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Theorem poimirlem27 32606
 Description: Lemma for poimir 32612 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 12633 . . . . . 6 (0...𝐾) ∈ Fin
2 fzfi 12633 . . . . . 6 (1...𝑁) ∈ Fin
3 mapfi 8145 . . . . . 6 (((0...𝐾) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((0...𝐾) ↑𝑚 (1...𝑁)) ∈ Fin)
41, 2, 3mp2an 704 . . . . 5 ((0...𝐾) ↑𝑚 (1...𝑁)) ∈ Fin
5 fzfi 12633 . . . . 5 (0...(𝑁 − 1)) ∈ Fin
6 mapfi 8145 . . . . 5 ((((0...𝐾) ↑𝑚 (1...𝑁)) ∈ Fin ∧ (0...(𝑁 − 1)) ∈ Fin) → (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1))) ∈ Fin)
74, 5, 6mp2an 704 . . . 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 11286 . . . 4 2 ∈ ℤ
109a1i 11 . . 3 (𝜑 → 2 ∈ ℤ)
11 fzofi 12635 . . . . . . . 8 (0..^𝐾) ∈ Fin
12 mapfi 8145 . . . . . . . 8 (((0..^𝐾) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((0..^𝐾) ↑𝑚 (1...𝑁)) ∈ Fin)
1311, 2, 12mp2an 704 . . . . . . 7 ((0..^𝐾) ↑𝑚 (1...𝑁)) ∈ Fin
14 mapfi 8145 . . . . . . . . 9 (((1...𝑁) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((1...𝑁) ↑𝑚 (1...𝑁)) ∈ Fin)
152, 2, 14mp2an 704 . . . . . . . 8 ((1...𝑁) ↑𝑚 (1...𝑁)) ∈ Fin
16 f1of 6050 . . . . . . . . . 10 (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑓:(1...𝑁)⟶(1...𝑁))
1716ss2abi 3637 . . . . . . . . 9 {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ {𝑓𝑓:(1...𝑁)⟶(1...𝑁)}
18 ovex 6577 . . . . . . . . . 10 (1...𝑁) ∈ V
1918, 18mapval 7756 . . . . . . . . 9 ((1...𝑁) ↑𝑚 (1...𝑁)) = {𝑓𝑓:(1...𝑁)⟶(1...𝑁)}
2017, 19sseqtr4i 3601 . . . . . . . 8 {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ ((1...𝑁) ↑𝑚 (1...𝑁))
21 ssfi 8065 . . . . . . . 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 704 . . . . . . 7 {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin
23 xpfi 8116 . . . . . . 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 704 . . . . . 6 (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin
25 fzfi 12633 . . . . . 6 (0...𝑁) ∈ Fin
26 xpfi 8116 . . . . . 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 704 . . . . 5 ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin
28 rabfi 8070 . . . . 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 13009 . . . . 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 11356 . . . 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 2979 . . . . 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 1830 . . . . . 6 𝑡(𝜑𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1))))
35 nfcv 2751 . . . . . . 7 𝑡2
36 nfcv 2751 . . . . . . 7 𝑡
37 nfcv 2751 . . . . . . . 8 𝑡#
38 nfrab1 3099 . . . . . . . 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 6110 . . . . . . 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 4629 . . . . . 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 3889 . . . . . . . . . . . 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 14833 . . . . . . . . . . . . . . . . 17 (2 ∈ ℤ → 2 ∥ 2)
439, 42ax-mp 5 . . . . . . . . . . . . . . . 16 2 ∥ 2
44 vex 3176 . . . . . . . . . . . . . . . . . . 19 𝑠 ∈ V
45 hashsng 13020 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ V → (#‘{𝑠}) = 1)
4644, 45ax-mp 5 . . . . . . . . . . . . . . . . . 18 (#‘{𝑠}) = 1
4746oveq2i 6560 . . . . . . . . . . . . . . . . 17 (1 + (#‘{𝑠})) = (1 + 1)
48 df-2 10956 . . . . . . . . . . . . . . . . 17 2 = (1 + 1)
4947, 48eqtr4i 2635 . . . . . . . . . . . . . . . 16 (1 + (#‘{𝑠})) = 2
5043, 49breqtrri 4610 . . . . . . . . . . . . . . 15 2 ∥ (1 + (#‘{𝑠}))
51 rabfi 8070 . . . . . . . . . . . . . . . . . . . 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 8077 . . . . . . . . . . . . . . . . . . . 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 7923 . . . . . . . . . . . . . . . . . . 19 {𝑠} ∈ Fin
55 incom 3767 . . . . . . . . . . . . . . . . . . . 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 3992 . . . . . . . . . . . . . . . . . . . 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 2632 . . . . . . . . . . . . . . . . . . 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 13032 . . . . . . . . . . . . . . . . . . 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 1416 . . . . . . . . . . . . . . . . . 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 4282 . . . . . . . . . . . . . . . . . . 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 6107 . . . . . . . . . . . . . . . . . 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 2658 . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . 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 762 . . . . . . . . . . . . . . . . . . 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 6103 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 = 𝑢 → (2nd𝑡) = (2nd𝑢))
6766breq2d 4595 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = 𝑢 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑢)))
6867ifbid 4058 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑢 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑢), 𝑦, (𝑦 + 1)))
6968csbeq1d 3506 . . . . . . . . . . . . . . . . . . . . . . . 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 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 = 𝑢 → (1st𝑡) = (1st𝑢))
7170fveq2d 6107 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = 𝑢 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑢)))
7270fveq2d 6107 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑡 = 𝑢 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑢)))
7372imaeq1d 5384 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑡 = 𝑢 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑢)) “ (1...𝑗)))
7473xpeq1d 5062 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 = 𝑢 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑢)) “ (1...𝑗)) × {1}))
7572imaeq1d 5384 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑡 = 𝑢 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)))
7675xpeq1d 5062 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 = 𝑢 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))
7774, 76uneq12d 3730 . . . . . . . . . . . . . . . . . . . . . . . . . 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})))
7871, 77oveq12d 6567 . . . . . . . . . . . . . . . . . . . . . . . . 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}))))
7978csbeq2dv 3944 . . . . . . . . . . . . . . . . . . . . . . . 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}))))
8069, 79eqtrd 2644 . . . . . . . . . . . . . . . . . . . . . . 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}))))
8180mpteq2dv 4673 . . . . . . . . . . . . . . . . . . . . . 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})))))
82 breq1 4586 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑤 → (𝑦 < (2nd𝑢) ↔ 𝑤 < (2nd𝑢)))
83 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑤𝑦 = 𝑤)
84 oveq1 6556 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑤 → (𝑦 + 1) = (𝑤 + 1))
