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Theorem poimirlem27 35056
Description: Lemma for poimir 35062 showing that the difference between admissible faces in the whole cube and admissible faces on the back face is even. Equation (7) of [Kulpa] p. 548. (Contributed by Brendan Leahy, 21-Aug-2020.)
Hypotheses
Ref Expression
poimir.0 (𝜑𝑁 ∈ ℕ)
poimirlem28.1 (𝑝 = ((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶)
poimirlem28.2 ((𝜑𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁))
poimirlem28.3 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 0)) → 𝐵 < 𝑛)
poimirlem28.4 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 𝐾)) → 𝐵 ≠ (𝑛 − 1))
Assertion
Ref Expression
poimirlem27 (𝜑 → 2 ∥ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)})))
Distinct variable groups:   𝑓,𝑖,𝑗,𝑛,𝑝,𝑠,𝑡   𝜑,𝑗,𝑛   𝑗,𝑁,𝑛   𝜑,𝑖,𝑝,𝑠,𝑡   𝐵,𝑓,𝑖,𝑗,𝑛,𝑠,𝑡   𝑓,𝐾,𝑖,𝑗,𝑛,𝑝,𝑠,𝑡   𝑓,𝑁,𝑖,𝑝,𝑠,𝑡   𝐶,𝑖,𝑛,𝑝,𝑡
Allowed substitution hints:   𝜑(𝑓)   𝐵(𝑝)   𝐶(𝑓,𝑗,𝑠)

Proof of Theorem poimirlem27
Dummy variables 𝑚 𝑞 𝑢 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fzfi 13346 . . . . . 6 (0...𝐾) ∈ Fin
2 fzfi 13346 . . . . . 6 (1...𝑁) ∈ Fin
3 mapfi 8819 . . . . . 6 (((0...𝐾) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((0...𝐾) ↑m (1...𝑁)) ∈ Fin)
41, 2, 3mp2an 691 . . . . 5 ((0...𝐾) ↑m (1...𝑁)) ∈ Fin
5 fzfi 13346 . . . . 5 (0...(𝑁 − 1)) ∈ Fin
6 mapfi 8819 . . . . 5 ((((0...𝐾) ↑m (1...𝑁)) ∈ Fin ∧ (0...(𝑁 − 1)) ∈ Fin) → (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1))) ∈ Fin)
74, 5, 6mp2an 691 . . . 4 (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1))) ∈ Fin
87a1i 11 . . 3 (𝜑 → (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1))) ∈ Fin)
9 2z 12013 . . . 4 2 ∈ ℤ
109a1i 11 . . 3 (𝜑 → 2 ∈ ℤ)
11 fzofi 13348 . . . . . . . 8 (0..^𝐾) ∈ Fin
12 mapfi 8819 . . . . . . . 8 (((0..^𝐾) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((0..^𝐾) ↑m (1...𝑁)) ∈ Fin)
1311, 2, 12mp2an 691 . . . . . . 7 ((0..^𝐾) ↑m (1...𝑁)) ∈ Fin
14 mapfi 8819 . . . . . . . . 9 (((1...𝑁) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((1...𝑁) ↑m (1...𝑁)) ∈ Fin)
152, 2, 14mp2an 691 . . . . . . . 8 ((1...𝑁) ↑m (1...𝑁)) ∈ Fin
16 f1of 6608 . . . . . . . . . 10 (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑓:(1...𝑁)⟶(1...𝑁))
1716ss2abi 4029 . . . . . . . . 9 {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ {𝑓𝑓:(1...𝑁)⟶(1...𝑁)}
18 ovex 7184 . . . . . . . . . 10 (1...𝑁) ∈ V
1918, 18mapval 8416 . . . . . . . . 9 ((1...𝑁) ↑m (1...𝑁)) = {𝑓𝑓:(1...𝑁)⟶(1...𝑁)}
2017, 19sseqtrri 3990 . . . . . . . 8 {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ ((1...𝑁) ↑m (1...𝑁))
21 ssfi 8737 . . . . . . . 8 ((((1...𝑁) ↑m (1...𝑁)) ∈ Fin ∧ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ ((1...𝑁) ↑m (1...𝑁))) → {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin)
2215, 20, 21mp2an 691 . . . . . . 7 {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin
23 xpfi 8788 . . . . . . 7 ((((0..^𝐾) ↑m (1...𝑁)) ∈ Fin ∧ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin) → (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin)
2413, 22, 23mp2an 691 . . . . . 6 (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin
25 fzfi 13346 . . . . . 6 (0...𝑁) ∈ Fin
26 xpfi 8788 . . . . . 6 (((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin ∧ (0...𝑁) ∈ Fin) → ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin)
2724, 25, 26mp2an 691 . . . . 5 ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin
28 rabfi 8742 . . . . 5 (((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))} ∈ Fin)
2927, 28ax-mp 5 . . . 4 {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))} ∈ Fin
30 hashcl 13724 . . . . 5 ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))} ∈ Fin → (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))}) ∈ ℕ0)
3130nn0zd 12084 . . . 4 ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))} ∈ Fin → (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))}) ∈ ℤ)
3229, 31mp1i 13 . . 3 ((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) → (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))}) ∈ ℤ)
33 dfrex2 3233 . . . . 5 (∃𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) ↔ ¬ ∀𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))
34 nfv 1916 . . . . . 6 𝑡(𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1))))
35 nfcv 2982 . . . . . . 7 𝑡2
36 nfcv 2982 . . . . . . 7 𝑡
37 nfcv 2982 . . . . . . . 8 𝑡
38 nfrab1 3375 . . . . . . . 8 𝑡{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))}
3937, 38nffv 6673 . . . . . . 7 𝑡(♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))})
4035, 36, 39nfbr 5100 . . . . . 6 𝑡2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))})
41 neq0 4292 . . . . . . . . . . . 12 (¬ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))} = ∅ ↔ ∃𝑠 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))})
42 iddvds 15625 . . . . . . . . . . . . . . . . 17 (2 ∈ ℤ → 2 ∥ 2)
439, 42ax-mp 5 . . . . . . . . . . . . . . . 16 2 ∥ 2
44 vex 3483 . . . . . . . . . . . . . . . . . . 19 𝑠 ∈ V
45 hashsng 13737 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ V → (♯‘{𝑠}) = 1)
4644, 45ax-mp 5 . . . . . . . . . . . . . . . . . 18 (♯‘{𝑠}) = 1
4746oveq2i 7162 . . . . . . . . . . . . . . . . 17 (1 + (♯‘{𝑠})) = (1 + 1)
48 df-2 11699 . . . . . . . . . . . . . . . . 17 2 = (1 + 1)
4947, 48eqtr4i 2850 . . . . . . . . . . . . . . . 16 (1 + (♯‘{𝑠})) = 2
5043, 49breqtrri 5080 . . . . . . . . . . . . . . 15 2 ∥ (1 + (♯‘{𝑠}))
51 rabfi 8742 . . . . . . . . . . . . . . . . . . . 20 (((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))} ∈ Fin)
52 diffi 8749 . . . . . . . . . . . . . . . . . . . 20 ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))} ∈ Fin → ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))} ∖ {𝑠}) ∈ Fin)
5327, 51, 52mp2b 10 . . . . . . . . . . . . . . . . . . 19 ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))} ∖ {𝑠}) ∈ Fin
54 snfi 8592 . . . . . . . . . . . . . . . . . . 19 {𝑠} ∈ Fin
55 incom 4163 . . . . . . . . . . . . . . . . . . . 20 (({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))} ∖ {𝑠}) ∩ {𝑠}) = ({𝑠} ∩ ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))} ∖ {𝑠}))
56 disjdif 4404 . . . . . . . . . . . . . . . . . . . 20 ({𝑠} ∩ ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))} ∖ {𝑠})) = ∅
5755, 56eqtri 2847 . . . . . . . . . . . . . . . . . . 19 (({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))} ∖ {𝑠}) ∩ {𝑠}) = ∅
58 hashun 13750 . . . . . . . . . . . . . . . . . . 19 ((({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))} ∖ {𝑠}) ∈ Fin ∧ {𝑠} ∈ Fin ∧ (({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))} ∖ {𝑠}) ∩ {𝑠}) = ∅) → (♯‘(({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))} ∖ {𝑠}) ∪ {𝑠})) = ((♯‘({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))} ∖ {𝑠})) + (♯‘{𝑠})))
5953, 54, 57, 58mp3an 1458 . . . . . . . . . . . . . . . . . 18 (♯‘(({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))} ∖ {𝑠}) ∪ {𝑠})) = ((♯‘({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))} ∖ {𝑠})) + (♯‘{𝑠}))
60 difsnid 4727 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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)) ∣ 𝑥 = (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}))))} ∖ {𝑠}) ∪ {𝑠}) = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))})
6160fveq2d 6667 . . . . . . . . . . . . . . . . . 18 (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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)) ∣ 𝑥 = (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}))))} ∖ {𝑠}) ∪ {𝑠})) = (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))}))
6259, 61syl5eqr 2873 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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)) ∣ 𝑥 = (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}))))} ∖ {𝑠})) + (♯‘{𝑠})) = (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))}))
6362adantl 485 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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)) ∣ 𝑥 = (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}))))} ∖ {𝑠})) + (♯‘{𝑠})) = (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))}))
64 poimir.0 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑁 ∈ ℕ)
6564ad3antrrr 729 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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 ∈ ℕ)
66 fveq2 6663 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 = 𝑢 → (2nd𝑡) = (2nd𝑢))
6766breq2d 5065 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = 𝑢 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑢)))
6867ifbid 4472 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑢 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑢), 𝑦, (𝑦 + 1)))
6968csbeq1d 3870 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑢if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑢), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))
70 2fveq3 6668 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = 𝑢 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑢)))
71 2fveq3 6668 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑡 = 𝑢 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑢)))
7271imaeq1d 5917 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑡 = 𝑢 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑢)) “ (1...𝑗)))
7372xpeq1d 5572 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 = 𝑢 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑢)) “ (1...𝑗)) × {1}))
