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Theorem poimirlem28 33748
Description: Lemma for poimir 33753, a variant of Sperner's lemma. (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))
poimirlem28.5 (𝜑𝐾 ∈ ℕ)
Assertion
Ref Expression
poimirlem28 (𝜑 → ∃𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)
Distinct variable groups:   𝑓,𝑖,𝑗,𝑛,𝑝,𝑠   𝜑,𝑗,𝑛   𝑗,𝑁,𝑛   𝜑,𝑖,𝑝,𝑠   𝐵,𝑓,𝑖,𝑗,𝑛,𝑠   𝑓,𝐾,𝑖,𝑗,𝑛,𝑝,𝑠   𝑓,𝑁,𝑖,𝑝,𝑠   𝐶,𝑖,𝑛,𝑝
Allowed substitution hints:   𝜑(𝑓)   𝐵(𝑝)   𝐶(𝑓,𝑗,𝑠)

Proof of Theorem poimirlem28
Dummy variables 𝑘 𝑚 𝑞 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 poimir.0 . . . . . 6 (𝜑𝑁 ∈ ℕ)
21nnnn0d 11615 . . . . 5 (𝜑𝑁 ∈ ℕ0)
31nnred 11318 . . . . . 6 (𝜑𝑁 ∈ ℝ)
43leidd 10877 . . . . 5 (𝜑𝑁𝑁)
52, 2, 43jca 1151 . . . 4 (𝜑 → (𝑁 ∈ ℕ0𝑁 ∈ ℕ0𝑁𝑁))
6 oveq2 6880 . . . . . . . . . . . . . . . 16 (𝑘 = 0 → (1...𝑘) = (1...0))
7 fz10 12583 . . . . . . . . . . . . . . . 16 (1...0) = ∅
86, 7syl6eq 2854 . . . . . . . . . . . . . . 15 (𝑘 = 0 → (1...𝑘) = ∅)
98oveq2d 6888 . . . . . . . . . . . . . 14 (𝑘 = 0 → ((0..^𝐾) ↑𝑚 (1...𝑘)) = ((0..^𝐾) ↑𝑚 ∅))
10 fzofi 12995 . . . . . . . . . . . . . . . 16 (0..^𝐾) ∈ Fin
11 map0e 8128 . . . . . . . . . . . . . . . 16 ((0..^𝐾) ∈ Fin → ((0..^𝐾) ↑𝑚 ∅) = 1𝑜)
1210, 11ax-mp 5 . . . . . . . . . . . . . . 15 ((0..^𝐾) ↑𝑚 ∅) = 1𝑜
13 df1o2 7807 . . . . . . . . . . . . . . 15 1𝑜 = {∅}
1412, 13eqtri 2826 . . . . . . . . . . . . . 14 ((0..^𝐾) ↑𝑚 ∅) = {∅}
159, 14syl6eq 2854 . . . . . . . . . . . . 13 (𝑘 = 0 → ((0..^𝐾) ↑𝑚 (1...𝑘)) = {∅})
16 eqidd 2805 . . . . . . . . . . . . . . . . 17 (𝑘 = 0 → 𝑓 = 𝑓)
1716, 8, 8f1oeq123d 6347 . . . . . . . . . . . . . . . 16 (𝑘 = 0 → (𝑓:(1...𝑘)–1-1-onto→(1...𝑘) ↔ 𝑓:∅–1-1-onto→∅))
18 eqid 2804 . . . . . . . . . . . . . . . . 17 ∅ = ∅
19 f1o00 6385 . . . . . . . . . . . . . . . . 17 (𝑓:∅–1-1-onto→∅ ↔ (𝑓 = ∅ ∧ ∅ = ∅))
2018, 19mpbiran2 692 . . . . . . . . . . . . . . . 16 (𝑓:∅–1-1-onto→∅ ↔ 𝑓 = ∅)
2117, 20syl6bb 278 . . . . . . . . . . . . . . 15 (𝑘 = 0 → (𝑓:(1...𝑘)–1-1-onto→(1...𝑘) ↔ 𝑓 = ∅))
2221abbidv 2923 . . . . . . . . . . . . . 14 (𝑘 = 0 → {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)} = {𝑓𝑓 = ∅})
23 df-sn 4369 . . . . . . . . . . . . . 14 {∅} = {𝑓𝑓 = ∅}
2422, 23syl6eqr 2856 . . . . . . . . . . . . 13 (𝑘 = 0 → {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)} = {∅})
2515, 24xpeq12d 5339 . . . . . . . . . . . 12 (𝑘 = 0 → (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) = ({∅} × {∅}))
26 0ex 4982 . . . . . . . . . . . . 13 ∅ ∈ V
2726, 26xpsn 6628 . . . . . . . . . . . 12 ({∅} × {∅}) = {⟨∅, ∅⟩}
2825, 27syl6req 2855 . . . . . . . . . . 11 (𝑘 = 0 → {⟨∅, ∅⟩} = (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}))
29 elsni 4385 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ {⟨∅, ∅⟩} → 𝑠 = ⟨∅, ∅⟩)
3026, 26op1std 7406 . . . . . . . . . . . . . . . . . . 19 (𝑠 = ⟨∅, ∅⟩ → (1st𝑠) = ∅)
3129, 30syl 17 . . . . . . . . . . . . . . . . . 18 (𝑠 ∈ {⟨∅, ∅⟩} → (1st𝑠) = ∅)
3231oveq1d 6887 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ {⟨∅, ∅⟩} → ((1st𝑠) ∘𝑓 + ∅) = (∅ ∘𝑓 + ∅))
33 f0 6299 . . . . . . . . . . . . . . . . . . . 20 ∅:∅⟶∅
34 ffn 6254 . . . . . . . . . . . . . . . . . . . 20 (∅:∅⟶∅ → ∅ Fn ∅)
3533, 34mp1i 13 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ {⟨∅, ∅⟩} → ∅ Fn ∅)
3626a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ {⟨∅, ∅⟩} → ∅ ∈ V)
37 inidm 4017 . . . . . . . . . . . . . . . . . . 19 (∅ ∩ ∅) = ∅
38 0fv 6445 . . . . . . . . . . . . . . . . . . . 20 (∅‘𝑛) = ∅
3938a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝑠 ∈ {⟨∅, ∅⟩} ∧ 𝑛 ∈ ∅) → (∅‘𝑛) = ∅)
4035, 35, 36, 36, 37, 39, 39offval 7132 . . . . . . . . . . . . . . . . . 18 (𝑠 ∈ {⟨∅, ∅⟩} → (∅ ∘𝑓 + ∅) = (𝑛 ∈ ∅ ↦ (∅ + ∅)))
41 mpt0 6230 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ∅ ↦ (∅ + ∅)) = ∅
4240, 41syl6eq 2854 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ {⟨∅, ∅⟩} → (∅ ∘𝑓 + ∅) = ∅)
4332, 42eqtrd 2838 . . . . . . . . . . . . . . . 16 (𝑠 ∈ {⟨∅, ∅⟩} → ((1st𝑠) ∘𝑓 + ∅) = ∅)
4443uneq1d 3963 . . . . . . . . . . . . . . 15 (𝑠 ∈ {⟨∅, ∅⟩} → (((1st𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) = (∅ ∪ ((1...𝑁) × {0})))
45 uncom 3954 . . . . . . . . . . . . . . . 16 (∅ ∪ ((1...𝑁) × {0})) = (((1...𝑁) × {0}) ∪ ∅)
46 un0 4163 . . . . . . . . . . . . . . . 16 (((1...𝑁) × {0}) ∪ ∅) = ((1...𝑁) × {0})
4745, 46eqtri 2826 . . . . . . . . . . . . . . 15 (∅ ∪ ((1...𝑁) × {0})) = ((1...𝑁) × {0})
4844, 47syl6req 2855 . . . . . . . . . . . . . 14 (𝑠 ∈ {⟨∅, ∅⟩} → ((1...𝑁) × {0}) = (((1st𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})))
4948csbeq1d 3733 . . . . . . . . . . . . 13 (𝑠 ∈ {⟨∅, ∅⟩} → ((1...𝑁) × {0}) / 𝑝𝐵 = (((1st𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝𝐵)
5049eqeq2d 2814 . . . . . . . . . . . 12 (𝑠 ∈ {⟨∅, ∅⟩} → (0 = ((1...𝑁) × {0}) / 𝑝𝐵 ↔ 0 = (((1st𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝𝐵))
51 oveq2 6880 . . . . . . . . . . . . . . 15 (𝑘 = 0 → (0...𝑘) = (0...0))
52 0z 11652 . . . . . . . . . . . . . . . 16 0 ∈ ℤ
53 fzsn 12604 . . . . . . . . . . . . . . . 16 (0 ∈ ℤ → (0...0) = {0})
5452, 53ax-mp 5 . . . . . . . . . . . . . . 15 (0...0) = {0}
5551, 54syl6eq 2854 . . . . . . . . . . . . . 14 (𝑘 = 0 → (0...𝑘) = {0})
56 oveq2 6880 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 0 → ((𝑗 + 1)...𝑘) = ((𝑗 + 1)...0))
5756imaeq2d 5674 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 0 → ((2nd𝑠) “ ((𝑗 + 1)...𝑘)) = ((2nd𝑠) “ ((𝑗 + 1)...0)))
5857xpeq1d 5337 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 0 → (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}) = (((2nd𝑠) “ ((𝑗 + 1)...0)) × {0}))
5958uneq2d 3964 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 0 → ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0})) = ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...0)) × {0})))
6059oveq2d 6888 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) = ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...0)) × {0}))))
61 oveq1 6879 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 0 → (𝑘 + 1) = (0 + 1))
62 0p1e1 11412 . . . . . . . . . . . . . . . . . . . . . 22 (0 + 1) = 1
6361, 62syl6eq 2854 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 0 → (𝑘 + 1) = 1)
6463oveq1d 6887 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 0 → ((𝑘 + 1)...𝑁) = (1...𝑁))
6564xpeq1d 5337 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (((𝑘 + 1)...𝑁) × {0}) = ((1...𝑁) × {0}))
6660, 65uneq12d 3965 . . . . . . . . . . . . . . . . . 18 (𝑘 = 0 → (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪ ((1...𝑁) × {0})))
6766csbeq1d 3733 . . . . . . . . . . . . . . . . 17 (𝑘 = 0 → (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪ ((1...𝑁) × {0})) / 𝑝𝐵)
6867eqeq2d 2814 . . . . . . . . . . . . . . . 16 (𝑘 = 0 → (𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪ ((1...𝑁) × {0})) / 𝑝𝐵))
6955, 68rexeqbidv 3340 . . . . . . . . . . . . . . 15 (𝑘 = 0 → (∃𝑗 ∈ (0...𝑘)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵 ↔ ∃𝑗 ∈ {0}𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪ ((1...𝑁) × {0})) / 𝑝𝐵))
70 c0ex 10317 . . . . . . . . . . . . . . . 16 0 ∈ V
71 oveq2 6880 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 = 0 → (1...𝑗) = (1...0))
7271, 7syl6eq 2854 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 = 0 → (1...𝑗) = ∅)
7372imaeq2d 5674 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 = 0 → ((2nd𝑠) “ (1...𝑗)) = ((2nd𝑠) “ ∅))
74 ima0 5689 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((2nd𝑠) “ ∅) = ∅
7573, 74syl6eq 2854 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 = 0 → ((2nd𝑠) “ (1...𝑗)) = ∅)
7675xpeq1d 5337 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = 0 → (((2nd𝑠) “ (1...𝑗)) × {1}) = (∅ × {1}))
77 0xp 5399 . . . . . . . . . . . . . . . . . . . . . . 23 (∅ × {1}) = ∅
7876, 77syl6eq 2854 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 0 → (((2nd𝑠) “ (1...𝑗)) × {1}) = ∅)
79 oveq1 6879 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 = 0 → (𝑗 + 1) = (0 + 1))
8079, 62syl6eq 2854 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 = 0 → (𝑗 + 1) = 1)
8180oveq1d 6887 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 = 0 → ((𝑗 + 1)...0) = (1...0))
8281, 7syl6eq 2854 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 = 0 → ((𝑗 + 1)...0) = ∅)
8382imaeq2d 5674 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 = 0 → ((2nd𝑠) “ ((𝑗 + 1)...0)) = ((2nd𝑠) “ ∅))
8483, 74syl6eq 2854 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 = 0 → ((2nd𝑠) “ ((𝑗 + 1)...0)) = ∅)
8584xpeq1d 5337 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = 0 → (((2nd𝑠) “ ((𝑗 + 1)...0)) × {0}) = (∅ × {0}))
86 0xp 5399 . . . . . . . . . . . . . . . . . . . . . . 23 (∅ × {0}) = ∅
8785, 86syl6eq 2854 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 0 → (((2nd𝑠) “ ((𝑗 + 1)...0)) × {0}) = ∅)