8582, 83, 84ifbieq12d 4063 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 = 𝑤 → if(𝑦 < (2nd𝑢), 𝑦, (𝑦 + 1)) = if(𝑤 < (2nd𝑢), 𝑤, (𝑤 + 1)))
8685csbeq1d 3506 . . . . . . . . . . . . . . . . . . . . . . . 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}))))
87 oveq2 6557 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 = 𝑖 → (1...𝑗) = (1...𝑖))
8887imaeq2d 5385 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 = 𝑖 → ((2nd ‘(1st𝑢)) “ (1...𝑗)) = ((2nd ‘(1st𝑢)) “ (1...𝑖)))
8988xpeq1d 5062 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 = 𝑖 → (((2nd ‘(1st𝑢)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑢)) “ (1...𝑖)) × {1}))
90 oveq1 6556 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 = 𝑖 → (𝑗 + 1) = (𝑖 + 1))
9190oveq1d 6564 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 = 𝑖 → ((𝑗 + 1)...𝑁) = ((𝑖 + 1)...𝑁))
9291imaeq2d 5385 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 = 𝑖 → ((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑢)) “ ((𝑖 + 1)...𝑁)))
9392xpeq1d 5062 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 = 𝑖 → (((2nd ‘(1st𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑢)) “ ((𝑖 + 1)...𝑁)) × {0}))
9489, 93uneq12d 3730 . . . . . . . . . . . . . . . . . . . . . . . . . 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})))
9594oveq2d 6565 . . . . . . . . . . . . . . . . . . . . . . . . 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}))))
9695cbvcsbv 3505 . . . . . . . . . . . . . . . . . . . . . . . 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})))
9786, 96syl6eq 2660 . . . . . . . . . . . . . . . . . . . . . . 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}))))
9897cbvmptv 4678 . . . . . . . . . . . . . . . . . . . . . 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}))))
9981, 98syl6eq 2660 . . . . . . . . . . . . . . . . . . . . 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})))))
10099eqeq2d 2620 . . . . . . . . . . . . . . . . . . . 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}))))))
101100cbvrabv 3172 . . . . . . . . . . . . . . . . . . 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}))))}
102 elmapi 7765 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1))) → 𝑥:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)))
103102ad3antlr 763 . . . . . . . . . . . . . . . . . . 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)))
104 simpr 476 . . . . . . . . . . . . . . . . . . 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}))))})
105 simpl 472 . . . . . . . . . . . . . . . . . . . . . 22 ((∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾) → ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0)
106105ralimi 2936 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0)
107106ad2antlr 759 . . . . . . . . . . . . . . . . . . . 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)
108 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑚 → (𝑝𝑛) = (𝑝𝑚))
109108neeq1d 2841 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑚 → ((𝑝𝑛) ≠ 0 ↔ (𝑝𝑚) ≠ 0))
110109rexbidv 3034 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑚 → (∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝑥(𝑝𝑚) ≠ 0))
111 fveq1 6102 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑝 = 𝑞 → (𝑝𝑚) = (𝑞𝑚))
112111neeq1d 2841 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑝 = 𝑞 → ((𝑝𝑚) ≠ 0 ↔ (𝑞𝑚) ≠ 0))
113112cbvrexv 3148 . . . . . . . . . . . . . . . . . . . . . 22 (∃𝑝 ∈ ran 𝑥(𝑝𝑚) ≠ 0 ↔ ∃𝑞 ∈ ran 𝑥(𝑞𝑚) ≠ 0)
114110, 113syl6bb 275 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑚 → (∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ ∃𝑞 ∈ ran 𝑥(𝑞𝑚) ≠ 0))
115114rspccva 3281 . . . . . . . . . . . . . . . . . . . 20 ((∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ 𝑚 ∈ (1...𝑁)) → ∃𝑞 ∈ ran 𝑥(𝑞𝑚) ≠ 0)
116107, 115sylan 487 . . . . . . . . . . . . . . . . . . 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)
117 simpr 476 . . . . . . . . . . . . . . . . . . . . . 22 ((∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾) → ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)
118117ralimi 2936 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾)
119118ad2antlr 759 . . . . . . . . . . . . . . . . . . . 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)
120108neeq1d 2841 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑚 → ((𝑝𝑛) ≠ 𝐾 ↔ (𝑝𝑚) ≠ 𝐾))
121120rexbidv 3034 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑚 → (∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾 ↔ ∃𝑝 ∈ ran 𝑥(𝑝𝑚) ≠ 𝐾))
122111neeq1d 2841 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑝 = 𝑞 → ((𝑝𝑚) ≠ 𝐾 ↔ (𝑞𝑚) ≠ 𝐾))
123122cbvrexv 3148 . . . . . . . . . . . . . . . . . . . . . 22 (∃𝑝 ∈ ran 𝑥(𝑝𝑚) ≠ 𝐾 ↔ ∃𝑞 ∈ ran 𝑥(𝑞𝑚) ≠ 𝐾)
124121, 123syl6bb 275 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑚 → (∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾 ↔ ∃𝑞 ∈ ran 𝑥(𝑞𝑚) ≠ 𝐾))
125124rspccva 3281 . . . . . . . . . . . . . . . . . . . 20 ((∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾𝑚 ∈ (1...𝑁)) → ∃𝑞 ∈ ran 𝑥(𝑞𝑚) ≠ 𝐾)
126119, 125sylan 487 . . . . . . . . . . . . . . . . . . 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)
12765, 101, 103, 104, 116, 126poimirlem22 32601 . . . . . . . . . . . . . . . . . 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)
128 eldifsn 4260 . . . . . . . . . . . . . . . . . . . 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))
129128eubii 2480 . . . . . . . . . . . . . . . . . . 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))
13053elexi 3186 . . . . . . . . . . . . . . . . . . . 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
131 euhash1 13069 . . . . . . . . . . . . . . . . . . . 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})))
132130, 131ax-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}))
133 df-reu 2903 . . . . . . . . . . . . . . . . . . 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))
134129, 132, 1333bitr4ri 292 . . . . . . . . . . . . . . . . . 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)
135127, 134sylib 207 . . . . . . . . . . . . . . . . 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)
136135oveq1d 6564 . . . . . . . . . . . . . . . 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})))
13763, 136eqtr3d 2646 . . . . . . . . . . . . . . 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})))
13850, 137syl5breqr 4621 . . . . . . . . . . . . . 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}))))}))
139138ex 449 . . . . . . . . . . . . 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}))))})))
140139exlimdv 1848 . . . . . . . . . . . 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}))))})))
14141, 140syl5bi 231 . . . . . . . . . . 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}))))})))