7471imaeq1d 5917 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑡 = 𝑢 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)))
7574xpeq1d 5572 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 = 𝑢 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))
7673, 75uneq12d 4126 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = 𝑢 → ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})))
7770, 76oveq12d 7169 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑢 → ((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑢)) ∘f + ((((2nd ‘(1st𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))))
7877csbeq2dv 3873 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑢if(𝑦 < (2nd𝑢), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑢), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑢)) ∘f + ((((2nd ‘(1st𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))))
7969, 78eqtrd 2859 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑢if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑢), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑢)) ∘f + ((((2nd ‘(1st𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))))
8079mpteq2dv 5149 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = 𝑢 → (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 < (2nd𝑢), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑢)) ∘f + ((((2nd ‘(1st𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})))))
81 breq1 5056 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑤 → (𝑦 < (2nd𝑢) ↔ 𝑤 < (2nd𝑢)))
82 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑤𝑦 = 𝑤)
83 oveq1 7158 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑤 → (𝑦 + 1) = (𝑤 + 1))
8481, 82, 83ifbieq12d 4477 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 = 𝑤 → if(𝑦 < (2nd𝑢), 𝑦, (𝑦 + 1)) = if(𝑤 < (2nd𝑢), 𝑤, (𝑤 + 1)))
8584csbeq1d 3870 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 𝑤if(𝑦 < (2nd𝑢), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑢)) ∘f + ((((2nd ‘(1st𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑤 < (2nd𝑢), 𝑤, (𝑤 + 1)) / 𝑗((1st ‘(1st𝑢)) ∘f + ((((2nd ‘(1st𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))))
86 oveq2 7159 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 = 𝑖 → (1...𝑗) = (1...𝑖))
8786imaeq2d 5918 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 = 𝑖 → ((2nd ‘(1st𝑢)) “ (1...𝑗)) = ((2nd ‘(1st𝑢)) “ (1...𝑖)))
8887xpeq1d 5572 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 = 𝑖 → (((2nd ‘(1st𝑢)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑢)) “ (1...𝑖)) × {1}))
89 oveq1 7158 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 = 𝑖 → (𝑗 + 1) = (𝑖 + 1))
9089oveq1d 7166 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 = 𝑖 → ((𝑗 + 1)...𝑁) = ((𝑖 + 1)...𝑁))
9190imaeq2d 5918 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 = 𝑖 → ((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑢)) “ ((𝑖 + 1)...𝑁)))
9291xpeq1d 5572 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 = 𝑖 → (((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑢)) “ ((𝑖 + 1)...𝑁)) × {0}))
9388, 92uneq12d 4126 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 = 𝑖 → ((((2nd ‘(1st𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑖 + 1)...𝑁)) × {0})))
9493oveq2d 7167 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 = 𝑖 → ((1st ‘(1st𝑢)) ∘f + ((((2nd ‘(1st𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑢)) ∘f + ((((2nd ‘(1st𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑖 + 1)...𝑁)) × {0}))))
9594cbvcsbv 3878 . . . . . . . . . . . . . . . . . . . . . . . 24 if(𝑤 < (2nd𝑢), 𝑤, (𝑤 + 1)) / 𝑗((1st ‘(1st𝑢)) ∘f + ((((2nd ‘(1st𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑤 < (2nd𝑢), 𝑤, (𝑤 + 1)) / 𝑖((1st ‘(1st𝑢)) ∘f + ((((2nd ‘(1st𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑖 + 1)...𝑁)) × {0})))
9685, 95syl6eq 2875 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = 𝑤if(𝑦 < (2nd𝑢), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑢)) ∘f + ((((2nd ‘(1st𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑤 < (2nd𝑢), 𝑤, (𝑤 + 1)) / 𝑖((1st ‘(1st𝑢)) ∘f + ((((2nd ‘(1st𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑖 + 1)...𝑁)) × {0}))))
9796cbvmptv 5156 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑢), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑢)) ∘f + ((((2nd ‘(1st𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑤 ∈ (0...(𝑁 − 1)) ↦ if(𝑤 < (2nd𝑢), 𝑤, (𝑤 + 1)) / 𝑖((1st ‘(1st𝑢)) ∘f + ((((2nd ‘(1st𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑖 + 1)...𝑁)) × {0}))))
9880, 97syl6eq 2875 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = 𝑢 → (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})))) = (𝑤 ∈ (0...(𝑁 − 1)) ↦ if(𝑤 < (2nd𝑢), 𝑤, (𝑤 + 1)) / 𝑖((1st ‘(1st𝑢)) ∘f + ((((2nd ‘(1st𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑖 + 1)...𝑁)) × {0})))))
9998eqeq2d 2835 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑢 → (𝑥 = (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})))) ↔ 𝑥 = (𝑤 ∈ (0...(𝑁 − 1)) ↦ if(𝑤 < (2nd𝑢), 𝑤, (𝑤 + 1)) / 𝑖((1st ‘(1st𝑢)) ∘f + ((((2nd ‘(1st𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑖 + 1)...𝑁)) × {0}))))))
10099cbvrabv 3477 . . . . . . . . . . . . . . . . . . 19 {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))} = {𝑢 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑤 ∈ (0...(𝑁 − 1)) ↦ if(𝑤 < (2nd𝑢), 𝑤, (𝑤 + 1)) / 𝑖((1st ‘(1st𝑢)) ∘f + ((((2nd ‘(1st𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd ‘(1st𝑢)) “ ((𝑖 + 1)...𝑁)) × {0}))))}
101 elmapi 8426 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1))) → 𝑥:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))
102101ad3antlr 730 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))}) → 𝑥:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))
103 simpr 488 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))}) → 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))})
104 simpl 486 . . . . . . . . . . . . . . . . . . . . . 22 ((∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾) → ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0)
105104ralimi 3155 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0)
106105ad2antlr 726 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0)
107 fveq2 6663 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑚 → (𝑝𝑛) = (𝑝𝑚))
108107neeq1d 3073 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑚 → ((𝑝𝑛) ≠ 0 ↔ (𝑝𝑚) ≠ 0))
109108rexbidv 3289 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑚 → (∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝑥(𝑝𝑚) ≠ 0))
110 fveq1 6662 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑝 = 𝑞 → (𝑝𝑚) = (𝑞𝑚))
111110neeq1d 3073 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑝 = 𝑞 → ((𝑝𝑚) ≠ 0 ↔ (𝑞𝑚) ≠ 0))
112111cbvrexvw 3435 . . . . . . . . . . . . . . . . . . . . . 22 (∃𝑝 ∈ ran 𝑥(𝑝𝑚) ≠ 0 ↔ ∃𝑞 ∈ ran 𝑥(𝑞𝑚) ≠ 0)
113109, 112syl6bb 290 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑚 → (∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ ∃𝑞 ∈ ran 𝑥(𝑞𝑚) ≠ 0))
114113rspccva 3608 . . . . . . . . . . . . . . . . . . . 20 ((∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ 𝑚 ∈ (1...𝑁)) → ∃𝑞 ∈ ran 𝑥(𝑞𝑚) ≠ 0)
115106, 114sylan 583 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))}) ∧ 𝑚 ∈ (1...𝑁)) → ∃𝑞 ∈ ran 𝑥(𝑞𝑚) ≠ 0)
116 simpr 488 . . . . . . . . . . . . . . . . . . . . . 22 ((∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾) → ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)
117116ralimi 3155 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)
118117ad2antlr 726 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)
119107neeq1d 3073 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑚 → ((𝑝𝑛) ≠ 𝐾 ↔ (𝑝𝑚) ≠ 𝐾))
120119rexbidv 3289 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑚 → (∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾 ↔ ∃𝑝 ∈ ran 𝑥(𝑝𝑚) ≠ 𝐾))
121110neeq1d 3073 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑝 = 𝑞 → ((𝑝𝑚) ≠ 𝐾 ↔ (𝑞𝑚) ≠ 𝐾))
122121cbvrexvw 3435 . . . . . . . . . . . . . . . . . . . . . 22 (∃𝑝 ∈ ran 𝑥(𝑝𝑚) ≠ 𝐾 ↔ ∃𝑞 ∈ ran 𝑥(𝑞𝑚) ≠ 𝐾)
123120, 122syl6bb 290 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑚 → (∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾 ↔ ∃𝑞 ∈ ran 𝑥(𝑞𝑚) ≠ 𝐾))
124123rspccva 3608 . . . . . . . . . . . . . . . . . . . 20 ((∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾𝑚 ∈ (1...𝑁)) → ∃𝑞 ∈ ran 𝑥(𝑞𝑚) ≠ 𝐾)
125118, 124sylan 583 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))}) ∧ 𝑚 ∈ (1...𝑁)) → ∃𝑞 ∈ ran 𝑥(𝑞𝑚) ≠ 𝐾)
12665, 100, 102, 103, 115, 125poimirlem22 35051 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → ∃!𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))}𝑧𝑠)
127 eldifsn 4704 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))} ∖ {𝑠}) ↔ (𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))} ∧ 𝑧𝑠))
128127eubii 2671 . . . . . . . . . . . . . . . . . . 19 (∃!𝑧 𝑧 ∈ ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))} ∖ {𝑠}) ↔ ∃!𝑧(𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))} ∧ 𝑧𝑠))
12953elexi 3499 . . . . . . . . . . . . . . . . . . . 20 ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))} ∖ {𝑠}) ∈ V
130 euhash1 13788 . . . . . . . . . . . . . . . . . . . 20 (({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))} ∖ {𝑠}) ∈ V → ((♯‘({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))} ∖ {𝑠})) = 1 ↔ ∃!𝑧 𝑧 ∈ ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))} ∖ {𝑠})))
131129, 130ax-mp 5 . . . . . . . . . . . . . . . . . . 19 ((♯‘({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))} ∖ {𝑠})) = 1 ↔ ∃!𝑧 𝑧 ∈ ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))} ∖ {𝑠}))