8878, 87uneq12d 3965 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 0 → ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...0)) × {0})) = (∅ ∪ ∅))
89 un0 4163 . . . . . . . . . . . . . . . . . . . . 21 (∅ ∪ ∅) = ∅
9088, 89syl6eq 2854 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 0 → ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...0)) × {0})) = ∅)
9190oveq2d 6888 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 0 → ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...0)) × {0}))) = ((1st𝑠) ∘𝑓 + ∅))
9291uneq1d 3963 . . . . . . . . . . . . . . . . . 18 (𝑗 = 0 → (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪ ((1...𝑁) × {0})) = (((1st𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})))
9392csbeq1d 3733 . . . . . . . . . . . . . . . . 17 (𝑗 = 0 → (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪ ((1...𝑁) × {0})) / 𝑝𝐵 = (((1st𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝𝐵)
9493eqeq2d 2814 . . . . . . . . . . . . . . . 16 (𝑗 = 0 → (𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪ ((1...𝑁) × {0})) / 𝑝𝐵𝑖 = (((1st𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝𝐵))
9570, 94rexsn 4414 . . . . . . . . . . . . . . 15 (∃𝑗 ∈ {0}𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪ ((1...𝑁) × {0})) / 𝑝𝐵𝑖 = (((1st𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝𝐵)
9669, 95syl6bb 278 . . . . . . . . . . . . . 14 (𝑘 = 0 → (∃𝑗 ∈ (0...𝑘)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵𝑖 = (((1st𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝𝐵))
9755, 96raleqbidv 3339 . . . . . . . . . . . . 13 (𝑘 = 0 → (∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵 ↔ ∀𝑖 ∈ {0}𝑖 = (((1st𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝𝐵))
98 eqeq1 2808 . . . . . . . . . . . . . 14 (𝑖 = 0 → (𝑖 = (((1st𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝𝐵 ↔ 0 = (((1st𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝𝐵))
9970, 98ralsn 4413 . . . . . . . . . . . . 13 (∀𝑖 ∈ {0}𝑖 = (((1st𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝𝐵 ↔ 0 = (((1st𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝𝐵)
10097, 99syl6rbb 279 . . . . . . . . . . . 12 (𝑘 = 0 → (0 = (((1st𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝𝐵 ↔ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵))
10150, 100sylan9bbr 502 . . . . . . . . . . 11 ((𝑘 = 0 ∧ 𝑠 ∈ {⟨∅, ∅⟩}) → (0 = ((1...𝑁) × {0}) / 𝑝𝐵 ↔ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵))
10228, 101rabeqbidva 3384 . . . . . . . . . 10 (𝑘 = 0 → {𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ((1...𝑁) × {0}) / 𝑝𝐵} = {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵})
103102eqcomd 2810 . . . . . . . . 9 (𝑘 = 0 → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵} = {𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ((1...𝑁) × {0}) / 𝑝𝐵})
104103fveq2d 6410 . . . . . . . 8 (𝑘 = 0 → (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵}) = (♯‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ((1...𝑁) × {0}) / 𝑝𝐵}))
105104breq2d 4854 . . . . . . 7 (𝑘 = 0 → (2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵}) ↔ 2 ∥ (♯‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ((1...𝑁) × {0}) / 𝑝𝐵})))
106105notbid 309 . . . . . 6 (𝑘 = 0 → (¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵}) ↔ ¬ 2 ∥ (♯‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ((1...𝑁) × {0}) / 𝑝𝐵})))
107106imbi2d 331 . . . . 5 (𝑘 = 0 → ((𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵})) ↔ (𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ((1...𝑁) × {0}) / 𝑝𝐵}))))
108 oveq2 6880 . . . . . . . . . . . 12 (𝑘 = 𝑚 → (1...𝑘) = (1...𝑚))
109108oveq2d 6888 . . . . . . . . . . 11 (𝑘 = 𝑚 → ((0..^𝐾) ↑𝑚 (1...𝑘)) = ((0..^𝐾) ↑𝑚 (1...𝑚)))
110 eqidd 2805 . . . . . . . . . . . . 13 (𝑘 = 𝑚𝑓 = 𝑓)
111110, 108, 108f1oeq123d 6347 . . . . . . . . . . . 12 (𝑘 = 𝑚 → (𝑓:(1...𝑘)–1-1-onto→(1...𝑘) ↔ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)))
112111abbidv 2923 . . . . . . . . . . 11 (𝑘 = 𝑚 → {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)} = {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)})
113109, 112xpeq12d 5339 . . . . . . . . . 10 (𝑘 = 𝑚 → (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) = (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}))
114 oveq2 6880 . . . . . . . . . . 11 (𝑘 = 𝑚 → (0...𝑘) = (0...𝑚))
115 oveq2 6880 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑚 → ((𝑗 + 1)...𝑘) = ((𝑗 + 1)...𝑚))
116115imaeq2d 5674 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑚 → ((2nd𝑠) “ ((𝑗 + 1)...𝑘)) = ((2nd𝑠) “ ((𝑗 + 1)...𝑚)))
117116xpeq1d 5337 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚 → (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}) = (((2nd𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))
118117uneq2d 3964 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0})) = ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑚)) × {0})))
119118oveq2d 6888 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚 → ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) = ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))))
120 oveq1 6879 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚 → (𝑘 + 1) = (𝑚 + 1))
121120oveq1d 6887 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → ((𝑘 + 1)...𝑁) = ((𝑚 + 1)...𝑁))
122121xpeq1d 5337 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚 → (((𝑘 + 1)...𝑁) × {0}) = (((𝑚 + 1)...𝑁) × {0}))
123119, 122uneq12d 3965 . . . . . . . . . . . . . 14 (𝑘 = 𝑚 → (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})))
124123csbeq1d 3733 . . . . . . . . . . . . 13 (𝑘 = 𝑚(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝𝐵)
125124eqeq2d 2814 . . . . . . . . . . . 12 (𝑘 = 𝑚 → (𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝𝐵))
126114, 125rexeqbidv 3340 . . . . . . . . . . 11 (𝑘 = 𝑚 → (∃𝑗 ∈ (0...𝑘)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵 ↔ ∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝𝐵))
127114, 126raleqbidv 3339 . . . . . . . . . 10 (𝑘 = 𝑚 → (∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵 ↔ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝𝐵))
128113, 127rabeqbidv 3383 . . . . . . . . 9 (𝑘 = 𝑚 → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵} = {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝𝐵})
129128fveq2d 6410 . . . . . . . 8 (𝑘 = 𝑚 → (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵}) = (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝𝐵}))
130129breq2d 4854 . . . . . . 7 (𝑘 = 𝑚 → (2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵}) ↔ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝𝐵})))
131130notbid 309 . . . . . 6 (𝑘 = 𝑚 → (¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵}) ↔ ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝𝐵})))
132131imbi2d 331 . . . . 5 (𝑘 = 𝑚 → ((𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵})) ↔ (𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝𝐵}))))
133 oveq2 6880 . . . . . . . . . . . 12 (𝑘 = (𝑚 + 1) → (1...𝑘) = (1...(𝑚 + 1)))
134133oveq2d 6888 . . . . . . . . . . 11 (𝑘 = (𝑚 + 1) → ((0..^𝐾) ↑𝑚 (1...𝑘)) = ((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))))
135 eqidd 2805 . . . . . . . . . . . . 13 (𝑘 = (𝑚 + 1) → 𝑓 = 𝑓)
136135, 133, 133f1oeq123d 6347 . . . . . . . . . . . 12 (𝑘 = (𝑚 + 1) → (𝑓:(1...𝑘)–1-1-onto→(1...𝑘) ↔ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))))
137136abbidv 2923 . . . . . . . . . . 11 (𝑘 = (𝑚 + 1) → {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)} = {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))})
138134, 137xpeq12d 5339 . . . . . . . . . 10 (𝑘 = (𝑚 + 1) → (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) = (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}))
139 oveq2 6880 . . . . . . . . . . 11 (𝑘 = (𝑚 + 1) → (0...𝑘) = (0...(𝑚 + 1)))
140 oveq2 6880 . . . . . . . . . . . . . . . . . . 19 (𝑘 = (𝑚 + 1) → ((𝑗 + 1)...𝑘) = ((𝑗 + 1)...(𝑚 + 1)))
141140imaeq2d 5674 . . . . . . . . . . . . . . . . . 18 (𝑘 = (𝑚 + 1) → ((2nd𝑠) “ ((𝑗 + 1)...𝑘)) = ((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))))
142141xpeq1d 5337 . . . . . . . . . . . . . . . . 17 (𝑘 = (𝑚 + 1) → (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}) = (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))
143142uneq2d 3964 . . . . . . . . . . . . . . . 16 (𝑘 = (𝑚 + 1) → ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0})) = ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0})))
144143oveq2d 6888 . . . . . . . . . . . . . . 15 (𝑘 = (𝑚 + 1) → ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) = ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))))
145 oveq1 6879 . . . . . . . . . . . . . . . . 17 (𝑘 = (𝑚 + 1) → (𝑘 + 1) = ((𝑚 + 1) + 1))
146145oveq1d 6887 . . . . . . . . . . . . . . . 16 (𝑘 = (𝑚 + 1) → ((𝑘 + 1)...𝑁) = (((𝑚 + 1) + 1)...𝑁))