142 dvds0 14835 . . . . . . . . . . . . . 14 (2 ∈ ℤ → 2 ∥ 0)
1439, 142ax-mp 5 . . . . . . . . . . . . 13 2 ∥ 0
144 hash0 13019 . . . . . . . . . . . . 13 (#‘∅) = 0
145143, 144breqtrri 4610 . . . . . . . . . . . 12 2 ∥ (#‘∅)
146 fveq2 6103 . . . . . . . . . . . 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}))))}) = (#‘∅))
147145, 146syl5breqr 4621 . . . . . . . . . . 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}))))}))
148141, 147pm2.61d2 171 . . . . . . . . . 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}))))}))
149148ex 449 . . . . . . . . 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}))))})))
150149adantld 482 . . . . . . . 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}))))})))
151 iba 523 . . . . . . . . . . 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)))))
152151rabbidv 3164 . . . . . . . . . 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)))})
153152fveq2d 6107 . . . . . . . . 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)))}))
154153breq2d 4595 . . . . . . . 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)))})))
155150, 154mpbidi 230 . . . . . . 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)))})))
156155a1d 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)))}))))
15734, 40, 156rexlimd 3008 . . . . 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)))})))
15833, 157syl5bir 232 . . . 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)))})))
159 simpr 476 . . . . . . . . 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)))
160159con3i 149 . . . . . . . 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))))
161160ralimi 2936 . . . . . . 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))))
162 rabeq0 3911 . . . . . . 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))))
163161, 162sylibr 223 . . . . . 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)))} = ∅)
164163fveq2d 6107 . . . . 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)))}) = (#‘∅))
165145, 164syl5breqr 4621 . . . 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)))}))
166158, 165pm2.61d2 171 . . 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)))}))
1678, 10, 32, 166fsumdvds 14868 . 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)))}))
168 rabfi 8070 . . . . 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)
16927, 168ax-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
170 simp1 1054 . . . . . . 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)
171 sneq 4135 . . . . . . . . . . . . 13 ((2nd𝑡) = 𝑁 → {(2nd𝑡)} = {𝑁})
172171difeq2d 3690 . . . . . . . . . . . 12 ((2nd𝑡) = 𝑁 → ((0...𝑁) ∖ {(2nd𝑡)}) = ((0...𝑁) ∖ {𝑁}))
173 difun2 4000 . . . . . . . . . . . . 13 (((0...(𝑁 − 1)) ∪ {𝑁}) ∖ {𝑁}) = ((0...(𝑁 − 1)) ∖ {𝑁})
17464nnnn0d 11228 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ ℕ0)
175 nn0uz 11598 . . . . . . . . . . . . . . . . . 18 0 = (ℤ‘0)
176174, 175syl6eleq 2698 . . . . . . . . . . . . . . . . 17 (𝜑𝑁 ∈ (ℤ‘0))
177 fzm1 12289 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ (ℤ‘0) → (𝑛 ∈ (0...𝑁) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 = 𝑁)))
178176, 177syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑛 ∈ (0...𝑁) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 = 𝑁)))
179 elun 3715 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ((0...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 ∈ {𝑁}))
180 velsn 4141 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ {𝑁} ↔ 𝑛 = 𝑁)
181180orbi2i 540 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 ∈ {𝑁}) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 = 𝑁))
182179, 181bitri 263 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ((0...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 = 𝑁))
183178, 182syl6bbr 277 . . . . . . . . . . . . . . 15 (𝜑 → (𝑛 ∈ (0...𝑁) ↔ 𝑛 ∈ ((0...(𝑁 − 1)) ∪ {𝑁})))
184183eqrdv 2608 . . . . . . . . . . . . . 14 (𝜑 → (0...𝑁) = ((0...(𝑁 − 1)) ∪ {𝑁}))
185184difeq1d 3689 . . . . . . . . . . . . 13 (𝜑 → ((0...𝑁) ∖ {𝑁}) = (((0...(𝑁 − 1)) ∪ {𝑁}) ∖ {𝑁}))
18664nnzd 11357 . . . . . . . . . . . . . . 15 (𝜑𝑁 ∈ ℤ)
187 uzid 11578 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ𝑁))
188 uznfz 12292 . . . . . . . . . . . . . . 15 (𝑁 ∈ (ℤ𝑁) → ¬ 𝑁 ∈ (0...(𝑁 − 1)))
189186, 187, 1883syl 18 . . . . . . . . . . . . . 14 (𝜑 → ¬ 𝑁 ∈ (0...(𝑁 − 1)))
190 disjsn 4192 . . . . . . . . . . . . . . 15 (((0...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (0...(𝑁 − 1)))
191 disj3 3973 . . . . . . . . . . . . . . 15 (((0...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ (0...(𝑁 − 1)) = ((0...(𝑁 − 1)) ∖ {𝑁}))
192190, 191bitr3i 265 . . . . . . . . . . . . . 14 𝑁 ∈ (0...(𝑁 − 1)) ↔ (0...(𝑁 − 1)) = ((0...(𝑁 − 1)) ∖ {𝑁}))
193189, 192sylib 207 . . . . . . . . . . . . 13 (𝜑 → (0...(𝑁 − 1)) = ((0...(𝑁 − 1)) ∖ {𝑁}))
194173, 185, 1933eqtr4a 2670 . . . . . . . . . . . 12 (𝜑 → ((0...𝑁) ∖ {𝑁}) = (0...(𝑁 − 1)))
195172, 194sylan9eqr 2666 . . . . . . . . . . 11 ((𝜑 ∧ (2nd𝑡) = 𝑁) → ((0...𝑁) ∖ {(2nd𝑡)}) = (0...(𝑁 − 1)))
196195rexeqdv 3122 . . . . . . . . . 10 ((𝜑 ∧ (2nd𝑡) = 𝑁) → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶))
197196biimprd 237 . . . . . . . . 9 ((𝜑 ∧ (2nd𝑡) = 𝑁) → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 → ∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶))
198197ralimdv 2946 . . . . . . . 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))
199198expimpd 627 . . . . . . 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))
200170, 199sylan2i 685 . . . . . 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))
201200adantr 480 . . . . 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))
202201ss2rabdv 3646 . . . 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})
203 hashssdif 13061 . . . 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))})))
204169, 202, 203sylancr 694 . . 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))})))
20564adantr 480 . . . . . . . . . 10 ((𝜑𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → 𝑁 ∈ ℕ)
206 poimirlem28.1 . . . . . . . . . 10 (𝑝 = ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶)
207 poimirlem28.2 . . . . . . . . . . 11 ((𝜑𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁))
208207adantlr 747 . . . . . . . . . 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))
209 xp1st 7089 . . . . . . . . . . . 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)}))
210 xp1st 7089 . . . . . . . . . . . 12 ((1st𝑡) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st𝑡)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
211 elmapi 7765 . . . . . . . . . . . 12 ((1st ‘(1st𝑡)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st ‘(1st𝑡)):(1...𝑁)⟶(0..^𝐾))
212209, 210, 2113syl 18 . . . . . . . . . . 11 (𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st ‘(1st𝑡)):(1...𝑁)⟶(0..^𝐾))