132 df-reu 3140 . . . . . . . . . . . . . . . . . . 19 (∃!𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))}𝑧𝑠 ↔ ∃!𝑧(𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))} ∧ 𝑧𝑠))
133128, 131, 1323bitr4ri 307 . . . . . . . . . . . . . . . . . 18 (∃!𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))}𝑧𝑠 ↔ (♯‘({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))} ∖ {𝑠})) = 1)
134126, 133sylib 221 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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)) ∣ 𝑥 = (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}))))} ∖ {𝑠})) = 1)
135134oveq1d 7166 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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)) ∣ 𝑥 = (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}))))} ∖ {𝑠})) + (♯‘{𝑠})) = (1 + (♯‘{𝑠})))
13663, 135eqtr3d 2861 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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)) ∣ 𝑥 = (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}))))}) = (1 + (♯‘{𝑠})))
13750, 136breqtrrid 5091 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))}) → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))}))
138137ex 416 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) → (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))} → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))})))
139138exlimdv 1935 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) → (∃𝑠 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))} → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))})))
14041, 139syl5bi 245 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) → (¬ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))} = ∅ → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))})))
141 dvds0 15627 . . . . . . . . . . . . . 14 (2 ∈ ℤ → 2 ∥ 0)
1429, 141ax-mp 5 . . . . . . . . . . . . 13 2 ∥ 0
143 hash0 13735 . . . . . . . . . . . . 13 (♯‘∅) = 0
144142, 143breqtrri 5080 . . . . . . . . . . . 12 2 ∥ (♯‘∅)
145 fveq2 6663 . . . . . . . . . . . 12 ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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)) ∣ 𝑥 = (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}))))}) = (♯‘∅))
146144, 145breqtrrid 5091 . . . . . . . . . . 11 ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))} = ∅ → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))}))
147140, 146pm2.61d2 184 . . . . . . . . . 10 (((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))}))
148147ex 416 . . . . . . . . 9 ((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) → (∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾) → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))})))
149148adantld 494 . . . . . . . 8 ((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) → (((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))})))
150 iba 531 . . . . . . . . . . 11 (((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) → (𝑥 = (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})))) ↔ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))))
151150rabbidv 3465 . . . . . . . . . 10 (((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))})
152151fveq2d 6667 . . . . . . . . 9 (((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) → (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))}))
153152breq2d 5065 . . . . . . . 8 (((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) → (2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))}) ↔ 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))})))
154149, 153mpbidi 244 . . . . . . 7 ((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) → (((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))})))
155154a1d 25 . . . . . 6 ((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) → (𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))}))))
15634, 40, 155rexlimd 3309 . . . . 5 ((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) → (∃𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))})))
15733, 156syl5bir 246 . . . 4 ((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) → (¬ ∀𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))})))
158 simpr 488 . . . . . . . . 9 ((𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾))) → ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))
159158con3i 157 . . . . . . . 8 (¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) → ¬ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾))))
160159ralimi 3155 . . . . . . 7 (∀𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) → ∀𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾))))
161 rabeq0 4321 . . . . . . 7 ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))} = ∅ ↔ ∀𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾))))
162160, 161sylibr 237 . . . . . 6 (∀𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))} = ∅)
163162fveq2d 6667 . . . . 5 (∀𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) → (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))}) = (♯‘∅))
164144, 163breqtrrid 5091 . . . 4 (∀𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)) → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))}))
165157, 164pm2.61d2 184 . . 3 ((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) → 2 ∥ (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))}))
1668, 10, 32, 165fsumdvds 15660 . 2 (𝜑 → 2 ∥ Σ𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))(♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))}))
167 rabfi 8742 . . . . 5 (((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶} ∈ Fin)
16827, 167ax-mp 5 . . . 4 {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶} ∈ Fin
169 simp1 1133 . . . . . . 7 ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶)
170 sneq 4560 . . . . . . . . . . . . 13 ((2nd𝑡) = 𝑁 → {(2nd𝑡)} = {𝑁})
171170difeq2d 4085 . . . . . . . . . . . 12 ((2nd𝑡) = 𝑁 → ((0...𝑁) ∖ {(2nd𝑡)}) = ((0...𝑁) ∖ {𝑁}))
172 difun2 4412 . . . . . . . . . . . . 13 (((0...(𝑁 − 1)) ∪ {𝑁}) ∖ {𝑁}) = ((0...(𝑁 − 1)) ∖ {𝑁})
17364nnnn0d 11954 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ ℕ0)
174 nn0uz 12279 . . . . . . . . . . . . . . . . . 18 0 = (ℤ‘0)
175173, 174eleqtrdi 2926 . . . . . . . . . . . . . . . . 17 (𝜑𝑁 ∈ (ℤ‘0))
176 fzm1 12993 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ (ℤ‘0) → (𝑛 ∈ (0...𝑁) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 = 𝑁)))
177175, 176syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑛 ∈ (0...𝑁) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 = 𝑁)))
178 elun 4111 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ((0...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 ∈ {𝑁}))
179 velsn 4566 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ {𝑁} ↔ 𝑛 = 𝑁)
180179orbi2i 910 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 ∈ {𝑁}) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 = 𝑁))
181178, 180bitri 278 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ((0...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 = 𝑁))
182177, 181syl6bbr 292 . . . . . . . . . . . . . . 15 (𝜑 → (𝑛 ∈ (0...𝑁) ↔ 𝑛 ∈ ((0...(𝑁 − 1)) ∪ {𝑁})))
183182eqrdv 2822 . . . . . . . . . . . . . 14 (𝜑 → (0...𝑁) = ((0...(𝑁 − 1)) ∪ {𝑁}))
184183difeq1d 4084 . . . . . . . . . . . . 13 (𝜑 → ((0...𝑁) ∖ {𝑁}) = (((0...(𝑁 − 1)) ∪ {𝑁}) ∖ {𝑁}))
18564nnzd 12085 . . . . . . . . . . . . . . 15 (𝜑𝑁 ∈ ℤ)
186 uzid 12257 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ𝑁))
187 uznfz 12996 . . . . . . . . . . . . . . 15 (𝑁 ∈ (ℤ𝑁) → ¬ 𝑁 ∈ (0...(𝑁 − 1)))
188185, 186, 1873syl 18 . . . . . . . . . . . . . 14 (𝜑 → ¬ 𝑁 ∈ (0...(𝑁 − 1)))
189 disjsn 4632 . . . . . . . . . . . . . . 15 (((0...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (0...(𝑁 − 1)))
190 disj3 4386 . . . . . . . . . . . . . . 15 (((0...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ (0...(𝑁 − 1)) = ((0...(𝑁 − 1)) ∖ {𝑁}))
191189, 190bitr3i 280 . . . . . . . . . . . . . 14 𝑁 ∈ (0...(𝑁 − 1)) ↔ (0...(𝑁 − 1)) = ((0...(𝑁 − 1)) ∖ {𝑁}))
192188, 191sylib 221 . . . . . . . . . . . . 13 (𝜑 → (0...(𝑁 − 1)) = ((0...(𝑁 − 1)) ∖ {𝑁}))
193172, 184, 1923eqtr4a 2885 . . . . . . . . . . . 12 (𝜑 → ((0...𝑁) ∖ {𝑁}) = (0...(𝑁 − 1)))
194171, 193sylan9eqr 2881 . . . . . . . . . . 11 ((𝜑 ∧ (2nd𝑡) = 𝑁) → ((0...𝑁) ∖ {(2nd𝑡)}) = (0...(𝑁 − 1)))
195194rexeqdv 3403 . . . . . . . . . 10 ((𝜑 ∧ (2nd𝑡) = 𝑁) → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶))
196195biimprd 251 . . . . . . . . 9 ((𝜑 ∧ (2nd𝑡) = 𝑁) → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 → ∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶))
197196ralimdv 3173 . . . . . . . 8 ((𝜑 ∧ (2nd𝑡) = 𝑁) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶))
198197expimpd 457 . . . . . . 7 (𝜑 → (((2nd𝑡) = 𝑁 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶))
199169, 198sylan2i 608 . . . . . 6 (𝜑 → (((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶))
200199adantr 484 . . . . 5 ((𝜑𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶))
201200ss2rabdv 4038 . . . 4 (𝜑 → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} ⊆ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶})
202 hashssdif 13780 . . . 4 (({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶} ∈ Fin ∧ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} ⊆ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶}) → (♯‘({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))})) = ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶}) − (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))})))