147146xpeq1d 5337 . . . . . . . . . . . . . . 15 (𝑘 = (𝑚 + 1) → (((𝑘 + 1)...𝑁) × {0}) = ((((𝑚 + 1) + 1)...𝑁) × {0}))
148144, 147uneq12d 3965 . . . . . . . . . . . . . 14 (𝑘 = (𝑚 + 1) → (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})))
149148csbeq1d 3733 . . . . . . . . . . . . 13 (𝑘 = (𝑚 + 1) → (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵)
150149eqeq2d 2814 . . . . . . . . . . . 12 (𝑘 = (𝑚 + 1) → (𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵))
151139, 150rexeqbidv 3340 . . . . . . . . . . 11 (𝑘 = (𝑚 + 1) → (∃𝑗 ∈ (0...𝑘)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵 ↔ ∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵))
152139, 151raleqbidv 3339 . . . . . . . . . 10 (𝑘 = (𝑚 + 1) → (∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵 ↔ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵))
153138, 152rabeqbidv 3383 . . . . . . . . 9 (𝑘 = (𝑚 + 1) → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵} = {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵})
154153fveq2d 6410 . . . . . . . 8 (𝑘 = (𝑚 + 1) → (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵}) = (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}))
155154breq2d 4854 . . . . . . 7 (𝑘 = (𝑚 + 1) → (2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵}) ↔ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵})))
156155notbid 309 . . . . . 6 (𝑘 = (𝑚 + 1) → (¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵}) ↔ ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵})))
157156imbi2d 331 . . . . 5 (𝑘 = (𝑚 + 1) → ((𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵})) ↔ (𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}))))
158 oveq2 6880 . . . . . . . . . . . 12 (𝑘 = 𝑁 → (1...𝑘) = (1...𝑁))
159158oveq2d 6888 . . . . . . . . . . 11 (𝑘 = 𝑁 → ((0..^𝐾) ↑𝑚 (1...𝑘)) = ((0..^𝐾) ↑𝑚 (1...𝑁)))
160 eqidd 2805 . . . . . . . . . . . . 13 (𝑘 = 𝑁𝑓 = 𝑓)
161160, 158, 158f1oeq123d 6347 . . . . . . . . . . . 12 (𝑘 = 𝑁 → (𝑓:(1...𝑘)–1-1-onto→(1...𝑘) ↔ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)))
162161abbidv 2923 . . . . . . . . . . 11 (𝑘 = 𝑁 → {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)} = {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
163159, 162xpeq12d 5339 . . . . . . . . . 10 (𝑘 = 𝑁 → (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) = (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
164 oveq2 6880 . . . . . . . . . . 11 (𝑘 = 𝑁 → (0...𝑘) = (0...𝑁))
165 oveq2 6880 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑁 → ((𝑗 + 1)...𝑘) = ((𝑗 + 1)...𝑁))
166165imaeq2d 5674 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑁 → ((2nd𝑠) “ ((𝑗 + 1)...𝑘)) = ((2nd𝑠) “ ((𝑗 + 1)...𝑁)))
167166xpeq1d 5337 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑁 → (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}) = (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))
168167uneq2d 3964 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑁 → ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0})) = ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
169168oveq2d 6888 . . . . . . . . . . . . . . 15 (𝑘 = 𝑁 → ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) = ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))))
170 oveq1 6879 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑁 → (𝑘 + 1) = (𝑁 + 1))
171170oveq1d 6887 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑁 → ((𝑘 + 1)...𝑁) = ((𝑁 + 1)...𝑁))
172171xpeq1d 5337 . . . . . . . . . . . . . . 15 (𝑘 = 𝑁 → (((𝑘 + 1)...𝑁) × {0}) = (((𝑁 + 1)...𝑁) × {0}))
173169, 172uneq12d 3965 . . . . . . . . . . . . . 14 (𝑘 = 𝑁 → (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})))
174173csbeq1d 3733 . . . . . . . . . . . . 13 (𝑘 = 𝑁(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝𝐵)
175174eqeq2d 2814 . . . . . . . . . . . 12 (𝑘 = 𝑁 → (𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝𝐵))
176164, 175rexeqbidv 3340 . . . . . . . . . . 11 (𝑘 = 𝑁 → (∃𝑗 ∈ (0...𝑘)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵 ↔ ∃𝑗 ∈ (0...𝑁)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝𝐵))
177164, 176raleqbidv 3339 . . . . . . . . . 10 (𝑘 = 𝑁 → (∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵 ↔ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝𝐵))
178163, 177rabeqbidv 3383 . . . . . . . . 9 (𝑘 = 𝑁 → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵} = {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝𝐵})
179178fveq2d 6410 . . . . . . . 8 (𝑘 = 𝑁 → (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵}) = (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝𝐵}))
180179breq2d 4854 . . . . . . 7 (𝑘 = 𝑁 → (2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵}) ↔ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝𝐵})))
181180notbid 309 . . . . . 6 (𝑘 = 𝑁 → (¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵}) ↔ ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝𝐵})))
182181imbi2d 331 . . . . 5 (𝑘 = 𝑁 → ((𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝𝐵})) ↔ (𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝𝐵}))))
183 n2dvds1 15322 . . . . . . 7 ¬ 2 ∥ 1
184 opex 5120 . . . . . . . . . 10 ⟨∅, ∅⟩ ∈ V
185 hashsng 13375 . . . . . . . . . 10 (⟨∅, ∅⟩ ∈ V → (♯‘{⟨∅, ∅⟩}) = 1)
186184, 185ax-mp 5 . . . . . . . . 9 (♯‘{⟨∅, ∅⟩}) = 1
187 nnuz 11939 . . . . . . . . . . . . . . . . 17 ℕ = (ℤ‘1)
1881, 187syl6eleq 2893 . . . . . . . . . . . . . . . 16 (𝜑𝑁 ∈ (ℤ‘1))
189 eluzfz1 12569 . . . . . . . . . . . . . . . 16 (𝑁 ∈ (ℤ‘1) → 1 ∈ (1...𝑁))
190188, 189syl 17 . . . . . . . . . . . . . . 15 (𝜑 → 1 ∈ (1...𝑁))
191 poimirlem28.5 . . . . . . . . . . . . . . . . 17 (𝜑𝐾 ∈ ℕ)
192191nnnn0d 11615 . . . . . . . . . . . . . . . 16 (𝜑𝐾 ∈ ℕ0)
193 0elfz 12658 . . . . . . . . . . . . . . . 16 (𝐾 ∈ ℕ0 → 0 ∈ (0...𝐾))
194 fconst6g 6307 . . . . . . . . . . . . . . . 16 (0 ∈ (0...𝐾) → ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾))
195192, 193, 1943syl 18 . . . . . . . . . . . . . . 15 (𝜑 → ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾))
19670fvconst2 6692 . . . . . . . . . . . . . . . 16 (1 ∈ (1...𝑁) → (((1...𝑁) × {0})‘1) = 0)
197190, 196syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (((1...𝑁) × {0})‘1) = 0)
198190, 195, 1973jca 1151 . . . . . . . . . . . . . 14 (𝜑 → (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) = 0))
199 nfv 2005 . . . . . . . . . . . . . . . 16 𝑝(𝜑 ∧ (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) = 0))
200 nfcsb1v 3742 . . . . . . . . . . . . . . . . 17 𝑝((1...𝑁) × {0}) / 𝑝𝐵
201200nfeq1 2960 . . . . . . . . . . . . . . . 16 𝑝((1...𝑁) × {0}) / 𝑝𝐵 = 0
202199, 201nfim 1987 . . . . . . . . . . . . . . 15 𝑝((𝜑 ∧ (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) = 0)) → ((1...𝑁) × {0}) / 𝑝𝐵 = 0)
203 ovex 6904 . . . . . . . . . . . . . . . 16 (1...𝑁) ∈ V
204 snex 5096 . . . . . . . . . . . . . . . 16 {0} ∈ V
205203, 204xpex 7190 . . . . . . . . . . . . . . 15 ((1...𝑁) × {0}) ∈ V
206 feq1 6235 . . . . . . . . . . . . . . . . . 18 (𝑝 = ((1...𝑁) × {0}) → (𝑝:(1...𝑁)⟶(0...𝐾) ↔ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾)))
207 fveq1 6405 . . . . . . . . . . . . . . . . . . 19 (𝑝 = ((1...𝑁) × {0}) → (𝑝‘1) = (((1...𝑁) × {0})‘1))
208207eqeq1d 2806 . . . . . . . . . . . . . . . . . 18 (𝑝 = ((1...𝑁) × {0}) → ((𝑝‘1) = 0 ↔ (((1...𝑁) × {0})‘1) = 0))
209206, 2083anbi23d 1556 . . . . . . . . . . . . . . . . 17 (𝑝 = ((1...𝑁) × {0}) → ((1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0) ↔ (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) = 0)))
210209anbi2d 616 . . . . . . . . . . . . . . . 16 (𝑝 = ((1...𝑁) × {0}) → ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)) ↔ (𝜑 ∧ (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) = 0))))
211 csbeq1a 3735 . . . . . . . . . . . . . . . . 17 (𝑝 = ((1...𝑁) × {0}) → 𝐵 = ((1...𝑁) × {0}) / 𝑝𝐵)
212211eqeq1d 2806 . . . . . . . . . . . . . . . 16 (𝑝 = ((1...𝑁) × {0}) → (𝐵 = 0 ↔ ((1...𝑁) × {0}) / 𝑝𝐵 = 0))
213210, 212imbi12d 335 . . . . . . . . . . . . . . 15 (𝑝 = ((1...𝑁) × {0}) → (((𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)) → 𝐵 = 0) ↔ ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) = 0)) → ((1...𝑁) × {0}) / 𝑝𝐵 = 0)))
214 1ex 10319 . . . . . . . . . . . . . . . . 17 1 ∈ V
215 eleq1 2871 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 1 → (𝑛 ∈ (1...𝑁) ↔ 1 ∈ (1...𝑁)))
216 fveqeq2 6415 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 1 → ((𝑝𝑛) = 0 ↔ (𝑝‘1) = 0))
217215, 2163anbi13d 1555 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 1 → ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 0) ↔ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)))
218217anbi2d 616 . . . . . . . . . . . . . . . . . 18 (𝑛 = 1 → ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 0)) ↔ (𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0))))
219 breq2 4846 . . . . . . . . . . . . . . . . . 18 (𝑛 = 1 → (𝐵 < 𝑛𝐵 < 1))
