 (𝑁 ∈ (1...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1))
333331, 332nsyl 135 . . . . . . . . . . . 12 (𝜑 → ¬ 𝑁 ∈ (1...(𝑁 − 1)))
334 disjsn 4237 . . . . . . . . . . . 12 (((1...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (1...(𝑁 − 1)))
335333, 334sylibr 224 . . . . . . . . . . 11 (𝜑 → ((1...(𝑁 − 1)) ∩ {𝑁}) = ∅)
336 1re 10024 . . . . . . . . . . . . . . . 16 1 ∈ ℝ
337336ltp1i 10912 . . . . . . . . . . . . . . 15 1 < (1 + 1)
338299zrei 11368 . . . . . . . . . . . . . . . 16 (1 + 1) ∈ ℝ
339336, 338ltnlei 10143 . . . . . . . . . . . . . . 15 (1 < (1 + 1) ↔ ¬ (1 + 1) ≤ 1)
340337, 339mpbi 220 . . . . . . . . . . . . . 14 ¬ (1 + 1) ≤ 1
341 elfzle1 12329 . . . . . . . . . . . . . 14 (1 ∈ ((1 + 1)...𝑁) → (1 + 1) ≤ 1)
342340, 341mto 188 . . . . . . . . . . . . 13 ¬ 1 ∈ ((1 + 1)...𝑁)
343 disjsn 4237 . . . . . . . . . . . . 13 ((((1 + 1)...𝑁) ∩ {1}) = ∅ ↔ ¬ 1 ∈ ((1 + 1)...𝑁))
344342, 343mpbir 221 . . . . . . . . . . . 12 (((1 + 1)...𝑁) ∩ {1}) = ∅
345344a1i 11 . . . . . . . . . . 11 (𝜑 → (((1 + 1)...𝑁) ∩ {1}) = ∅)
346 f1oun 6143 . . . . . . . . . . 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 1325 . . . . . . . . . 10 (𝜑 → ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {⟨𝑁, 1⟩}):((1...(𝑁 − 1)) ∪ {𝑁})–1-1-onto→(((1 + 1)...𝑁) ∪ {1}))
348 elex 3207 . . . . . . . . . . . . . 14 (𝑁 ∈ ℕ → 𝑁 ∈ V)
3491, 348syl 17 . . . . . . . . . . . . 13 (𝜑𝑁 ∈ V)
350129a1i 11 . . . . . . . . . . . . 13 (𝜑 → 1 ∈ V)
351159, 97eqeltrd 2699 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘1))
352 uzid 11687 . . . . . . . . . . . . . . . . 17 ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)))
353 peano2uz 11726 . . . . . . . . . . . . . . . . 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 2700 . . . . . . . . . . . . . . 15 (𝜑𝑁 ∈ (ℤ‘(𝑁 − 1)))
356 fzsplit2 12351 . . . . . . . . . . . . . . 15 ((((𝑁 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ‘(𝑁 − 1))) → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
357351, 355, 356syl2anc 692 . . . . . . . . . . . . . 14 (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
358159oveq1d 6650 . . . . . . . . . . . . . . . 16 (𝜑 → (((𝑁 − 1) + 1)...𝑁) = (𝑁...𝑁))
359 fzsn 12368 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
36011, 359syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑁...𝑁) = {𝑁})
361358, 360eqtrd 2654 . . . . . . . . . . . . . . 15 (𝜑 → (((𝑁 − 1) + 1)...𝑁) = {𝑁})
362361uneq2d 3759 . . . . . . . . . . . . . 14 (𝜑 → ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = ((1...(𝑁 − 1)) ∪ {𝑁}))
363357, 362eqtr2d 2655 . . . . . . . . . . . . 13 (𝜑 → ((1...(𝑁 − 1)) ∪ {𝑁}) = (1...𝑁))
364 iftrue 4083 . . . . . . . . . . . . . 14 (𝑛 = 𝑁 → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = 1)
365364adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑛 = 𝑁) → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = 1)
366349, 350, 363, 365fmptapd 6422 . . . . . . . . . . . 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 2687 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑁 → (𝑛 ∈ (1...(𝑁 − 1)) ↔ 𝑁 ∈ (1...(𝑁 − 1))))
368367notbid 308 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑁 → (¬ 𝑛 ∈ (1...(𝑁 − 1)) ↔ ¬ 𝑁 ∈ (1...(𝑁 − 1))))
369333, 368syl5ibrcom 237 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑛 = 𝑁 → ¬ 𝑛 ∈ (1...(𝑁 − 1))))
370369necon2ad 2806 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛𝑁))
371370imp 445 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (1...(𝑁 − 1))) → 𝑛𝑁)
372 ifnefalse 4089 . . . . . . . . . . . . . . 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 4735 . . . . . . . . . . . . 13 (𝜑 → (𝑛 ∈ (1...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)))
375374uneq1d 3758 . . . . . . . . . . . 12 (𝜑 → ((𝑛 ∈ (1...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ∪ {⟨𝑁, 1⟩}) = ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {⟨𝑁, 1⟩}))
376366, 375eqtr3d 2656 . . . . . . . . . . 11 (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {⟨𝑁, 1⟩}))