203168, 201, 202sylancr 590 . . 3 (𝜑 → (♯‘({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))})) = ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶}) − (♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))})))
20464adantr 484 . . . . . . . . . 10 ((𝜑𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → 𝑁 ∈ ℕ)
205 poimirlem28.1 . . . . . . . . . 10 (𝑝 = ((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶)
206 poimirlem28.2 . . . . . . . . . . 11 ((𝜑𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁))
207206adantlr 714 . . . . . . . . . 10 (((𝜑𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁))
208 xp1st 7718 . . . . . . . . . . . 12 (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)}))
209 xp1st 7718 . . . . . . . . . . . 12 ((1st𝑡) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st𝑡)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
210 elmapi 8426 . . . . . . . . . . . 12 ((1st ‘(1st𝑡)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st ‘(1st𝑡)):(1...𝑁)⟶(0..^𝐾))
211208, 209, 2103syl 18 . . . . . . . . . . 11 (𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st ‘(1st𝑡)):(1...𝑁)⟶(0..^𝐾))
212211adantl 485 . . . . . . . . . 10 ((𝜑𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (1st ‘(1st𝑡)):(1...𝑁)⟶(0..^𝐾))
213 xp2nd 7719 . . . . . . . . . . . . 13 ((1st𝑡) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘(1st𝑡)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
214 fvex 6676 . . . . . . . . . . . . . 14 (2nd ‘(1st𝑡)) ∈ V
215 f1oeq1 6597 . . . . . . . . . . . . . 14 (𝑓 = (2nd ‘(1st𝑡)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑡)):(1...𝑁)–1-1-onto→(1...𝑁)))
216214, 215elab 3653 . . . . . . . . . . . . 13 ((2nd ‘(1st𝑡)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑡)):(1...𝑁)–1-1-onto→(1...𝑁))
217213, 216sylib 221 . . . . . . . . . . . 12 ((1st𝑡) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘(1st𝑡)):(1...𝑁)–1-1-onto→(1...𝑁))
218208, 217syl 17 . . . . . . . . . . 11 (𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd ‘(1st𝑡)):(1...𝑁)–1-1-onto→(1...𝑁))
219218adantl 485 . . . . . . . . . 10 ((𝜑𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (2nd ‘(1st𝑡)):(1...𝑁)–1-1-onto→(1...𝑁))
220 xp2nd 7719 . . . . . . . . . . 11 (𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd𝑡) ∈ (0...𝑁))
221220adantl 485 . . . . . . . . . 10 ((𝜑𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (2nd𝑡) ∈ (0...𝑁))
222204, 205, 207, 212, 219, 221poimirlem24 35053 . . . . . . . . 9 ((𝜑𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (∃𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))(𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0)) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = ⟨(1st ‘(1st𝑡)), (2nd ‘(1st𝑡))⟩ / 𝑠𝐶 ∧ ¬ ((2nd𝑡) = 𝑁 ∧ (((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)))))
223208adantl 485 . . . . . . . . . 10 ((𝜑𝑡 ∈ ((((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)}))
224 1st2nd2 7725 . . . . . . . . . . . . . . 15 ((1st𝑡) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st𝑡) = ⟨(1st ‘(1st𝑡)), (2nd ‘(1st𝑡))⟩)
225224csbeq1d 3870 . . . . . . . . . . . . . 14 ((1st𝑡) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st𝑡) / 𝑠𝐶 = ⟨(1st ‘(1st𝑡)), (2nd ‘(1st𝑡))⟩ / 𝑠𝐶)
226225eqeq2d 2835 . . . . . . . . . . . . 13 ((1st𝑡) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (𝑖 = (1st𝑡) / 𝑠𝐶𝑖 = ⟨(1st ‘(1st𝑡)), (2nd ‘(1st𝑡))⟩ / 𝑠𝐶))
227226rexbidv 3289 . . . . . . . . . . . 12 ((1st𝑡) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = ⟨(1st ‘(1st𝑡)), (2nd ‘(1st𝑡))⟩ / 𝑠𝐶))
228227ralbidv 3192 . . . . . . . . . . 11 ((1st𝑡) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = ⟨(1st ‘(1st𝑡)), (2nd ‘(1st𝑡))⟩ / 𝑠𝐶))
229228anbi1d 632 . . . . . . . . . 10 ((1st𝑡) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ¬ ((2nd𝑡) = 𝑁 ∧ (((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = ⟨(1st ‘(1st𝑡)), (2nd ‘(1st𝑡))⟩ / 𝑠𝐶 ∧ ¬ ((2nd𝑡) = 𝑁 ∧ (((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)))))
230223, 229syl 17 . . . . . . . . 9 ((𝜑𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ¬ ((2nd𝑡) = 𝑁 ∧ (((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = ⟨(1st ‘(1st𝑡)), (2nd ‘(1st𝑡))⟩ / 𝑠𝐶 ∧ ¬ ((2nd𝑡) = 𝑁 ∧ (((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)))))
231222, 230bitr4d 285 . . . . . . . 8 ((𝜑𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (∃𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))(𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0)) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ¬ ((2nd𝑡) = 𝑁 ∧ (((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)))))
232101frnd 6512 . . . . . . . . . . . . . 14 (𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1))) → ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁)))
233232anim2i 619 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) → (𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))))
234 dfss3 3941 . . . . . . . . . . . . . 14 ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ↔ ∀𝑛 ∈ (0...(𝑁 − 1))𝑛 ∈ ran (𝑝 ∈ ran 𝑥𝐵))
235 vex 3483 . . . . . . . . . . . . . . . 16 𝑛 ∈ V
236 eqid 2824 . . . . . . . . . . . . . . . . 17 (𝑝 ∈ ran 𝑥𝐵) = (𝑝 ∈ ran 𝑥𝐵)
237236elrnmpt 5816 . . . . . . . . . . . . . . . 16 (𝑛 ∈ V → (𝑛 ∈ ran (𝑝 ∈ ran 𝑥𝐵) ↔ ∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵))
238235, 237ax-mp 5 . . . . . . . . . . . . . . 15 (𝑛 ∈ ran (𝑝 ∈ ran 𝑥𝐵) ↔ ∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)
239238ralbii 3160 . . . . . . . . . . . . . 14 (∀𝑛 ∈ (0...(𝑁 − 1))𝑛 ∈ ran (𝑝 ∈ ran 𝑥𝐵) ↔ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)
240234, 239sylbb 222 . . . . . . . . . . . . 13 ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) → ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)
241 1eluzge0 12291 . . . . . . . . . . . . . . . . 17 1 ∈ (ℤ‘0)
242 fzss1 12952 . . . . . . . . . . . . . . . . 17 (1 ∈ (ℤ‘0) → (1...(𝑁 − 1)) ⊆ (0...(𝑁 − 1)))
243 ssralv 4019 . . . . . . . . . . . . . . . . 17 ((1...(𝑁 − 1)) ⊆ (0...(𝑁 − 1)) → (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 → ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵))
244241, 242, 243mp2b 10 . . . . . . . . . . . . . . . 16 (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 → ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)
24564nncnd 11652 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝑁 ∈ ℂ)
246 npcan1 11065 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
247245, 246syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
248 peano2zm 12024 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
249185, 248syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑁 − 1) ∈ ℤ)
250 uzid 12257 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)))
251 peano2uz 12300 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
252249, 250, 2513syl 18 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
253247, 252eqeltrrd 2917 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑁 ∈ (ℤ‘(𝑁 − 1)))
254 fzss2 12953 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ (ℤ‘(𝑁 − 1)) → (1...(𝑁 − 1)) ⊆ (1...𝑁))
255253, 254syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
256255sselda 3953 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ (1...(𝑁 − 1))) → 𝑛 ∈ (1...𝑁))
257256adantlr 714 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...(𝑁 − 1))) → 𝑛 ∈ (1...𝑁))
258 simplr 768 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁)))
259 ssel2 3948 . . . . . . . . . . . . . . . . . . . . . . 23 ((ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → 𝑝 ∈ ((0...𝐾) ↑m (1...𝑁)))
260 elmapi 8426 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑝 ∈ ((0...𝐾) ↑m (1...𝑁)) → 𝑝:(1...𝑁)⟶(0...𝐾))
261259, 260syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → 𝑝:(1...𝑁)⟶(0...𝐾))
262258, 261sylan 583 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → 𝑝:(1...𝑁)⟶(0...𝐾))
263 poimirlem28.3 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 0)) → 𝐵 < 𝑛)
264 elfzelz 12913 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℤ)
265264zred 12086 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℝ)
266265ltnrd 10774 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑛 ∈ (1...𝑁) → ¬ 𝑛 < 𝑛)
267 breq1 5056 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑛 = 𝐵 → (𝑛 < 𝑛𝐵 < 𝑛))
268267notbid 321 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑛 = 𝐵 → (¬ 𝑛 < 𝑛 ↔ ¬ 𝐵 < 𝑛))
269266, 268syl5ibcom 248 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑛 ∈ (1...𝑁) → (𝑛 = 𝐵 → ¬ 𝐵 < 𝑛))
270269necon2ad 3029 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 ∈ (1...𝑁) → (𝐵 < 𝑛𝑛𝐵))
2712703ad2ant1 1130 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 0) → (𝐵 < 𝑛𝑛𝐵))
272271adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 0)) → (𝐵 < 𝑛𝑛𝐵))
273263, 272mpd 15 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 0)) → 𝑛𝐵)
2742733exp2 1351 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑛 ∈ (1...𝑁) → (𝑝:(1...𝑁)⟶(0...𝐾) → ((𝑝𝑛) = 0 → 𝑛𝐵))))
275274imp31 421 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → ((𝑝𝑛) = 0 → 𝑛𝐵))
276275necon2d 3037 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → (𝑛 = 𝐵 → (𝑝𝑛) ≠ 0))
277276adantllr 718 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → (𝑛 = 𝐵 → (𝑝𝑛) ≠ 0))
278262, 277syldan 594 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → (𝑛 = 𝐵 → (𝑝𝑛) ≠ 0))
279278reximdva 3266 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → (∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 → ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0))