220218, 219imbi12d 335 . . . . . . . . . . . . . . . . 17 (𝑛 = 1 → (((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 0)) → 𝐵 < 𝑛) ↔ ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)) → 𝐵 < 1)))
221 poimirlem28.3 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 0)) → 𝐵 < 𝑛)
222214, 220, 221vtocl 3450 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)) → 𝐵 < 1)
223 poimirlem28.2 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁))
224 elfznn0 12654 . . . . . . . . . . . . . . . . . 18 (𝐵 ∈ (0...𝑁) → 𝐵 ∈ ℕ0)
225 nn0lt10b 11703 . . . . . . . . . . . . . . . . . 18 (𝐵 ∈ ℕ0 → (𝐵 < 1 ↔ 𝐵 = 0))
226223, 224, 2253syl 18 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝:(1...𝑁)⟶(0...𝐾)) → (𝐵 < 1 ↔ 𝐵 = 0))
2272263ad2antr2 1233 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)) → (𝐵 < 1 ↔ 𝐵 = 0))
228222, 227mpbid 223 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)) → 𝐵 = 0)
229202, 205, 213, 228vtoclf 3449 . . . . . . . . . . . . . 14 ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) = 0)) → ((1...𝑁) × {0}) / 𝑝𝐵 = 0)
230198, 229mpdan 670 . . . . . . . . . . . . 13 (𝜑((1...𝑁) × {0}) / 𝑝𝐵 = 0)
231230eqcomd 2810 . . . . . . . . . . . 12 (𝜑 → 0 = ((1...𝑁) × {0}) / 𝑝𝐵)
232231ralrimivw 3153 . . . . . . . . . . 11 (𝜑 → ∀𝑠 ∈ {⟨∅, ∅⟩}0 = ((1...𝑁) × {0}) / 𝑝𝐵)
233 rabid2 3305 . . . . . . . . . . 11 ({⟨∅, ∅⟩} = {𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ((1...𝑁) × {0}) / 𝑝𝐵} ↔ ∀𝑠 ∈ {⟨∅, ∅⟩}0 = ((1...𝑁) × {0}) / 𝑝𝐵)
234232, 233sylibr 225 . . . . . . . . . 10 (𝜑 → {⟨∅, ∅⟩} = {𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ((1...𝑁) × {0}) / 𝑝𝐵})
235234fveq2d 6410 . . . . . . . . 9 (𝜑 → (♯‘{⟨∅, ∅⟩}) = (♯‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ((1...𝑁) × {0}) / 𝑝𝐵}))
236186, 235syl5eqr 2852 . . . . . . . 8 (𝜑 → 1 = (♯‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ((1...𝑁) × {0}) / 𝑝𝐵}))
237236breq2d 4854 . . . . . . 7 (𝜑 → (2 ∥ 1 ↔ 2 ∥ (♯‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ((1...𝑁) × {0}) / 𝑝𝐵})))
238183, 237mtbii 317 . . . . . 6 (𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ((1...𝑁) × {0}) / 𝑝𝐵}))
239238a1i 11 . . . . 5 (𝑁 ∈ ℕ0 → (𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ((1...𝑁) × {0}) / 𝑝𝐵})))
240 2z 11673 . . . . . . . . . . . . 13 2 ∈ ℤ
241 fzfi 12993 . . . . . . . . . . . . . . . . 17 (1...(𝑚 + 1)) ∈ Fin
242 mapfi 8499 . . . . . . . . . . . . . . . . 17 (((0..^𝐾) ∈ Fin ∧ (1...(𝑚 + 1)) ∈ Fin) → ((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) ∈ Fin)
24310, 241, 242mp2an 675 . . . . . . . . . . . . . . . 16 ((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) ∈ Fin
244 ovex 6904 . . . . . . . . . . . . . . . . . . 19 (1...(𝑚 + 1)) ∈ V
245244, 244mapval 8102 . . . . . . . . . . . . . . . . . 18 ((1...(𝑚 + 1)) ↑𝑚 (1...(𝑚 + 1))) = {𝑓𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))}
246 mapfi 8499 . . . . . . . . . . . . . . . . . . 19 (((1...(𝑚 + 1)) ∈ Fin ∧ (1...(𝑚 + 1)) ∈ Fin) → ((1...(𝑚 + 1)) ↑𝑚 (1...(𝑚 + 1))) ∈ Fin)
247241, 241, 246mp2an 675 . . . . . . . . . . . . . . . . . 18 ((1...(𝑚 + 1)) ↑𝑚 (1...(𝑚 + 1))) ∈ Fin
248245, 247eqeltrri 2880 . . . . . . . . . . . . . . . . 17 {𝑓𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))} ∈ Fin
249 f1of 6351 . . . . . . . . . . . . . . . . . 18 (𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1)) → 𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1)))
250249ss2abi 3869 . . . . . . . . . . . . . . . . 17 {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))} ⊆ {𝑓𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))}
251 ssfi 8417 . . . . . . . . . . . . . . . . 17 (({𝑓𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))} ∈ Fin ∧ {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))} ⊆ {𝑓𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))}) → {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))} ∈ Fin)
252248, 250, 251mp2an 675 . . . . . . . . . . . . . . . 16 {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))} ∈ Fin
253 xpfi 8468 . . . . . . . . . . . . . . . 16 ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) ∈ Fin ∧ {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))} ∈ Fin) → (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∈ Fin)
254243, 252, 253mp2an 675 . . . . . . . . . . . . . . 15 (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∈ Fin
255 rabfi 8422 . . . . . . . . . . . . . . 15 ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∈ Fin → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵} ∈ Fin)
256 hashcl 13363 . . . . . . . . . . . . . . 15 ({𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵} ∈ Fin → (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) ∈ ℕ0)
257254, 255, 256mp2b 10 . . . . . . . . . . . . . 14 (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) ∈ ℕ0
258257nn0zi 11666 . . . . . . . . . . . . 13 (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) ∈ ℤ
259 rabfi 8422 . . . . . . . . . . . . . . 15 ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∈ Fin → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))} ∈ Fin)
260 hashcl 13363 . . . . . . . . . . . . . . 15 ({𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))} ∈ Fin → (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ∈ ℕ0)
261254, 259, 260mp2b 10 . . . . . . . . . . . . . 14 (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ∈ ℕ0
262261nn0zi 11666 . . . . . . . . . . . . 13 (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ∈ ℤ
263240, 258, 2623pm3.2i 1431 . . . . . . . . . . . 12 (2 ∈ ℤ ∧ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) ∈ ℤ ∧ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ∈ ℤ)
264 nn0p1nn 11596 . . . . . . . . . . . . . . . 16 (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℕ)
265264ad2antrl 710 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → (𝑚 + 1) ∈ ℕ)
266 uneq1 3957 . . . . . . . . . . . . . . . 16 (𝑞 = ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})))
267266csbeq1d 3733 . . . . . . . . . . . . . . 15 (𝑞 = ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵)
26870fconst 6304 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑚 + 1) + 1)...𝑁) × {0}):(((𝑚 + 1) + 1)...𝑁)⟶{0}
269268jctr 516 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑞:(1...(𝑚 + 1))⟶(0...𝐾) → (𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ ((((𝑚 + 1) + 1)...𝑁) × {0}):(((𝑚 + 1) + 1)...𝑁)⟶{0}))
270264nnred 11318 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℝ)
271270ltp1d 11237 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 ∈ ℕ0 → (𝑚 + 1) < ((𝑚 + 1) + 1))
272 fzdisj 12589 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑚 + 1) < ((𝑚 + 1) + 1) → ((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅)
273271, 272syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 ∈ ℕ0 → ((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅)
274 fun 6279 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ ((((𝑚 + 1) + 1)...𝑁) × {0}):(((𝑚 + 1) + 1)...𝑁)⟶{0}) ∧ ((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁))⟶((0...𝐾) ∪ {0}))
275269, 273, 274syl2anr 586 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑚 ∈ ℕ0𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁))⟶((0...𝐾) ∪ {0}))
276275adantlr 697 . . . . . . . . . . . . . . . . . . . . 21 (((𝑚 ∈ ℕ0𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁))⟶((0...𝐾) ∪ {0}))
277276adantl 469 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ ((𝑚 ∈ ℕ0𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁))⟶((0...𝐾) ∪ {0}))
278264peano2nnd 11320 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑚 ∈ ℕ0 → ((𝑚 + 1) + 1) ∈ ℕ)
279278, 187syl6eleq 2893 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑚 ∈ ℕ0 → ((𝑚 + 1) + 1) ∈ (ℤ‘1))
280279ad2antrl 710 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → ((𝑚 + 1) + 1) ∈ (ℤ‘1))
281 nn0z 11664 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑚 ∈ ℕ0𝑚 ∈ ℤ)
2821nnzd 11745 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑𝑁 ∈ ℤ)
283 zltp1le 11691 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑚 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑚 < 𝑁 ↔ (𝑚 + 1) ≤ 𝑁))
284281, 282, 283syl2anr 586 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑚 ∈ ℕ0) → (𝑚 < 𝑁 ↔ (𝑚 + 1) ≤ 𝑁))
285284biimpa 464 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) → (𝑚 + 1) ≤ 𝑁)
286285anasss 454 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → (𝑚 + 1) ≤ 𝑁)
287281peano2zd 11749 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℤ)
288287adantr 468 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑚 ∈ ℕ0𝑚 < 𝑁) → (𝑚 + 1) ∈ ℤ)
289 eluz 11916 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑚 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ‘(𝑚 + 1)) ↔ (𝑚 + 1) ≤ 𝑁))