377357, 362eqtrd 2654 . . . . . . . . . . 11 (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁}))
378 uzid 11687 . . . . . . . . . . . . . 14 (1 ∈ ℤ → 1 ∈ (ℤ‘1))
379 peano2uz 11726 . . . . . . . . . . . . . 14 (1 ∈ (ℤ‘1) → (1 + 1) ∈ (ℤ‘1))
380286, 378, 379mp2b 10 . . . . . . . . . . . . 13 (1 + 1) ∈ (ℤ‘1)
381 fzsplit2 12351 . . . . . . . . . . . . 13 (((1 + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ‘1)) → (1...𝑁) = ((1...1) ∪ ((1 + 1)...𝑁)))
382380, 97, 381sylancr 694 . . . . . . . . . . . 12 (𝜑 → (1...𝑁) = ((1...1) ∪ ((1 + 1)...𝑁)))
383 fzsn 12368 . . . . . . . . . . . . . . 15 (1 ∈ ℤ → (1...1) = {1})
384286, 383ax-mp 5 . . . . . . . . . . . . . 14 (1...1) = {1}
385384uneq1i 3755 . . . . . . . . . . . . 13 ((1...1) ∪ ((1 + 1)...𝑁)) = ({1} ∪ ((1 + 1)...𝑁))
386385equncomi 3751 . . . . . . . . . . . 12 ((1...1) ∪ ((1 + 1)...𝑁)) = (((1 + 1)...𝑁) ∪ {1})
387382, 386syl6eq 2670 . . . . . . . . . . 11 (𝜑 → (1...𝑁) = (((1 + 1)...𝑁) ∪ {1}))
388376, 377, 387f1oeq123d 6120 . . . . . . . . . 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 247 . . . . . . . . 9 (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)–1-1-onto→(1...𝑁))
390 f1oco 6146 . . . . . . . . 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 692 . . . . . . . 8 (𝜑 → ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–1-1-onto→(1...𝑁))
392 f1oeq1 6114 . . . . . . . . 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 3344 . . . . . . . 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 224 . . . . . . 7 (𝜑 → ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
395 opelxpi 5138 . . . . . . 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 692 . . . . . 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 11336 . . . . . . 7 (𝜑𝑁 ∈ ℕ0)
398 nn0fz0 12421 . . . . . . 7 (𝑁 ∈ ℕ0𝑁 ∈ (0...𝑁))
399397, 398sylib 208 . . . . . 6 (𝜑𝑁 ∈ (0...𝑁))
400 opelxpi 5138 . . . . . 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 692 . . . . 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 3356 . . . . 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 692 . . . 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 247 . . 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 2710 . 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 6178 . . . . . . 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 2622 . . . . . 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 235 . . . . 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 11050 . . . . . 6 (𝜑𝑁 ≠ 0)
410 neeq1 2853 . . . . . 6 ((2nd𝑇) = 𝑁 → ((2nd𝑇) ≠ 0 ↔ 𝑁 ≠ 0))
411409, 410syl5ibrcom 237 . . . . 5 (𝜑 → ((2nd𝑇) = 𝑁 → (2nd𝑇) ≠ 0))
412408, 411syld 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))
413412necon2d 2814 . . 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 2853 . . 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 3304 . 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 692 1 (𝜑 → ∃𝑧𝑆 𝑧𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036  wal 1479   = wceq 1481  wcel 1988  {cab 2606  wne 2791  wral 2909  wrex 2910  ∃!wreu 2911  {crab 2913  Vcvv 3195  csb 3526  cun 3565  cin 3566  wss 3567  c0 3907  ifcif 4077  {csn 4168  cop 4174   class class class wbr 4644  cmpt 4720   × cxp 5102  ccnv 5103  ran crn 5105  cima 5107  ccom 5108  Fun wfun 5870   Fn wfn 5871  wf 5872  ontowfo 5874  1-1-ontowf1o 5875  cfv 5876  (class class class)co 6635  𝑓 cof 6880  1st c1st 7151  2nd c2nd 7152  𝑚 cmap 7842  cc 9919  cr 9920  0cc0 9921  1c1 9922   + caddc 9924   < clt 10059  cle 10060  cmin 10251  cn 11005  0cn0 11277  cz 11362  cuz 11672  ...cfz 12311  ..^cfzo 12449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-of 6882  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-er 7727  df-map 7844  df-en 7941  df-dom 7942  df-sdom 7943  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-n0 11278  df-z 11363  df-uz 11673  df-fz 12312  df-fzo 12450
This theorem is referenced by:  poimirlem18  33398
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