280257, 279syldan 594 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 → ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0))
281280ralimdva 3172 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) → (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 → ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0))
282281imp 410 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0)
283244, 282sylan2 595 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0)
284283biantrurd 536 . . . . . . . . . . . . . 14 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0)))
285 nnuz 12280 . . . . . . . . . . . . . . . . . . . . . 22 ℕ = (ℤ‘1)
28664, 285eleqtrdi 2926 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑁 ∈ (ℤ‘1))
287 fzm1 12993 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ (ℤ‘1) → (𝑛 ∈ (1...𝑁) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁)))
288286, 287syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑛 ∈ (1...𝑁) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁)))
289 elun 4111 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 ∈ {𝑁}))
290179orbi2i 910 . . . . . . . . . . . . . . . . . . . . 21 ((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 ∈ {𝑁}) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁))
291289, 290bitri 278 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁))
292288, 291syl6bbr 292 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑛 ∈ (1...𝑁) ↔ 𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁})))
293292eqrdv 2822 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁}))
294293raleqdv 3402 . . . . . . . . . . . . . . . . 17 (𝜑 → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ ∀𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁})∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0))
295 ralunb 4153 . . . . . . . . . . . . . . . . 17 (∀𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁})∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0))
296294, 295syl6bb 290 . . . . . . . . . . . . . . . 16 (𝜑 → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0)))
297 fveq2 6663 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑁 → (𝑝𝑛) = (𝑝𝑁))
298297neeq1d 3073 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑁 → ((𝑝𝑛) ≠ 0 ↔ (𝑝𝑁) ≠ 0))
299298rexbidv 3289 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑁 → (∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0))
300299ralsng 4598 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ → (∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0))
30164, 300syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0))
302301anbi2d 631 . . . . . . . . . . . . . . . 16 (𝜑 → ((∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0) ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0)))
303296, 302bitrd 282 . . . . . . . . . . . . . . 15 (𝜑 → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0)))
304303ad2antrr 725 . . . . . . . . . . . . . 14 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0)))
305 0z 11991 . . . . . . . . . . . . . . . . . . . . . . 23 0 ∈ ℤ
306 1z 12011 . . . . . . . . . . . . . . . . . . . . . . 23 1 ∈ ℤ
307 fzshftral 13001 . . . . . . . . . . . . . . . . . . . . . . 23 ((0 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ ∧ 1 ∈ ℤ) → (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ ((0 + 1)...((𝑁 − 1) + 1))[(𝑚 − 1) / 𝑛]𝑝 ∈ ran 𝑥 𝑛 = 𝐵))
308305, 306, 307mp3an13 1449 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑁 − 1) ∈ ℤ → (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ ((0 + 1)...((𝑁 − 1) + 1))[(𝑚 − 1) / 𝑛]𝑝 ∈ ran 𝑥 𝑛 = 𝐵))
309185, 248, 3083syl 18 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ ((0 + 1)...((𝑁 − 1) + 1))[(𝑚 − 1) / 𝑛]𝑝 ∈ ran 𝑥 𝑛 = 𝐵))
310 0p1e1 11758 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 + 1) = 1
311310a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (0 + 1) = 1)
312311, 247oveq12d 7169 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((0 + 1)...((𝑁 − 1) + 1)) = (1...𝑁))
313312raleqdv 3402 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (∀𝑚 ∈ ((0 + 1)...((𝑁 − 1) + 1))[(𝑚 − 1) / 𝑛]𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ (1...𝑁)[(𝑚 − 1) / 𝑛]𝑝 ∈ ran 𝑥 𝑛 = 𝐵))
314309, 313bitrd 282 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ (1...𝑁)[(𝑚 − 1) / 𝑛]𝑝 ∈ ran 𝑥 𝑛 = 𝐵))
315 ovex 7184 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 − 1) ∈ V
316 eqeq1 2828 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = (𝑚 − 1) → (𝑛 = 𝐵 ↔ (𝑚 − 1) = 𝐵))
317316rexbidv 3289 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = (𝑚 − 1) → (∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵))
318315, 317sbcie 3798 . . . . . . . . . . . . . . . . . . . . . 22 ([(𝑚 − 1) / 𝑛]𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵)
319318ralbii 3160 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑚 ∈ (1...𝑁)[(𝑚 − 1) / 𝑛]𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵)
320 oveq1 7158 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 = 𝑛 → (𝑚 − 1) = (𝑛 − 1))
321320eqeq1d 2826 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = 𝑛 → ((𝑚 − 1) = 𝐵 ↔ (𝑛 − 1) = 𝐵))
322321rexbidv 3289 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑛 → (∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵 ↔ ∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵))
323322cbvralvw 3434 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑚 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵 ↔ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵)
324319, 323bitri 278 . . . . . . . . . . . . . . . . . . . 20 (∀𝑚 ∈ (1...𝑁)[(𝑚 − 1) / 𝑛]𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵)
325314, 324syl6bb 290 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵))
326325biimpa 480 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵)
327326adantlr 714 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵)
328 poimirlem28.4 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 𝐾)) → 𝐵 ≠ (𝑛 − 1))
329328necomd 3069 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 𝐾)) → (𝑛 − 1) ≠ 𝐵)
3303293exp2 1351 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑛 ∈ (1...𝑁) → (𝑝:(1...𝑁)⟶(0...𝐾) → ((𝑝𝑛) = 𝐾 → (𝑛 − 1) ≠ 𝐵))))
331330imp31 421 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → ((𝑝𝑛) = 𝐾 → (𝑛 − 1) ≠ 𝐵))
332331necon2d 3037 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → ((𝑛 − 1) = 𝐵 → (𝑝𝑛) ≠ 𝐾))
333332adantllr 718 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → ((𝑛 − 1) = 𝐵 → (𝑝𝑛) ≠ 𝐾))
334262, 333syldan 594 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → ((𝑛 − 1) = 𝐵 → (𝑝𝑛) ≠ 𝐾))
335334reximdva 3266 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → (∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵 → ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾))
336335ralimdva 3172 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵 → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾))
337336imp 410 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)
338327, 337syldan 594 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)
339338biantrud 535 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))
340 r19.26 3165 . . . . . . . . . . . . . . 15 (∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾) ↔ (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾))
341339, 340syl6bbr 292 . . . . . . . . . . . . . 14 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))
342284, 304, 3413bitr2d 310 . . . . . . . . . . . . 13 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑m (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0 ↔ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))
343233, 240, 342syl2an 598 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) ∧ (0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵)) → (∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0 ↔ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))
344343pm5.32da 582 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) → (((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0) ↔ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾))))
345344anbi2d 631 . . . . . . . . . 10 ((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) → ((𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0)) ↔ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))))
346345rexbidva 3288 . . . . . . . . 9 (𝜑 → (∃𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))(𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0)) ↔ ∃𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))(𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))))
347346adantr 484 . . . . . . . 8 ((𝜑𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (∃𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))(𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0)) ↔ ∃𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))(𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))))
348193rexeqdv 3403 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶))
349348biimpd 232 . . . . . . . . . . . . . . . . . 18 (𝜑 → (∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = (1st𝑡) / 𝑠𝐶 → ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶))
350349ralimdv 3173 . . . . . . . . . . . . . . . . 17 (𝜑 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = (1st𝑡) / 𝑠𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶))
351171rexeqdv 3403 . . . . . . . . . . . . . . . . . . 19 ((2nd𝑡) = 𝑁 → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = (1st𝑡) / 𝑠𝐶))