290288, 282, 289syl2anr 586 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → (𝑁 ∈ (ℤ‘(𝑚 + 1)) ↔ (𝑚 + 1) ≤ 𝑁))
291286, 290mpbird 248 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → 𝑁 ∈ (ℤ‘(𝑚 + 1)))
292 fzsplit2 12587 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑚 + 1) + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ‘(𝑚 + 1))) → (1...𝑁) = ((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁)))
293280, 291, 292syl2anc 575 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → (1...𝑁) = ((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁)))
294293eqcomd 2810 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → ((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁)) = (1...𝑁))
295192, 193syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → 0 ∈ (0...𝐾))
296295snssd 4528 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → {0} ⊆ (0...𝐾))
297 ssequn2 3983 . . . . . . . . . . . . . . . . . . . . . . . 24 ({0} ⊆ (0...𝐾) ↔ ((0...𝐾) ∪ {0}) = (0...𝐾))
298296, 297sylib 209 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((0...𝐾) ∪ {0}) = (0...𝐾))
299298adantr 468 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → ((0...𝐾) ∪ {0}) = (0...𝐾))
300294, 299feq23d 6249 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁))⟶((0...𝐾) ∪ {0}) ↔ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)))
301300adantrr 699 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ ((𝑚 ∈ ℕ0𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁))⟶((0...𝐾) ∪ {0}) ↔ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)))
302277, 301mpbid 223 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ((𝑚 ∈ ℕ0𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
303 nfv 2005 . . . . . . . . . . . . . . . . . . . . 21 𝑝(𝜑 ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
304 nfcsb1v 3742 . . . . . . . . . . . . . . . . . . . . . 22 𝑝(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵
305304nfel1 2961 . . . . . . . . . . . . . . . . . . . . 21 𝑝(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ (0...𝑁)
306303, 305nfim 1987 . . . . . . . . . . . . . . . . . . . 20 𝑝((𝜑 ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ (0...𝑁))
307 vex 3392 . . . . . . . . . . . . . . . . . . . . 21 𝑞 ∈ V
308 ovex 6904 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑚 + 1) + 1)...𝑁) ∈ V
309308, 204xpex 7190 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑚 + 1) + 1)...𝑁) × {0}) ∈ V
310307, 309unex 7184 . . . . . . . . . . . . . . . . . . . 20 (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) ∈ V
311 feq1 6235 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝑝:(1...𝑁)⟶(0...𝐾) ↔ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)))
312311anbi2d 616 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝜑𝑝:(1...𝑁)⟶(0...𝐾)) ↔ (𝜑 ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))))
313 csbeq1a 3735 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → 𝐵 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵)
314313eleq1d 2868 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝐵 ∈ (0...𝑁) ↔ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ (0...𝑁)))
315312, 314imbi12d 335 . . . . . . . . . . . . . . . . . . . 20 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (((𝜑𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁)) ↔ ((𝜑 ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ (0...𝑁))))
316306, 310, 315, 223vtoclf 3449 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ (0...𝑁))
317302, 316syldan 581 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑚 ∈ ℕ0𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ (0...𝑁))
318317anassrs 455 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ (0...𝑁))
319 elfznn0 12654 . . . . . . . . . . . . . . . . 17 ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ (0...𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ ℕ0)
320318, 319syl 17 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ ℕ0)
321264nnnn0d 11615 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℕ0)
322321adantr 468 . . . . . . . . . . . . . . . . 17 ((𝑚 ∈ ℕ0𝑚 < 𝑁) → (𝑚 + 1) ∈ ℕ0)
323322ad2antlr 709 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑚 + 1) ∈ ℕ0)
324 leloe 10407 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑚 + 1) ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑚 + 1) ≤ 𝑁 ↔ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)))
325270, 3, 324syl2anr 586 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑚 ∈ ℕ0) → ((𝑚 + 1) ≤ 𝑁 ↔ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)))
326284, 325bitrd 270 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑚 ∈ ℕ0) → (𝑚 < 𝑁 ↔ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)))
327326biimpd 220 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ ℕ0) → (𝑚 < 𝑁 → ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)))
328327imdistani 560 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) → ((𝜑𝑚 ∈ ℕ0) ∧ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)))
329328anasss 454 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → ((𝜑𝑚 ∈ ℕ0) ∧ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)))
330 simplll 782 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → 𝜑)
331278nnge1d 11347 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 ∈ ℕ0 → 1 ≤ ((𝑚 + 1) + 1))
332331ad2antlr 709 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 1 ≤ ((𝑚 + 1) + 1))
333 zltp1le 11691 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑚 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑚 + 1) < 𝑁 ↔ ((𝑚 + 1) + 1) ≤ 𝑁))
334287, 282, 333syl2anr 586 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑚 ∈ ℕ0) → ((𝑚 + 1) < 𝑁 ↔ ((𝑚 + 1) + 1) ≤ 𝑁))
335334biimpa 464 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ≤ 𝑁)
336287peano2zd 11749 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑚 ∈ ℕ0 → ((𝑚 + 1) + 1) ∈ ℤ)
337 1z 11671 . . . . . . . . . . . . . . . . . . . . . . . . . 26 1 ∈ ℤ
338 elfz 12553 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝑚 + 1) + 1) ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((𝑚 + 1) + 1) ∈ (1...𝑁) ↔ (1 ≤ ((𝑚 + 1) + 1) ∧ ((𝑚 + 1) + 1) ≤ 𝑁)))
339337, 338mp3an2 1566 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑚 + 1) + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((𝑚 + 1) + 1) ∈ (1...𝑁) ↔ (1 ≤ ((𝑚 + 1) + 1) ∧ ((𝑚 + 1) + 1) ≤ 𝑁)))
340336, 282, 339syl2anr 586 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑚 ∈ ℕ0) → (((𝑚 + 1) + 1) ∈ (1...𝑁) ↔ (1 ≤ ((𝑚 + 1) + 1) ∧ ((𝑚 + 1) + 1) ≤ 𝑁)))
341340adantr 468 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → (((𝑚 + 1) + 1) ∈ (1...𝑁) ↔ (1 ≤ ((𝑚 + 1) + 1) ∧ ((𝑚 + 1) + 1) ≤ 𝑁)))
342332, 335, 341mpbir2and 695 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ∈ (1...𝑁))
343342adantlr 697 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ∈ (1...𝑁))
344 nn0re 11566 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑚 ∈ ℕ0𝑚 ∈ ℝ)
345344ad2antlr 709 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑚 ∈ ℝ)
346270ad2antlr 709 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → (𝑚 + 1) ∈ ℝ)
3473ad2antrr 708 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑁 ∈ ℝ)
348344ltp1d 11237 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑚 ∈ ℕ0𝑚 < (𝑚 + 1))
349348ad2antlr 709 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑚 < (𝑚 + 1))
350 simpr 473 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → (𝑚 + 1) < 𝑁)
351345, 346, 347, 349, 350lttrd 10481 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑚 < 𝑁)
352351adantlr 697 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → 𝑚 < 𝑁)
353 anass 456 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) ↔ (𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)))
354302anassrs 455 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
355353, 354sylanb 572 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
356355an32s 634 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ 𝑚 < 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
357352, 356syldan 581 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
358 ffn 6254 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑞:(1...(𝑚 + 1))⟶(0...𝐾) → 𝑞 Fn (1...(𝑚 + 1)))
359358ad2antlr 709 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → 𝑞 Fn (1...(𝑚 + 1)))
360273ad3antlr 713 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅)
361 eluz 11916 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑚 + 1) + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ‘((𝑚 + 1) + 1)) ↔ ((𝑚 + 1) + 1) ≤ 𝑁))
362336, 282, 361syl2anr 586 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑚 ∈ ℕ0) → (𝑁 ∈ (ℤ‘((𝑚 + 1) + 1)) ↔ ((𝑚 + 1) + 1) ≤ 𝑁))
363362adantr 468 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → (𝑁 ∈ (ℤ‘((𝑚 + 1) + 1)) ↔ ((𝑚 + 1) + 1) ≤ 𝑁))