213212adantl 481 . . . . . . . . . 10 ((𝜑𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (1st ‘(1st𝑡)):(1...𝑁)⟶(0..^𝐾))
214 xp2nd 7090 . . . . . . . . . . . . 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)})
215 fvex 6113 . . . . . . . . . . . . . 14 (2nd ‘(1st𝑡)) ∈ V
216 f1oeq1 6040 . . . . . . . . . . . . . 14 (𝑓 = (2nd ‘(1st𝑡)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑡)):(1...𝑁)–1-1-onto→(1...𝑁)))
217215, 216elab 3319 . . . . . . . . . . . . 13 ((2nd ‘(1st𝑡)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑡)):(1...𝑁)–1-1-onto→(1...𝑁))
218214, 217sylib 207 . . . . . . . . . . . 12 ((1st𝑡) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘(1st𝑡)):(1...𝑁)–1-1-onto→(1...𝑁))
219209, 218syl 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))
220219adantl 481 . . . . . . . . . 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))
221 xp2nd 7090 . . . . . . . . . . 11 (𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd𝑡) ∈ (0...𝑁))
222221adantl 481 . . . . . . . . . 10 ((𝜑𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (2nd𝑡) ∈ (0...𝑁))
223205, 206, 208, 213, 220, 222poimirlem24 32603 . . . . . . . . 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)))))
224209adantl 481 . . . . . . . . . 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)}))
225 1st2nd2 7096 . . . . . . . . . . . . . . 15 ((1st𝑡) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st𝑡) = ⟨(1st ‘(1st𝑡)), (2nd ‘(1st𝑡))⟩)
226225csbeq1d 3506 . . . . . . . . . . . . . 14 ((1st𝑡) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st𝑡) / 𝑠𝐶 = ⟨(1st ‘(1st𝑡)), (2nd ‘(1st𝑡))⟩ / 𝑠𝐶)
227226eqeq2d 2620 . . . . . . . . . . . . 13 ((1st𝑡) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (𝑖 = (1st𝑡) / 𝑠𝐶𝑖 = ⟨(1st ‘(1st𝑡)), (2nd ‘(1st𝑡))⟩ / 𝑠𝐶))
228227rexbidv 3034 . . . . . . . . . . . 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))
229228ralbidv 2969 . . . . . . . . . . 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))
230229anbi1d 737 . . . . . . . . . 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)))))
231224, 230syl 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)))))
232223, 231bitr4d 270 . . . . . . . 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)))))
233 frn 5966 . . . . . . . . . . . . . . 15 (𝑥:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)) → ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁)))
234102, 233syl 17 . . . . . . . . . . . . . 14 (𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1))) → ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁)))
235234anim2i 591 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))) → (𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))))
236 dfss3 3558 . . . . . . . . . . . . . 14 ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) ↔ ∀𝑛 ∈ (0...(𝑁 − 1))𝑛 ∈ ran (𝑝 ∈ ran 𝑥𝐵))
237 vex 3176 . . . . . . . . . . . . . . . 16 𝑛 ∈ V
238 eqid 2610 . . . . . . . . . . . . . . . . 17 (𝑝 ∈ ran 𝑥𝐵) = (𝑝 ∈ ran 𝑥𝐵)
239238elrnmpt 5293 . . . . . . . . . . . . . . . 16 (𝑛 ∈ V → (𝑛 ∈ ran (𝑝 ∈ ran 𝑥𝐵) ↔ ∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵))
240237, 239ax-mp 5 . . . . . . . . . . . . . . 15 (𝑛 ∈ ran (𝑝 ∈ ran 𝑥𝐵) ↔ ∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)
241240ralbii 2963 . . . . . . . . . . . . . 14 (∀𝑛 ∈ (0...(𝑁 − 1))𝑛 ∈ ran (𝑝 ∈ ran 𝑥𝐵) ↔ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)
242236, 241sylbb 208 . . . . . . . . . . . . 13 ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥𝐵) → ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)
243 1eluzge0 11608 . . . . . . . . . . . . . . . . 17 1 ∈ (ℤ‘0)
244 fzss1 12251 . . . . . . . . . . . . . . . . 17 (1 ∈ (ℤ‘0) → (1...(𝑁 − 1)) ⊆ (0...(𝑁 − 1)))
245 ssralv 3629 . . . . . . . . . . . . . . . . 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))
246243, 244, 245mp2b 10 . . . . . . . . . . . . . . . 16 (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 → ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)
24764nncnd 10913 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝑁 ∈ ℂ)
248 npcan1 10334 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
249247, 248syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
250 peano2zm 11297 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
251186, 250syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑁 − 1) ∈ ℤ)
252 uzid 11578 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)))
253 peano2uz 11617 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
254251, 252, 2533syl 18 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
255249, 254eqeltrrd 2689 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑁 ∈ (ℤ‘(𝑁 − 1)))
256 fzss2 12252 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ (ℤ‘(𝑁 − 1)) → (1...(𝑁 − 1)) ⊆ (1...𝑁))
257255, 256syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
258257sselda 3568 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ (1...(𝑁 − 1))) → 𝑛 ∈ (1...𝑁))
259258adantlr 747 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...(𝑁 − 1))) → 𝑛 ∈ (1...𝑁))
260 simplr 788 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁)))
261 ssel2 3563 . . . . . . . . . . . . . . . . . . . . . . 23 ((ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → 𝑝 ∈ ((0...𝐾) ↑𝑚 (1...𝑁)))
262 elmapi 7765 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑝 ∈ ((0...𝐾) ↑𝑚 (1...𝑁)) → 𝑝:(1...𝑁)⟶(0...𝐾))
263261, 262syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → 𝑝:(1...𝑁)⟶(0...𝐾))
264260, 263sylan 487 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → 𝑝:(1...𝑁)⟶(0...𝐾))
265 poimirlem28.3 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 0)) → 𝐵 < 𝑛)
266 elfzelz 12213 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℤ)
267266zred 11358 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℝ)
268267ltnrd 10050 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑛 ∈ (1...𝑁) → ¬ 𝑛 < 𝑛)
269 breq1 4586 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑛 = 𝐵 → (𝑛 < 𝑛𝐵 < 𝑛))
270269notbid 307 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑛 = 𝐵 → (¬ 𝑛 < 𝑛 ↔ ¬ 𝐵 < 𝑛))
271268, 270syl5ibcom 234 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑛 ∈ (1...𝑁) → (𝑛 = 𝐵 → ¬ 𝐵 < 𝑛))
272271necon2ad 2797 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 ∈ (1...𝑁) → (𝐵 < 𝑛𝑛𝐵))
2732723ad2ant1 1075 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 0) → (𝐵 < 𝑛𝑛𝐵))
274273adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 0)) → (𝐵 < 𝑛𝑛𝐵))
275265, 274mpd 15 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 0)) → 𝑛𝐵)
2762753exp2 1277 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑛 ∈ (1...𝑁) → (𝑝:(1...𝑁)⟶(0...𝐾) → ((𝑝𝑛) = 0 → 𝑛𝐵))))
277276imp31 447 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → ((𝑝𝑛) = 0 → 𝑛𝐵))