352351ralbidv 3192 . . . . . . . . . . . . . . . . . 18 ((2nd𝑡) = 𝑁 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = (1st𝑡) / 𝑠𝐶))
353352imbi1d 345 . . . . . . . . . . . . . . . . 17 ((2nd𝑡) = 𝑁 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = (1st𝑡) / 𝑠𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶)))
354350, 353syl5ibrcom 250 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd𝑡) = 𝑁 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶)))
355354com23 86 . . . . . . . . . . . . . . 15 (𝜑 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 → ((2nd𝑡) = 𝑁 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶)))
356355imp 410 . . . . . . . . . . . . . 14 ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶) → ((2nd𝑡) = 𝑁 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶))
357356adantrd 495 . . . . . . . . . . . . 13 ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶) → (((2nd𝑡) = 𝑁 ∧ (((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶))
358357pm4.71rd 566 . . . . . . . . . . . 12 ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶) → (((2nd𝑡) = 𝑁 ∧ (((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((2nd𝑡) = 𝑁 ∧ (((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)))))
359 an12 644 . . . . . . . . . . . . 13 ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((2nd𝑡) = 𝑁 ∧ (((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))) ↔ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ (((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))))
360 3anass 1092 . . . . . . . . . . . . . 14 ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ (((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)))
361360anbi2i 625 . . . . . . . . . . . . 13 (((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)) ↔ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ (((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))))
362359, 361bitr4i 281 . . . . . . . . . . . 12 ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((2nd𝑡) = 𝑁 ∧ (((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))) ↔ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)))
363358, 362syl6bb 290 . . . . . . . . . . 11 ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶) → (((2nd𝑡) = 𝑁 ∧ (((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)) ↔ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))))
364363notbid 321 . . . . . . . . . 10 ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶) → (¬ ((2nd𝑡) = 𝑁 ∧ (((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)) ↔ ¬ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))))
365364pm5.32da 582 . . . . . . . . 9 (𝜑 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ¬ ((2nd𝑡) = 𝑁 ∧ (((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ¬ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)))))
366365adantr 484 . . . . . . . 8 ((𝜑𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ¬ ((2nd𝑡) = 𝑁 ∧ (((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ¬ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)))))
367231, 347, 3663bitr3d 312 . . . . . . 7 ((𝜑𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (∃𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))(𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾))) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ¬ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)))))
368367rabbidva 3463 . . . . . 6 (𝜑 → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∃𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))(𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))} = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ¬ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)))})
369 iunrab 4962 . . . . . 6 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))} = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∃𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))(𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))}
370 difrab 4262 . . . . . 6 ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}) = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ¬ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)))}
371368, 369, 3703eqtr4g 2884 . . . . 5 (𝜑 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))} = ({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}))
372371fveq2d 6667 . . . 4 (𝜑 → (♯‘ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))}) = (♯‘({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))})))
37327, 28mp1i 13 . . . . 5 ((𝜑𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))) → {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))} ∈ Fin)
374 simpl 486 . . . . . . . . . . . 12 ((𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾))) → 𝑥 = (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})))))
375374a1i 11 . . . . . . . . . . 11 (𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → ((𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾))) → 𝑥 = (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}))))))
376375ss2rabi 4039 . . . . . . . . . 10 {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))} ⊆ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))}
377376sseli 3949 . . . . . . . . 9 (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))} → 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))})
378 fveq2 6663 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑠 → (2nd𝑡) = (2nd𝑠))
379378breq2d 5065 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑠 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑠)))
380379ifbid 4472 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑠 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑠), 𝑦, (𝑦 + 1)))
381380csbeq1d 3870 . . . . . . . . . . . . . . 15 (𝑡 = 𝑠if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑠), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))
382 2fveq3 6668 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑠 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑠)))
383 2fveq3 6668 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑠 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑠)))
384383imaeq1d 5917 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑠 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑠)) “ (1...𝑗)))
385384xpeq1d 5572 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑠 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑠)) “ (1...𝑗)) × {1}))
386383imaeq1d 5917 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑠 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)))
387386xpeq1d 5572 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑠 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)) × {0}))
388385, 387uneq12d 4126 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑠 → ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))
389382, 388oveq12d 7169 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑠 → ((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑠)) ∘f + ((((2nd ‘(1st𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)) × {0}))))
390389csbeq2dv 3873 . . . . . . . . . . . . . . 15 (𝑡 = 𝑠if(𝑦 < (2nd𝑠), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑠), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑠)) ∘f + ((((2nd ‘(1st𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)) × {0}))))
391381, 390eqtrd 2859 . . . . . . . . . . . . . 14 (𝑡 = 𝑠if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑠), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑠)) ∘f + ((((2nd ‘(1st𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)) × {0}))))
392391mpteq2dv 5149 . . . . . . . . . . . . 13 (𝑡 = 𝑠 → (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 < (2nd𝑠), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑠)) ∘f + ((((2nd ‘(1st𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))))
393392eqeq2d 2835 . . . . . . . . . . . 12 (𝑡 = 𝑠 → (𝑥 = (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})))) ↔ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑠), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑠)) ∘f + ((((2nd ‘(1st𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
394 eqcom 2831 . . . . . . . . . . . 12 (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑠), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑠)) ∘f + ((((2nd ‘(1st𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑠), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑠)) ∘f + ((((2nd ‘(1st𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥)
395393, 394syl6bb 290 . . . . . . . . . . 11 (𝑡 = 𝑠 → (𝑥 = (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 < (2nd𝑠), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑠)) ∘f + ((((2nd ‘(1st𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥))
396395elrab 3666 . . . . . . . . . 10 (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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}))))} ↔ (𝑠 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑠), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑠)) ∘f + ((((2nd ‘(1st𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥))
397396simprbi 500 . . . . . . . . 9 (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (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 < (2nd𝑠), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑠)) ∘f + ((((2nd ‘(1st𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥)