364335, 363mpbird 248 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑁 ∈ (ℤ‘((𝑚 + 1) + 1)))
365 eluzfz1 12569 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ (ℤ‘((𝑚 + 1) + 1)) → ((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁))
366364, 365syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁))
367366adantlr 697 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁))
368 fnconstg 6306 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0 ∈ V → ((((𝑚 + 1) + 1)...𝑁) × {0}) Fn (((𝑚 + 1) + 1)...𝑁))
36970, 368ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑚 + 1) + 1)...𝑁) × {0}) Fn (((𝑚 + 1) + 1)...𝑁)
370 fvun2 6489 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑞 Fn (1...(𝑚 + 1)) ∧ ((((𝑚 + 1) + 1)...𝑁) × {0}) Fn (((𝑚 + 1) + 1)...𝑁) ∧ (((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅ ∧ ((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = (((((𝑚 + 1) + 1)...𝑁) × {0})‘((𝑚 + 1) + 1)))
371369, 370mp3an2 1566 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑞 Fn (1...(𝑚 + 1)) ∧ (((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅ ∧ ((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = (((((𝑚 + 1) + 1)...𝑁) × {0})‘((𝑚 + 1) + 1)))
372359, 360, 367, 371syl12anc 856 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = (((((𝑚 + 1) + 1)...𝑁) × {0})‘((𝑚 + 1) + 1)))
37370fvconst2 6692 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁) → (((((𝑚 + 1) + 1)...𝑁) × {0})‘((𝑚 + 1) + 1)) = 0)
374367, 373syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → (((((𝑚 + 1) + 1)...𝑁) × {0})‘((𝑚 + 1) + 1)) = 0)
375372, 374eqtrd 2838 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0)
376 nfv 2005 . . . . . . . . . . . . . . . . . . . . . . 23 𝑝(𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0))
377 nfcv 2946 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑝 <
378 nfcv 2946 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑝((𝑚 + 1) + 1)
379304, 377, 378nfbr 4889 . . . . . . . . . . . . . . . . . . . . . . 23 𝑝(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 < ((𝑚 + 1) + 1)
380376, 379nfim 1987 . . . . . . . . . . . . . . . . . . . . . 22 𝑝((𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 < ((𝑚 + 1) + 1))
381 fveq1 6405 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝑝‘((𝑚 + 1) + 1)) = ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)))
382381eqeq1d 2806 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝑝‘((𝑚 + 1) + 1)) = 0 ↔ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0))
383311, 3823anbi23d 1556 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘((𝑚 + 1) + 1)) = 0) ↔ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0)))
384383anbi2d 616 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘((𝑚 + 1) + 1)) = 0)) ↔ (𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0))))
385313breq1d 4852 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝐵 < ((𝑚 + 1) + 1) ↔ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 < ((𝑚 + 1) + 1)))
386384, 385imbi12d 335 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (((𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘((𝑚 + 1) + 1)) = 0)) → 𝐵 < ((𝑚 + 1) + 1)) ↔ ((𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 < ((𝑚 + 1) + 1))))
387 ovex 6904 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑚 + 1) + 1) ∈ V
388 eleq1 2871 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = ((𝑚 + 1) + 1) → (𝑛 ∈ (1...𝑁) ↔ ((𝑚 + 1) + 1) ∈ (1...𝑁)))
389 fveqeq2 6415 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = ((𝑚 + 1) + 1) → ((𝑝𝑛) = 0 ↔ (𝑝‘((𝑚 + 1) + 1)) = 0))
390388, 3893anbi13d 1555 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = ((𝑚 + 1) + 1) → ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 0) ↔ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘((𝑚 + 1) + 1)) = 0)))
391390anbi2d 616 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = ((𝑚 + 1) + 1) → ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 0)) ↔ (𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘((𝑚 + 1) + 1)) = 0))))
392 breq2 4846 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = ((𝑚 + 1) + 1) → (𝐵 < 𝑛𝐵 < ((𝑚 + 1) + 1)))
393391, 392imbi12d 335 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = ((𝑚 + 1) + 1) → (((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 0)) → 𝐵 < 𝑛) ↔ ((𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘((𝑚 + 1) + 1)) = 0)) → 𝐵 < ((𝑚 + 1) + 1))))
394387, 393, 221vtocl 3450 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘((𝑚 + 1) + 1)) = 0)) → 𝐵 < ((𝑚 + 1) + 1))
395380, 310, 386, 394vtoclf 3449 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 < ((𝑚 + 1) + 1))
396330, 343, 357, 375, 395syl13anc 1484 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 < ((𝑚 + 1) + 1))
397353, 318sylanb 572 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ (0...𝑁))
398397an32s 634 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ 𝑚 < 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ (0...𝑁))
399 elfzelz 12563 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ (0...𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ ℤ)
400398, 399syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ 𝑚 < 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ ℤ)
401352, 400syldan 581 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ ℤ)
402287ad3antlr 713 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → (𝑚 + 1) ∈ ℤ)
403 zleltp1 11692 . . . . . . . . . . . . . . . . . . . . 21 (((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ ℤ ∧ (𝑚 + 1) ∈ ℤ) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ≤ (𝑚 + 1) ↔ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 < ((𝑚 + 1) + 1)))
404401, 402, 403syl2anc 575 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ≤ (𝑚 + 1) ↔ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 < ((𝑚 + 1) + 1)))
405396, 404mpbird 248 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ≤ (𝑚 + 1))
406348ad2antlr 709 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → 𝑚 < (𝑚 + 1))
407 breq2 4846 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑚 + 1) = 𝑁 → (𝑚 < (𝑚 + 1) ↔ 𝑚 < 𝑁))
408407biimpac 466 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑚 < (𝑚 + 1) ∧ (𝑚 + 1) = 𝑁) → 𝑚 < 𝑁)
409406, 408sylan 571 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) = 𝑁) → 𝑚 < 𝑁)
410 elfzle2 12566 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ (0...𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵𝑁)
411398, 410syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ 𝑚 < 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵𝑁)
412409, 411syldan 581 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) = 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵𝑁)
413 simpr 473 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) = 𝑁) → (𝑚 + 1) = 𝑁)
414412, 413breqtrrd 4870 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) = 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ≤ (𝑚 + 1))
415405, 414jaodan 971 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ≤ (𝑚 + 1))
416415an32s 634 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑚 ∈ ℕ0) ∧ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ≤ (𝑚 + 1))
417329, 416sylan 571 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ≤ (𝑚 + 1))
418 elfz2nn0 12652 . . . . . . . . . . . . . . . 16 ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ (0...(𝑚 + 1)) ↔ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ ℕ0 ∧ (𝑚 + 1) ∈ ℕ0(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ≤ (𝑚 + 1)))
419320, 323, 417, 418syl3anbrc 1436 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ (0...(𝑚 + 1)))
420 fzss2 12602 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ (ℤ‘(𝑚 + 1)) → (1...(𝑚 + 1)) ⊆ (1...𝑁))
421291, 420syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → (1...(𝑚 + 1)) ⊆ (1...𝑁))
422421sselda 3796 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) ∧ 𝑛 ∈ (1...(𝑚 + 1))) → 𝑛 ∈ (1...𝑁))
4234223ad2antr1 1232 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞𝑛) = 0)) → 𝑛 ∈ (1...𝑁))
4243543ad2antr2 1233 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞𝑛) = 0)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
425358ad2antll 711 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → 𝑞 Fn (1...(𝑚 + 1)))
426273ad2antlr 709 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → ((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅)
427 simprl 778 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → 𝑛 ∈ (1...(𝑚 + 1)))
428 fvun1 6488 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑞 Fn (1...(𝑚 + 1)) ∧ ((((𝑚 + 1) + 1)...𝑁) × {0}) Fn (((𝑚 + 1) + 1)...𝑁) ∧ (((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅ ∧ 𝑛 ∈ (1...(𝑚 + 1)))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = (𝑞𝑛))