278277necon2d 2805 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → (𝑛 = 𝐵 → (𝑝𝑛) ≠ 0))
279278adantllr 751 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → (𝑛 = 𝐵 → (𝑝𝑛) ≠ 0))
280264, 279syldan 486 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → (𝑛 = 𝐵 → (𝑝𝑛) ≠ 0))
281280reximdva 3000 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → (∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 → ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0))
282259, 281syldan 486 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 → ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0))
283282ralimdva 2945 . . . . . . . . . . . . . . . . 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))
284283imp 444 . . . . . . . . . . . . . . . 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)
285246, 284sylan2 490 . . . . . . . . . . . . . . 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)
286285biantrurd 528 . . . . . . . . . . . . . 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)))
287 nnuz 11599 . . . . . . . . . . . . . . . . . . . . . 22 ℕ = (ℤ‘1)
28864, 287syl6eleq 2698 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑁 ∈ (ℤ‘1))
289 fzm1 12289 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ (ℤ‘1) → (𝑛 ∈ (1...𝑁) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁)))
290288, 289syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑛 ∈ (1...𝑁) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁)))
291 elun 3715 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 ∈ {𝑁}))
292180orbi2i 540 . . . . . . . . . . . . . . . . . . . . 21 ((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 ∈ {𝑁}) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁))
293291, 292bitri 263 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁))
294290, 293syl6bbr 277 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑛 ∈ (1...𝑁) ↔ 𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁})))
295294eqrdv 2608 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁}))
296295raleqdv 3121 . . . . . . . . . . . . . . . . 17 (𝜑 → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ ∀𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁})∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0))
297 ralunb 3756 . . . . . . . . . . . . . . . . 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))
298296, 297syl6bb 275 . . . . . . . . . . . . . . . 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)))
299 fveq2 6103 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑁 → (𝑝𝑛) = (𝑝𝑁))
300299neeq1d 2841 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑁 → ((𝑝𝑛) ≠ 0 ↔ (𝑝𝑁) ≠ 0))
301300rexbidv 3034 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑁 → (∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0))
302301ralsng 4165 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ → (∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0))
30364, 302syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0))
304303anbi2d 736 . . . . . . . . . . . . . . . 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)))
305298, 304bitrd 267 . . . . . . . . . . . . . . 15 (𝜑 → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝𝑁) ≠ 0)))
306305ad2antrr 758 . . . . . . . . . . . . . 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)))
307 0z 11265 . . . . . . . . . . . . . . . . . . . . . . 23 0 ∈ ℤ
308 1z 11284 . . . . . . . . . . . . . . . . . . . . . . 23 1 ∈ ℤ
309 fzshftral 12297 . . . . . . . . . . . . . . . . . . . . . . 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))
310307, 308, 309mp3an13 1407 . . . . . . . . . . . . . . . . . . . . . 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))
311186, 250, 3103syl 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))
312 0p1e1 11009 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 + 1) = 1
313312a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (0 + 1) = 1)
314313, 249oveq12d 6567 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((0 + 1)...((𝑁 − 1) + 1)) = (1...𝑁))
315314raleqdv 3121 . . . . . . . . . . . . . . . . . . . . 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))
316311, 315bitrd 267 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ (1...𝑁)[(𝑚 − 1) / 𝑛]𝑝 ∈ ran 𝑥 𝑛 = 𝐵))
317 ovex 6577 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 − 1) ∈ V
318 eqeq1 2614 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = (𝑚 − 1) → (𝑛 = 𝐵 ↔ (𝑚 − 1) = 𝐵))
319318rexbidv 3034 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = (𝑚 − 1) → (∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵))
320317, 319sbcie 3437 . . . . . . . . . . . . . . . . . . . . . 22 ([(𝑚 − 1) / 𝑛]𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵)
321320ralbii 2963 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑚 ∈ (1...𝑁)[(𝑚 − 1) / 𝑛]𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵)
322 oveq1 6556 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 = 𝑛 → (𝑚 − 1) = (𝑛 − 1))
323322eqeq1d 2612 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = 𝑛 → ((𝑚 − 1) = 𝐵 ↔ (𝑛 − 1) = 𝐵))
324323rexbidv 3034 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑛 → (∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵 ↔ ∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵))
325324cbvralv 3147 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑚 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵 ↔ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵)
326321, 325bitri 263 . . . . . . . . . . . . . . . . . . . 20 (∀𝑚 ∈ (1...𝑁)[(𝑚 − 1) / 𝑛]𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵)
327316, 326syl6bb 275 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵))
328327biimpa 500 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵)
329328adantlr 747 . . . . . . . . . . . . . . . . 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)
330 poimirlem28.4 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 𝐾)) → 𝐵 ≠ (𝑛 − 1))
331330necomd 2837 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 𝐾)) → (𝑛 − 1) ≠ 𝐵)
3323313exp2 1277 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑛 ∈ (1...𝑁) → (𝑝:(1...𝑁)⟶(0...𝐾) → ((𝑝𝑛) = 𝐾 → (𝑛 − 1) ≠ 𝐵))))
333332imp31 447 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → ((𝑝𝑛) = 𝐾 → (𝑛 − 1) ≠ 𝐵))
334333necon2d 2805 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → ((𝑛 − 1) = 𝐵 → (𝑝𝑛) ≠ 𝐾))
335334adantllr 751 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → ((𝑛 − 1) = 𝐵 → (𝑝𝑛) ≠ 𝐾))
336264, 335syldan 486 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → ((𝑛 − 1) = 𝐵 → (𝑝𝑛) ≠ 𝐾))
337336reximdva 3000 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → (∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵 → ∃𝑝 ∈ ran 𝑥(𝑝𝑛) ≠ 𝐾))
338337ralimdva 2945 . . . . . . . . . . . . . . . . . 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))