398377, 397syl 17 . . . . . . . 8 (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))} → (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑠), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑠)) ∘f + ((((2nd ‘(1st𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥)
399398rgen 3143 . . . . . . 7 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))} (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑠), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑠)) ∘f + ((((2nd ‘(1st𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥
400399rgenw 3145 . . . . . 6 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))∀𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))} (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑠), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑠)) ∘f + ((((2nd ‘(1st𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥
401 invdisj 5037 . . . . . 6 (∀𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))∀𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))} (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑠), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑠)) ∘f + ((((2nd ‘(1st𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥Disj 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))})
402400, 401mp1i 13 . . . . 5 (𝜑Disj 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))})
4038, 373, 402hashiun 15179 . . . 4 (𝜑 → (♯‘ 𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))}) = Σ𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))(♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))}))
404372, 403eqtr3d 2861 . . 3 (𝜑 → (♯‘({𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))})) = Σ𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))(♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (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})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)))}))
405 fo1st 7706 . . . . . . . . . . . . 13 1st :V–onto→V
406 fofun 6584 . . . . . . . . . . . . 13 (1st :V–onto→V → Fun 1st )
407405, 406ax-mp 5 . . . . . . . . . . . 12 Fun 1st
408 ssv 3977 . . . . . . . . . . . . 13 {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} ⊆ V
409 fof 6583 . . . . . . . . . . . . . . 15 (1st :V–onto→V → 1st :V⟶V)
410405, 409ax-mp 5 . . . . . . . . . . . . . 14 1st :V⟶V
411410fdmi 6516 . . . . . . . . . . . . 13 dom 1st = V
412408, 411sseqtrri 3990 . . . . . . . . . . . 12 {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} ⊆ dom 1st
413 fores 6593 . . . . . . . . . . . 12 ((Fun 1st ∧ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} ⊆ dom 1st ) → (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}))
414407, 412, 413mp2an 691 . . . . . . . . . . 11 (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))})
415 fveqeq2 6672 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑥 → ((2nd𝑡) = 𝑁 ↔ (2nd𝑥) = 𝑁))
416 fveq2 6663 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = 𝑥 → (1st𝑡) = (1st𝑥))
417416csbeq1d 3870 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = 𝑥(1st𝑡) / 𝑠𝐶 = (1st𝑥) / 𝑠𝐶)
418417eqeq2d 2835 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑥 → (𝑖 = (1st𝑡) / 𝑠𝐶𝑖 = (1st𝑥) / 𝑠𝐶))
419418rexbidv 3289 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑥 → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶))
420419ralbidv 3192 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑥 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶))
421 2fveq3 6668 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑥 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑥)))
422421fveq1d 6665 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑥 → ((1st ‘(1st𝑡))‘𝑁) = ((1st ‘(1st𝑥))‘𝑁))
423422eqeq1d 2826 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑥 → (((1st ‘(1st𝑡))‘𝑁) = 0 ↔ ((1st ‘(1st𝑥))‘𝑁) = 0))
424 2fveq3 6668 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑥 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑥)))
425424fveq1d 6665 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑥 → ((2nd ‘(1st𝑡))‘𝑁) = ((2nd ‘(1st𝑥))‘𝑁))
426425eqeq1d 2826 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑥 → (((2nd ‘(1st𝑡))‘𝑁) = 𝑁 ↔ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁))
427420, 423, 4263anbi123d 1433 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑥 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁)))
428415, 427anbi12d 633 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑥 → (((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)) ↔ ((2nd𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁))))
429428rexrab 3673 . . . . . . . . . . . . . . 15 (∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} (1st𝑥) = 𝑠 ↔ ∃𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁)) ∧ (1st𝑥) = 𝑠))
430 xp1st 7718 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st𝑥) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
431430anim1i 617 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁)) → ((1st𝑥) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁)))
432 eleq1 2903 . . . . . . . . . . . . . . . . . . . . 21 ((1st𝑥) = 𝑠 → ((1st𝑥) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ↔ 𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})))
433 csbeq1a 3880 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑠 = (1st𝑥) → 𝐶 = (1st𝑥) / 𝑠𝐶)
434433eqcoms 2832 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((1st𝑥) = 𝑠𝐶 = (1st𝑥) / 𝑠𝐶)
435434eqcomd 2830 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((1st𝑥) = 𝑠(1st𝑥) / 𝑠𝐶 = 𝐶)
436435eqeq2d 2835 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1st𝑥) = 𝑠 → (𝑖 = (1st𝑥) / 𝑠𝐶𝑖 = 𝐶))
437436rexbidv 3289 . . . . . . . . . . . . . . . . . . . . . . 23 ((1st𝑥) = 𝑠 → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶))
438437ralbidv 3192 . . . . . . . . . . . . . . . . . . . . . 22 ((1st𝑥) = 𝑠 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶))
439 fveq2 6663 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1st𝑥) = 𝑠 → (1st ‘(1st𝑥)) = (1st𝑠))
440439fveq1d 6665 . . . . . . . . . . . . . . . . . . . . . . 23 ((1st𝑥) = 𝑠 → ((1st ‘(1st𝑥))‘𝑁) = ((1st𝑠)‘𝑁))
441440eqeq1d 2826 . . . . . . . . . . . . . . . . . . . . . 22 ((1st𝑥) = 𝑠 → (((1st ‘(1st𝑥))‘𝑁) = 0 ↔ ((1st𝑠)‘𝑁) = 0))
442 fveq2 6663 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1st𝑥) = 𝑠 → (2nd ‘(1st𝑥)) = (2nd𝑠))
443442fveq1d 6665 . . . . . . . . . . . . . . . . . . . . . . 23 ((1st𝑥) = 𝑠 → ((2nd ‘(1st𝑥))‘𝑁) = ((2nd𝑠)‘𝑁))
444443eqeq1d 2826 . . . . . . . . . . . . . . . . . . . . . 22 ((1st𝑥) = 𝑠 → (((2nd ‘(1st𝑥))‘𝑁) = 𝑁 ↔ ((2nd𝑠)‘𝑁) = 𝑁))
445438, 441, 4443anbi123d 1433 . . . . . . . . . . . . . . . . . . . . 21 ((1st𝑥) = 𝑠 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)))
446432, 445anbi12d 633 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑥) = 𝑠 → (((1st𝑥) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁)) ↔ (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁))))
447431, 446syl5ibcom 248 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁)) → ((1st𝑥) = 𝑠 → (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁))))
448447adantrl 715 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ ((2nd𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁))) → ((1st𝑥) = 𝑠 → (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁))))
449448expimpd 457 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → ((((2nd𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁)) ∧ (1st𝑥) = 𝑠) → (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁))))
450449rexlimiv 3272 . . . . . . . . . . . . . . . 16 (∃𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁)) ∧ (1st𝑥) = 𝑠) → (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)))
451 nn0fz0 13011 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0𝑁 ∈ (0...𝑁))
452173, 451sylib 221 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑁 ∈ (0...𝑁))
453 opelxpi 5580 . . . . . . . . . . . . . . . . . . . 20 ((𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑁 ∈ (0...𝑁)) → ⟨𝑠, 𝑁⟩ ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
454452, 453sylan2 595 . . . . . . . . . . . . . . . . . . 19 ((𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝜑) → ⟨𝑠, 𝑁⟩ ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
455454ancoms 462 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) → ⟨𝑠, 𝑁⟩ ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
456 opelxp2 5585 . . . . . . . . . . . . . . . . . . . . 21 (⟨𝑠, 𝑁⟩ ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → 𝑁 ∈ (0...𝑁))
457 op2ndg 7699 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (2nd ‘⟨𝑠, 𝑁⟩) = 𝑁)
458457biantrurd 536 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶 ∧ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁) ↔ ((2nd ‘⟨𝑠, 𝑁⟩) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶 ∧ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁))))
459 op1stg 7698 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (1st ‘⟨𝑠, 𝑁⟩) = 𝑠)
460 csbeq1a 3880 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑠 = (1st ‘⟨𝑠, 𝑁⟩) → 𝐶 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶)
461460eqcoms 2832 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((1st ‘⟨𝑠, 𝑁⟩) = 𝑠𝐶 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶)
462461eqcomd 2830 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((1st ‘⟨𝑠, 𝑁⟩) = 𝑠(1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶 = 𝐶)
463459, 462syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶 = 𝐶)
464463eqeq2d 2835 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶𝑖 = 𝐶))