429369, 428mp3an2 1566 . . . . . . . . . . . . . . . . . . . . 21 ((𝑞 Fn (1...(𝑚 + 1)) ∧ (((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅ ∧ 𝑛 ∈ (1...(𝑚 + 1)))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = (𝑞𝑛))
430425, 426, 427, 429syl12anc 856 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = (𝑞𝑛))
431430adantlrr 703 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = (𝑞𝑛))
4324313adantr3 1205 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞𝑛) = 0)) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = (𝑞𝑛))
433 simpr3 1245 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞𝑛) = 0)) → (𝑞𝑛) = 0)
434432, 433eqtrd 2838 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞𝑛) = 0)) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0)
435423, 424, 4343jca 1151 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞𝑛) = 0)) → (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0))
436 nfv 2005 . . . . . . . . . . . . . . . . . . 19 𝑝(𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0))
437 nfcv 2946 . . . . . . . . . . . . . . . . . . . 20 𝑝𝑛
438304, 377, 437nfbr 4889 . . . . . . . . . . . . . . . . . . 19 𝑝(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 < 𝑛
439436, 438nfim 1987 . . . . . . . . . . . . . . . . . 18 𝑝((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 < 𝑛)
440 fveq1 6405 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝑝𝑛) = ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛))
441440eqeq1d 2806 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝑝𝑛) = 0 ↔ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0))
442311, 4413anbi23d 1556 . . . . . . . . . . . . . . . . . . . 20 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 0) ↔ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0)))
443442anbi2d 616 . . . . . . . . . . . . . . . . . . 19 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 0)) ↔ (𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0))))
444313breq1d 4852 . . . . . . . . . . . . . . . . . . 19 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝐵 < 𝑛(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 < 𝑛))
445443, 444imbi12d 335 . . . . . . . . . . . . . . . . . 18 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 0)) → 𝐵 < 𝑛) ↔ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 < 𝑛)))
446439, 310, 445, 221vtoclf 3449 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 < 𝑛)
447446adantlr 697 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 < 𝑛)
448435, 447syldan 581 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞𝑛) = 0)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 < 𝑛)
449 simp1 1159 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞𝑛) = 𝐾) → 𝑛 ∈ (1...(𝑚 + 1)))
450422anasss 454 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ((𝑚 ∈ ℕ0𝑚 < 𝑁) ∧ 𝑛 ∈ (1...(𝑚 + 1)))) → 𝑛 ∈ (1...𝑁))
451449, 450sylanr2 665 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑚 ∈ ℕ0𝑚 < 𝑁) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞𝑛) = 𝐾))) → 𝑛 ∈ (1...𝑁))
452 simp2 1160 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞𝑛) = 𝐾) → 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))
453452, 302sylanr2 665 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑚 ∈ ℕ0𝑚 < 𝑁) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞𝑛) = 𝐾))) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
4544303adantr3 1205 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞𝑛) = 𝐾)) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = (𝑞𝑛))
455 simpr3 1245 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞𝑛) = 𝐾)) → (𝑞𝑛) = 𝐾)
456454, 455eqtrd 2838 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞𝑛) = 𝐾)) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)
457456anasss 454 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞𝑛) = 𝐾))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)
458457adantrlr 705 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑚 ∈ ℕ0𝑚 < 𝑁) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞𝑛) = 𝐾))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)
459451, 453, 4583jca 1151 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑚 ∈ ℕ0𝑚 < 𝑁) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞𝑛) = 𝐾))) → (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾))
460 nfv 2005 . . . . . . . . . . . . . . . . . . 19 𝑝(𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾))
461 nfcv 2946 . . . . . . . . . . . . . . . . . . . 20 𝑝(𝑛 − 1)
462304, 461nfne 3076 . . . . . . . . . . . . . . . . . . 19 𝑝(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ≠ (𝑛 − 1)
463460, 462nfim 1987 . . . . . . . . . . . . . . . . . 18 𝑝((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ≠ (𝑛 − 1))
464440eqeq1d 2806 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝑝𝑛) = 𝐾 ↔ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾))
465311, 4643anbi23d 1556 . . . . . . . . . . . . . . . . . . . 20 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 𝐾) ↔ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)))
466465anbi2d 616 . . . . . . . . . . . . . . . . . . 19 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 𝐾)) ↔ (𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾))))
467313neeq1d 3035 . . . . . . . . . . . . . . . . . . 19 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝐵 ≠ (𝑛 − 1) ↔ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ≠ (𝑛 − 1)))
468466, 467imbi12d 335 . . . . . . . . . . . . . . . . . 18 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 𝐾)) → 𝐵 ≠ (𝑛 − 1)) ↔ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ≠ (𝑛 − 1))))
469 poimirlem28.4 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 𝐾)) → 𝐵 ≠ (𝑛 − 1))
470463, 310, 468, 469vtoclf 3449 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ≠ (𝑛 − 1))
471459, 470syldan 581 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑚 ∈ ℕ0𝑚 < 𝑁) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞𝑛) = 𝐾))) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ≠ (𝑛 − 1))
472471anassrs 455 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞𝑛) = 𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ≠ (𝑛 − 1))
473265, 267, 419, 448, 472poimirlem27 33747 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → 2 ∥ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})))
474265, 267, 419poimirlem26 33746 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → 2 ∥ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵})))
475 fzfi 12993 . . . . . . . . . . . . . . . . . . 19 (0...(𝑚 + 1)) ∈ Fin
476 xpfi 8468 . . . . . . . . . . . . . . . . . . 19 (((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∈ Fin ∧ (0...(𝑚 + 1)) ∈ Fin) → ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∈ Fin)
477254, 475, 476mp2an 675 . . . . . . . . . . . . . . . . . 18 ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∈ Fin
478 rabfi 8422 . . . . . . . . . . . . . . . . . 18 (((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∈ Fin → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵} ∈ Fin)
479 hashcl 13363 . . . . . . . . . . . . . . . . . 18 ({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵} ∈ Fin → (♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) ∈ ℕ0)
480477, 478, 479mp2b 10 . . . . . . . . . . . . . . . . 17 (♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) ∈ ℕ0
481480nn0zi 11666 . . . . . . . . . . . . . . . 16 (♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) ∈ ℤ
482 zsubcl 11683 . . . . . . . . . . . . . . . 16 (((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) ∈ ℤ ∧ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ∈ ℤ) → ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) ∈ ℤ)
483481, 262, 482mp2an 675 . . . . . . . . . . . . . . 15 ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) ∈ ℤ
484 zsubcl 11683 . . . . . . . . . . . . . . . 16 (((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) ∈ ℤ ∧ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) ∈ ℤ) → ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵})) ∈ ℤ)
485481, 258, 484mp2an 675 . . . . . . . . . . . . . . 15 ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵})) ∈ ℤ
486 dvds2sub 15237 . . . . . . . . . . . . . . 15 ((2 ∈ ℤ ∧ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) ∈ ℤ ∧ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵})) ∈ ℤ) → ((2 ∥ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) ∧ 2 ∥ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}))) → 2 ∥ (((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) − ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵})))))
487240, 483, 485, 486mp3an 1578 . . . . . . . . . . . . . 14 ((2 ∥ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) ∧ 2 ∥ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}))) → 2 ∥ (((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) − ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}))))