339338imp 444 . . . . . . . . . . . . . . . . 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)
340329, 339syldan 486 . . . . . . . . . . . . . . . 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)
341340biantrud 527 . . . . . . . . . . . . . . 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)))
342 r19.26 3046 . . . . . . . . . . . . . . 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))
343341, 342syl6bbr 277 . . . . . . . . . . . . . 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)))
344286, 306, 3433bitr2d 295 . . . . . . . . . . . . 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)))
345235, 242, 344syl2an 493 . . . . . . . . . . . 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)))
346345pm5.32da 671 . . . . . . . . . . 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))))
347346anbi2d 736 . . . . . . . . . 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)))))
348347rexbidva 3031 . . . . . . . . 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)))))
349348adantr 480 . . . . . . . 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)))))
350194rexeqdv 3122 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶))
351350biimpd 218 . . . . . . . . . . . . . . . . . 18 (𝜑 → (∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = (1st𝑡) / 𝑠𝐶 → ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶))
352351ralimdv 2946 . . . . . . . . . . . . . . . . 17 (𝜑 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = (1st𝑡) / 𝑠𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶))
353172rexeqdv 3122 . . . . . . . . . . . . . . . . . . 19 ((2nd𝑡) = 𝑁 → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = (1st𝑡) / 𝑠𝐶))
354353ralbidv 2969 . . . . . . . . . . . . . . . . . 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))
355354imbi1d 330 . . . . . . . . . . . . . . . . 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)))
356352, 355syl5ibrcom 236 . . . . . . . . . . . . . . . 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)))
357356com23 84 . . . . . . . . . . . . . . 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)))
358357imp 444 . . . . . . . . . . . . . 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))
359358adantrd 483 . . . . . . . . . . . . 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))
360359pm4.71rd 665 . . . . . . . . . . . 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)))))
361 an12 834 . . . . . . . . . . . . 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))))
362 3anass 1035 . . . . . . . . . . . . . 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)))
363362anbi2i 726 . . . . . . . . . . . . 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))))
364361, 363bitr4i 266 . . . . . . . . . . . 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)))
365360, 364syl6bb 275 . . . . . . . . . . 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))))
366365notbid 307 . . . . . . . . . 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))))
367366pm5.32da 671 . . . . . . . . 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)))))
368367adantr 480 . . . . . . . 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)))))
369232, 349, 3683bitr3d 297 . . . . . . 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)))))
370369rabbidva 3163 . . . . . 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)))})
371 iunrab 4503 . . . . . 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)))}
372 difrab 3860 . . . . . 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)))}
373370, 371, 3723eqtr4g 2669 . . . . 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))}))
374373fveq2d 6107 . . . 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))})))
37527, 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)
376 simpl 472 . . . . . . . . . . . 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})))))
377376a1i 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}))))))
378377ss2rabi 3647 . . . . . . . . . 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}))))}
379378sseli 3564 . . . . . . . . 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}))))})
380 fveq2 6103 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑠 → (2nd𝑡) = (2nd𝑠))
381380breq2d 4595 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑠 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑠)))
382381ifbid 4058 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑠 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑠), 𝑦, (𝑦 + 1)))
383382csbeq1d 3506 . . . . . . . . . . . . . . 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}))))
384 fveq2 6103 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑠 → (1st𝑡) = (1st𝑠))
385384fveq2d 6107 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑠 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑠)))
386384fveq2d 6107 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑠 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑠)))
387386imaeq1d 5384 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑠 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑠)) “ (1...𝑗)))
388387xpeq1d 5062 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑠 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑠)) “ (1...𝑗)) × {1}))
389386imaeq1d 5384 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑠 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)))
390389xpeq1d 5062 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑠 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑠)) “ ((𝑗 + 1)...𝑁)) × {0}))
391388, 390uneq12d 3730 . . . . . . . . . . . . . . . . 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})))
392385, 391oveq12d 6567 . . . . . . . . . . . . . . . 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}))))
393392csbeq2dv 3944 . . . . . . . . . . . . . . 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}))))
394383, 393eqtrd 2644 . . . . . . . . . . . . . 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}))))
395394mpteq2dv 4673 . . . . . . . . . . . . 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})))))
396395eqeq2d 2620 . . . . . . . . . . . 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}))))))
397 eqcom 2617 . . . . . . . . . . . 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)
398396, 397syl6bb 275 . . . . . . . . . . 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))
399398elrab 3331 . . . . . . . . . 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))
400399simprbi 479 . . . . . . . . 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)
401379, 400syl 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)
402401rgen 2906 . . . . . . 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
403402rgenw 2908 . . . . . 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