465464rexbidv 3289 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶))
466465ralbidv 3192 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶))
467459fveq2d 6667 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (1st ‘(1st ‘⟨𝑠, 𝑁⟩)) = (1st𝑠))
468467fveq1d 6665 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = ((1st𝑠)‘𝑁))
469468eqeq1d 2826 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ↔ ((1st𝑠)‘𝑁) = 0))
470459fveq2d 6667 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (2nd ‘(1st ‘⟨𝑠, 𝑁⟩)) = (2nd𝑠))
471470fveq1d 6665 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = ((2nd𝑠)‘𝑁))
472471eqeq1d 2826 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁 ↔ ((2nd𝑠)‘𝑁) = 𝑁))
473466, 469, 4723anbi123d 1433 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶 ∧ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)))
474459biantrud 535 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (((2nd ‘⟨𝑠, 𝑁⟩) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶 ∧ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁)) ↔ (((2nd ‘⟨𝑠, 𝑁⟩) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶 ∧ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁)) ∧ (1st ‘⟨𝑠, 𝑁⟩) = 𝑠)))
475458, 473, 4743bitr3d 312 . . . . . . . . . . . . . . . . . . . . 21 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁) ↔ (((2nd ‘⟨𝑠, 𝑁⟩) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶 ∧ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁)) ∧ (1st ‘⟨𝑠, 𝑁⟩) = 𝑠)))
47644, 456, 475sylancr 590 . . . . . . . . . . . . . . . . . . . 20 (⟨𝑠, 𝑁⟩ ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁) ↔ (((2nd ‘⟨𝑠, 𝑁⟩) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶 ∧ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁)) ∧ (1st ‘⟨𝑠, 𝑁⟩) = 𝑠)))
477476biimpa 480 . . . . . . . . . . . . . . . . . . 19 ((⟨𝑠, 𝑁⟩ ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)) → (((2nd ‘⟨𝑠, 𝑁⟩) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶 ∧ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁)) ∧ (1st ‘⟨𝑠, 𝑁⟩) = 𝑠))
478 fveqeq2 6672 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = ⟨𝑠, 𝑁⟩ → ((2nd𝑥) = 𝑁 ↔ (2nd ‘⟨𝑠, 𝑁⟩) = 𝑁))
479 fveq2 6663 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 = ⟨𝑠, 𝑁⟩ → (1st𝑥) = (1st ‘⟨𝑠, 𝑁⟩))
480479csbeq1d 3870 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = ⟨𝑠, 𝑁⟩ → (1st𝑥) / 𝑠𝐶 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶)
481480eqeq2d 2835 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = ⟨𝑠, 𝑁⟩ → (𝑖 = (1st𝑥) / 𝑠𝐶𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶))
482481rexbidv 3289 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = ⟨𝑠, 𝑁⟩ → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶))
483482ralbidv 3192 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = ⟨𝑠, 𝑁⟩ → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶))
484 2fveq3 6668 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = ⟨𝑠, 𝑁⟩ → (1st ‘(1st𝑥)) = (1st ‘(1st ‘⟨𝑠, 𝑁⟩)))
485484fveq1d 6665 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = ⟨𝑠, 𝑁⟩ → ((1st ‘(1st𝑥))‘𝑁) = ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁))
486485eqeq1d 2826 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = ⟨𝑠, 𝑁⟩ → (((1st ‘(1st𝑥))‘𝑁) = 0 ↔ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0))
487 2fveq3 6668 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = ⟨𝑠, 𝑁⟩ → (2nd ‘(1st𝑥)) = (2nd ‘(1st ‘⟨𝑠, 𝑁⟩)))
488487fveq1d 6665 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = ⟨𝑠, 𝑁⟩ → ((2nd ‘(1st𝑥))‘𝑁) = ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁))
489488eqeq1d 2826 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = ⟨𝑠, 𝑁⟩ → (((2nd ‘(1st𝑥))‘𝑁) = 𝑁 ↔ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁))
490483, 486, 4893anbi123d 1433 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = ⟨𝑠, 𝑁⟩ → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶 ∧ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁)))
491478, 490anbi12d 633 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = ⟨𝑠, 𝑁⟩ → (((2nd𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁)) ↔ ((2nd ‘⟨𝑠, 𝑁⟩) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶 ∧ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁))))
492 fveqeq2 6672 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = ⟨𝑠, 𝑁⟩ → ((1st𝑥) = 𝑠 ↔ (1st ‘⟨𝑠, 𝑁⟩) = 𝑠))
493491, 492anbi12d 633 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = ⟨𝑠, 𝑁⟩ → ((((2nd𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁)) ∧ (1st𝑥) = 𝑠) ↔ (((2nd ‘⟨𝑠, 𝑁⟩) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶 ∧ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁)) ∧ (1st ‘⟨𝑠, 𝑁⟩) = 𝑠)))
494493rspcev 3609 . . . . . . . . . . . . . . . . . . 19 ((⟨𝑠, 𝑁⟩ ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (((2nd ‘⟨𝑠, 𝑁⟩) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶 ∧ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁)) ∧ (1st ‘⟨𝑠, 𝑁⟩) = 𝑠)) → ∃𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁)) ∧ (1st𝑥) = 𝑠))
495477, 494syldan 594 . . . . . . . . . . . . . . . . . 18 ((⟨𝑠, 𝑁⟩ ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)) → ∃𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁)) ∧ (1st𝑥) = 𝑠))
496455, 495sylan 583 . . . . . . . . . . . . . . . . 17 (((𝜑𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)) → ∃𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁)) ∧ (1st𝑥) = 𝑠))
497496expl 461 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)) → ∃𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁)) ∧ (1st𝑥) = 𝑠)))
498450, 497impbid2 229 . . . . . . . . . . . . . . 15 (𝜑 → (∃𝑥 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ∧ ((1st ‘(1st𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁)) ∧ (1st𝑥) = 𝑠) ↔ (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁))))
499429, 498syl5bb 286 . . . . . . . . . . . . . 14 (𝜑 → (∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} (1st𝑥) = 𝑠 ↔ (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁))))
500499abbidv 2888 . . . . . . . . . . . . 13 (𝜑 → {𝑠 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} (1st𝑥) = 𝑠} = {𝑠 ∣ (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁))})
501 dfimafn 6721 . . . . . . . . . . . . . . 15 ((Fun 1st ∧ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} ⊆ dom 1st ) → (1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}) = {𝑦 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} (1st𝑥) = 𝑦})
502407, 412, 501mp2an 691 . . . . . . . . . . . . . 14 (1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}) = {𝑦 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} (1st𝑥) = 𝑦}
503 nfv 1916 . . . . . . . . . . . . . . . . . 18 𝑠(2nd𝑡) = 𝑁
504 nfcv 2982 . . . . . . . . . . . . . . . . . . . 20 𝑠(0...(𝑁 − 1))
505 nfcsb1v 3890 . . . . . . . . . . . . . . . . . . . . . 22 𝑠(1st𝑡) / 𝑠𝐶
506505nfeq2 2999 . . . . . . . . . . . . . . . . . . . . 21 𝑠 𝑖 = (1st𝑡) / 𝑠𝐶
507504, 506nfrex 3301 . . . . . . . . . . . . . . . . . . . 20 𝑠𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶
508504, 507nfralw 3219 . . . . . . . . . . . . . . . . . . 19 𝑠𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶
509 nfv 1916 . . . . . . . . . . . . . . . . . . 19 𝑠((1st ‘(1st𝑡))‘𝑁) = 0
510 nfv 1916 . . . . . . . . . . . . . . . . . . 19 𝑠((2nd ‘(1st𝑡))‘𝑁) = 𝑁
511508, 509, 510nf3an 1903 . . . . . . . . . . . . . . . . . 18 𝑠(∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)
512503, 511nfan 1901 . . . . . . . . . . . . . . . . 17 𝑠((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))
513 nfcv 2982 . . . . . . . . . . . . . . . . 17 𝑠((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))
514512, 513nfrabw 3376 . . . . . . . . . . . . . . . 16 𝑠{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}
515 nfv 1916 . . . . . . . . . . . . . . . 16 𝑠(1st𝑥) = 𝑦
516514, 515nfrex 3301 . . . . . . . . . . . . . . 15 𝑠𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} (1st𝑥) = 𝑦
517 nfv 1916 . . . . . . . . . . . . . . 15 𝑦𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} (1st𝑥) = 𝑠
518 eqeq2 2836 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑠 → ((1st𝑥) = 𝑦 ↔ (1st𝑥) = 𝑠))
519518rexbidv 3289 . . . . . . . . . . . . . . 15 (𝑦 = 𝑠 → (∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} (1st𝑥) = 𝑦 ↔ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} (1st𝑥) = 𝑠))
520516, 517, 519cbvabw 2893 . . . . . . . . . . . . . 14 {𝑦 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} (1st𝑥) = 𝑦} = {𝑠 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} (1st𝑥) = 𝑠}
521502, 520eqtri 2847 . . . . . . . . . . . . 13 (1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}) = {𝑠 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))} (1st𝑥) = 𝑠}
522 df-rab 3142 . . . . . . . . . . . . 13 {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)} = {𝑠 ∣ (𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁))}
523500, 521, 5223eqtr4g 2884 . . . . . . . . . . . 12 (𝜑 → (1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}) = {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)})
524 foeq3 6581 . . . . . . . . . . . 12 ((1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}) = {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)} → ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}) ↔ (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))}–onto→{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st𝑠)‘𝑁) = 0 ∧ ((2nd𝑠)‘𝑁) = 𝑁)}))
525523, 524syl 17 . . . . . . . . . . 11 (𝜑 → ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