404 invdisj 4571 . . . . . 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)))})
405403, 404mp1i 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)))})
4068, 375, 405hashiun 14395 . . . 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)))}))
407374, 406eqtr3d 2646 . . 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)))}))
408 fo1st 7079 . . . . . . . . . . . . 13 1st :V–onto→V
409 fofun 6029 . . . . . . . . . . . . 13 (1st :V–onto→V → Fun 1st )
410408, 409ax-mp 5 . . . . . . . . . . . 12 Fun 1st
411 ssv 3588 . . . . . . . . . . . . 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
412 fof 6028 . . . . . . . . . . . . . . 15 (1st :V–onto→V → 1st :V⟶V)
413408, 412ax-mp 5 . . . . . . . . . . . . . 14 1st :V⟶V
414413fdmi 5965 . . . . . . . . . . . . 13 dom 1st = V
415411, 414sseqtr4i 3601 . . . . . . . . . . . 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
416 fores 6037 . . . . . . . . . . . 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))}))
417410, 415, 416mp2an 704 . . . . . . . . . . 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))})
418 fveq2 6103 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑥 → (2nd𝑡) = (2nd𝑥))
419418eqeq1d 2612 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑥 → ((2nd𝑡) = 𝑁 ↔ (2nd𝑥) = 𝑁))
420 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = 𝑥 → (1st𝑡) = (1st𝑥))
421420csbeq1d 3506 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = 𝑥(1st𝑡) / 𝑠𝐶 = (1st𝑥) / 𝑠𝐶)
422421eqeq2d 2620 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑥 → (𝑖 = (1st𝑡) / 𝑠𝐶𝑖 = (1st𝑥) / 𝑠𝐶))
423422rexbidv 3034 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑥 → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶))
424423ralbidv 2969 . . . . . . . . . . . . . . . . . 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))
425420fveq2d 6107 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑥 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑥)))
426425fveq1d 6105 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑥 → ((1st ‘(1st𝑡))‘𝑁) = ((1st ‘(1st𝑥))‘𝑁))
427426eqeq1d 2612 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑥 → (((1st ‘(1st𝑡))‘𝑁) = 0 ↔ ((1st ‘(1st𝑥))‘𝑁) = 0))
428420fveq2d 6107 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑥 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑥)))
429428fveq1d 6105 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑥 → ((2nd ‘(1st𝑡))‘𝑁) = ((2nd ‘(1st𝑥))‘𝑁))
430429eqeq1d 2612 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑥 → (((2nd ‘(1st𝑡))‘𝑁) = 𝑁 ↔ ((2nd ‘(1st𝑥))‘𝑁) = 𝑁))
431424, 427, 4303anbi123d 1391 . . . . . . . . . . . . . . . . 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)))
432419, 431anbi12d 743 . . . . . . . . . . . . . . . 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))))
433432rexrab 3337 . . . . . . . . . . . . . . 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))
434 xp1st 7089 . . . . . . . . . . . . . . . . . . . . 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)}))
435434anim1i 590 . . . . . . . . . . . . . . . . . . . 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)))
436 eleq1 2676 . . . . . . . . . . . . . . . . . . . . 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)})))
437 csbeq1a 3508 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑠 = (1st𝑥) → 𝐶 = (1st𝑥) / 𝑠𝐶)
438437eqcoms 2618 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((1st𝑥) = 𝑠𝐶 = (1st𝑥) / 𝑠𝐶)
439438eqcomd 2616 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((1st𝑥) = 𝑠(1st𝑥) / 𝑠𝐶 = 𝐶)
440439eqeq2d 2620 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1st𝑥) = 𝑠 → (𝑖 = (1st𝑥) / 𝑠𝐶𝑖 = 𝐶))
441440rexbidv 3034 . . . . . . . . . . . . . . . . . . . . . . 23 ((1st𝑥) = 𝑠 → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶))
442441ralbidv 2969 . . . . . . . . . . . . . . . . . . . . . 22 ((1st𝑥) = 𝑠 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶))
443 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1st𝑥) = 𝑠 → (1st ‘(1st𝑥)) = (1st𝑠))
444443fveq1d 6105 . . . . . . . . . . . . . . . . . . . . . . 23 ((1st𝑥) = 𝑠 → ((1st ‘(1st𝑥))‘𝑁) = ((1st𝑠)‘𝑁))
445444eqeq1d 2612 . . . . . . . . . . . . . . . . . . . . . 22 ((1st𝑥) = 𝑠 → (((1st ‘(1st𝑥))‘𝑁) = 0 ↔ ((1st𝑠)‘𝑁) = 0))
446 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1st𝑥) = 𝑠 → (2nd ‘(1st𝑥)) = (2nd𝑠))
447446fveq1d 6105 . . . . . . . . . . . . . . . . . . . . . . 23 ((1st𝑥) = 𝑠 → ((2nd ‘(1st𝑥))‘𝑁) = ((2nd𝑠)‘𝑁))
448447eqeq1d 2612 . . . . . . . . . . . . . . . . . . . . . 22 ((1st𝑥) = 𝑠 → (((2nd ‘(1st𝑥))‘𝑁) = 𝑁 ↔ ((2nd𝑠)‘𝑁) = 𝑁))
449442, 445, 4483anbi123d 1391 . . . . . . . . . . . . . . . . . . . . 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)))
450436, 449anbi12d 743 . . . . . . . . . . . . . . . . . . . 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))))
451435, 450syl5ibcom 234 . . . . . . . . . . . . . . . . . . 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))))
452451adantrl 748 . . . . . . . . . . . . . . . . . 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))))
453452expimpd 627 . . . . . . . . . . . . . . . . 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))))
454453rexlimiv 3009 . . . . . . . . . . . . . . . 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)))
455 nn0fz0 12306 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0𝑁 ∈ (0...𝑁))
456174, 455sylib 207 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑁 ∈ (0...𝑁))
457 opelxpi 5072 . . . . . . . . . . . . . . . . . . . 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)))
458456, 457sylan2 490 . . . . . . . . . . . . . . . . . . 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)))
459458ancoms 468 . . . . . . . . . . . . . . . . . 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)))
460 opelxp2 5075 . . . . . . . . . . . . . . . . . . . . 21 (⟨𝑠, 𝑁⟩ ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → 𝑁 ∈ (0...𝑁))
461 op2ndg 7072 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (2nd ‘⟨𝑠, 𝑁⟩) = 𝑁)
462461biantrurd 528 . . . . . . . . . . . . . . . . . . . . . 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))))
463 op1stg 7071 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (1st ‘⟨𝑠, 𝑁⟩) = 𝑠)
464 csbeq1a 3508 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑠 = (1st ‘⟨𝑠, 𝑁⟩) → 𝐶 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶)
465464eqcoms 2618 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((1st ‘⟨𝑠, 𝑁⟩) = 𝑠𝐶 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶)
466465eqcomd 2616 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((1st ‘⟨𝑠, 𝑁⟩) = 𝑠(1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶 = 𝐶)
467463, 466syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶 = 𝐶)
468467eqeq2d 2620 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶𝑖 = 𝐶))
469468rexbidv 3034 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶))
470469ralbidv 2969 . . . . . . . . . . . . . . . . . . . . . . 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))
471463fveq2d 6107 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (1st ‘(1st ‘⟨𝑠, 𝑁⟩)) = (1st𝑠))
472471fveq1d 6105 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = ((1st𝑠)‘𝑁))
473472eqeq1d 2612 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ↔ ((1st𝑠)‘𝑁) = 0))
474463fveq2d 6107 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (2nd ‘(1st ‘⟨𝑠, 𝑁⟩)) = (2nd𝑠))
475474fveq1d 6105 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = ((2nd𝑠)‘𝑁))
476475eqeq1d 2612 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁 ↔ ((2nd𝑠)‘𝑁) = 𝑁))
477470, 473, 4763anbi123d 1391 . . . . . . . . . . . . . . . . . . . . . 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)))
478463biantrud 527 . . . . . . . . . . . . . . . . . . . . . 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)))
479462, 477, 4783bitr3d 297 . . . . . . . . . . . . . . . . . . . . 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)))
48044, 460, 479sylancr 694 . . . . . . . . . . . . . . . . . . . 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)))
481480biimpa 500 . . . . . . . . . . . . . . . . . . 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))
482 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = ⟨𝑠, 𝑁⟩ → (2nd𝑥) = (2nd ‘⟨𝑠, 𝑁⟩))
483482eqeq1d 2612 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = ⟨𝑠, 𝑁⟩ → ((2nd𝑥) = 𝑁 ↔ (2nd ‘⟨𝑠, 𝑁⟩) = 𝑁))
484 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 = ⟨𝑠, 𝑁⟩ → (1st𝑥) = (1st ‘⟨𝑠, 𝑁⟩))
485484csbeq1d 3506 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = ⟨𝑠, 𝑁⟩ → (1st𝑥) / 𝑠𝐶 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶)
486485eqeq2d 2620 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = ⟨𝑠, 𝑁⟩ → (𝑖 = (1st𝑥) / 𝑠𝐶𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶))
487486rexbidv 3034 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = ⟨𝑠, 𝑁⟩ → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑥) / 𝑠𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st ‘⟨𝑠, 𝑁⟩) / 𝑠𝐶))
488487ralbidv 2969 . . . . . . . . . . . . . . . . . . . . . . 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))
489484fveq2d 6107 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = ⟨𝑠, 𝑁⟩ → (1st ‘(1st𝑥)) = (1st ‘(1st ‘⟨𝑠, 𝑁⟩)))
490489fveq1d 6105 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = ⟨𝑠, 𝑁⟩ → ((1st ‘(1st𝑥))‘𝑁) = ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁))
491490eqeq1d 2612 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = ⟨𝑠, 𝑁⟩ → (((1st ‘(1st𝑥))‘𝑁) = 0 ↔ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0))
492484fveq2d 6107 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = ⟨𝑠, 𝑁⟩ → (2nd ‘(1st𝑥)) = (2nd ‘(1st ‘⟨𝑠, 𝑁⟩)))
493492fveq1d 6105 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = ⟨𝑠, 𝑁⟩ → ((2nd ‘(1st𝑥))‘𝑁) = ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁))
494493eqeq1d 2612 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = ⟨𝑠, 𝑁⟩ → (((2nd ‘(1st𝑥))‘𝑁) = 𝑁 ↔ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁))
495488, 491, 4943anbi123d 1391 . . . . . . . . . . . . . . . . . . . . . 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)))
496483, 495anbi12d 743 . . . . . . . . . . . . . . . . . . . . 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))))
497484eqeq1d 2612 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = ⟨𝑠, 𝑁⟩ → ((1st𝑥) = 𝑠 ↔ (1st ‘⟨𝑠, 𝑁⟩) = 𝑠))
498496, 497anbi12d 743 . . . . . . . . . . . . . . . . . . . 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)))
499498rspcev 3282 . . . . . . . . . . . . . . . . . . 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))
500481, 499syldan 486 . . . . . . . . . . . . . . . . . 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))
501459, 500sylan 487 . . . . . . . . . . . . . . . . 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))
502501expl 646 . . . . . . . . . . . . . . . 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)))
503454, 502impbid2 215 . . . . . . . . . . . . . . 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))))
504433, 503syl5bb 271 . . . . . . . . . . . . . 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))))
505504abbidv 2728 . . . . . . . . . . . . 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))})
506 dfimafn 6155 . . . . . . . . . . . . . . 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})
507410, 415, 506mp2an 704 . . . . . . . . . . . . . 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}
508 nfv 1830 . . . . . . . . . . . . . . . . . 18 𝑠(2nd𝑡) = 𝑁
509 nfcv 2751 . . . . . . . . . . . . . . . . . . . 20 𝑠(0...(𝑁 − 1))
510 nfcsb1v 3515 . . . . . . . . . . . . . . . . . . . . . 22 𝑠(1st𝑡) / 𝑠𝐶
511510nfeq2 2766 . . . . . . . . . . . . . . . . . . . . 21 𝑠 𝑖 = (1st𝑡) / 𝑠𝐶
512509, 511nfrex 2990 . . . . . . . . . . . . . . . . . . . 20 𝑠𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶
513509, 512nfral 2929 . . . . . . . . . . . . . . . . . . 19 𝑠𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶
514 nfv 1830 . . . . . . . . . . . . . . . . . . 19 𝑠((1st ‘(1st𝑡))‘𝑁) = 0
515 nfv 1830 . . . . . . . . . . . . . . . . . . 19 𝑠((2nd ‘(1st𝑡))‘𝑁) = 𝑁
516513, 514, 515nf3an 1819 . . . . . . . . . . . . . . . . . 18 𝑠(∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁)
517508, 516nfan 1816 . . . . . . . . . . . . . . . . 17 𝑠((2nd𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = (1st𝑡) / 𝑠𝐶 ∧ ((1st ‘(1st𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st𝑡))‘𝑁) = 𝑁))
518 nfcv 2751 . . . . . . . . . . . . . . . . 17 𝑠((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))
519517, 518nfrab 3100 . . . . . . . . . . . . . . . 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))}
520 nfv 1830 . . . . . . . . . . . . . . . 16 𝑠(1st𝑥) = 𝑦
521519, 520nfrex 2990 . . . . . . . . . . . . . . 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
522 nfv 1830 . . . . . . . . . . . . . . 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
523 eqeq2 2621 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑠 → ((1st𝑥) = 𝑦 ↔ (1st𝑥) = 𝑠))
524523rexbidv 3034 . . . . . . . . . . . . . . 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))
525521, 522, 524cbvab 2733 . . . . . . . . . . . . . 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}
526507, 525eqtri 2632 . . . . . . . . . . . . 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}
527 df-rab 2905 . . . . . . . . . . . . 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))}
528505, 526, 5273eqtr4g 2669 . . . . . . . . . . . 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)})
529 foeq3 6026 . . . . . . . . . . . 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