488473, 474, 487syl2anc 575 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → 2 ∥ (((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) − ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}))))
489480nn0cni 11569 . . . . . . . . . . . . . 14 (♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) ∈ ℂ
490261nn0cni 11569 . . . . . . . . . . . . . 14 (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ∈ ℂ
491257nn0cni 11569 . . . . . . . . . . . . . 14 (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) ∈ ℂ
492 nnncan1 10600 . . . . . . . . . . . . . 14 (((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) ∈ ℂ ∧ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ∈ ℂ ∧ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) ∈ ℂ) → (((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) − ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}))) = ((♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})))
493489, 490, 491, 492mp3an 1578 . . . . . . . . . . . . 13 (((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) − ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd𝑡)})𝑖 = (1st𝑡) / 𝑠(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}))) = ((♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}))
494488, 493syl6breq 4883 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → 2 ∥ ((♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})))
495 dvdssub2 15244 . . . . . . . . . . . 12 (((2 ∈ ℤ ∧ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) ∈ ℤ ∧ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ∈ ℤ) ∧ 2 ∥ ((♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}))) → (2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) ↔ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})))
496263, 494, 495sylancr 577 . . . . . . . . . . 11 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → (2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) ↔ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})))
497 nn0cn 11567 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ ℕ0𝑚 ∈ ℂ)
498 pncan1 10737 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ ℂ → ((𝑚 + 1) − 1) = 𝑚)
499497, 498syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑚 ∈ ℕ0 → ((𝑚 + 1) − 1) = 𝑚)
500499oveq2d 6888 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ ℕ0 → (0...((𝑚 + 1) − 1)) = (0...𝑚))
501500rexeqdv 3332 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ ℕ0 → (∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ↔ ∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵))
502500, 501raleqbidv 3339 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ ℕ0 → (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ↔ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵))
5035023anbi1d 1557 . . . . . . . . . . . . . . . 16 (𝑚 ∈ ℕ0 → ((∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1)) ↔ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))))
504503rabbidv 3377 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℕ0 → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))} = {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})
505504fveq2d 6410 . . . . . . . . . . . . . 14 (𝑚 ∈ ℕ0 → (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) = (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}))
506505ad2antrl 710 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) = (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}))
5071adantr 468 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → 𝑁 ∈ ℕ)
508191adantr 468 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → 𝐾 ∈ ℕ)
509 simprl 778 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → 𝑚 ∈ ℕ0)
510 simprr 780 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → 𝑚 < 𝑁)
511507, 508, 509, 510poimirlem4 33724 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝𝐵} ≈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})
512 fzfi 12993 . . . . . . . . . . . . . . . . . 18 (1...𝑚) ∈ Fin
513 mapfi 8499 . . . . . . . . . . . . . . . . . 18 (((0..^𝐾) ∈ Fin ∧ (1...𝑚) ∈ Fin) → ((0..^𝐾) ↑𝑚 (1...𝑚)) ∈ Fin)
51410, 512, 513mp2an 675 . . . . . . . . . . . . . . . . 17 ((0..^𝐾) ↑𝑚 (1...𝑚)) ∈ Fin
515 ovex 6904 . . . . . . . . . . . . . . . . . . . 20 (1...𝑚) ∈ V
516515, 515mapval 8102 . . . . . . . . . . . . . . . . . . 19 ((1...𝑚) ↑𝑚 (1...𝑚)) = {𝑓𝑓:(1...𝑚)⟶(1...𝑚)}
517 mapfi 8499 . . . . . . . . . . . . . . . . . . . 20 (((1...𝑚) ∈ Fin ∧ (1...𝑚) ∈ Fin) → ((1...𝑚) ↑𝑚 (1...𝑚)) ∈ Fin)
518512, 512, 517mp2an 675 . . . . . . . . . . . . . . . . . . 19 ((1...𝑚) ↑𝑚 (1...𝑚)) ∈ Fin
519516, 518eqeltrri 2880 . . . . . . . . . . . . . . . . . 18 {𝑓𝑓:(1...𝑚)⟶(1...𝑚)} ∈ Fin
520 f1of 6351 . . . . . . . . . . . . . . . . . . 19 (𝑓:(1...𝑚)–1-1-onto→(1...𝑚) → 𝑓:(1...𝑚)⟶(1...𝑚))
521520ss2abi 3869 . . . . . . . . . . . . . . . . . 18 {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)} ⊆ {𝑓𝑓:(1...𝑚)⟶(1...𝑚)}
522 ssfi 8417 . . . . . . . . . . . . . . . . . 18 (({𝑓𝑓:(1...𝑚)⟶(1...𝑚)} ∈ Fin ∧ {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)} ⊆ {𝑓𝑓:(1...𝑚)⟶(1...𝑚)}) → {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)} ∈ Fin)
523519, 521, 522mp2an 675 . . . . . . . . . . . . . . . . 17 {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)} ∈ Fin
524 xpfi 8468 . . . . . . . . . . . . . . . . 17 ((((0..^𝐾) ↑𝑚 (1...𝑚)) ∈ Fin ∧ {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)} ∈ Fin) → (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∈ Fin)
525514, 523, 524mp2an 675 . . . . . . . . . . . . . . . 16 (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∈ Fin
526 rabfi 8422 . . . . . . . . . . . . . . . 16 ((((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∈ Fin → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝𝐵} ∈ Fin)
527525, 526ax-mp 5 . . . . . . . . . . . . . . 15 {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝𝐵} ∈ Fin
528 rabfi 8422 . . . . . . . . . . . . . . . 16 ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∈ Fin → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))} ∈ Fin)
529254, 528ax-mp 5 . . . . . . . . . . . . . . 15 {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))} ∈ Fin
530 hashen 13353 . . . . . . . . . . . . . . 15 (({𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝𝐵} ∈ Fin ∧ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))} ∈ Fin) → ((♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝𝐵}) = (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ↔ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝𝐵} ≈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}))
531527, 529, 530mp2an 675 . . . . . . . . . . . . . 14 ((♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝𝐵}) = (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ↔ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝𝐵} ≈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})
532511, 531sylibr 225 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝𝐵}) = (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}))
533506, 532eqtr4d 2841 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) = (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝𝐵}))
534533breq2d 4854 . . . . . . . . . . 11 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → (2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ↔ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝𝐵})))
535496, 534bitrd 270 . . . . . . . . . 10 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → (2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) ↔ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝𝐵})))
536535biimpd 220 . . . . . . . . 9 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → (2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}) → 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝𝐵})))
537536con3d 149 . . . . . . . 8 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → (¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝𝐵}) → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵})))
538537expcom 400 . . . . . . 7 ((𝑚 ∈ ℕ0𝑚 < 𝑁) → (𝜑 → (¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝𝐵}) → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}))))
539538a2d 29 . . . . . 6 ((𝑚 ∈ ℕ0𝑚 < 𝑁) → ((𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝𝐵})) → (𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}))))
5405393adant1 1153 . . . . 5 ((𝑁 ∈ ℕ0𝑚 ∈ ℕ0𝑚 < 𝑁) → ((𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝𝐵})) → (𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵}))))
541107, 132, 157, 182, 239, 540fnn0ind 11740 . . . 4 ((𝑁 ∈ ℕ0𝑁 ∈ ℕ0𝑁𝑁) → (𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝𝐵})))
5425, 541mpcom 38 . . 3 (𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝𝐵}))
543 dvds0 15218 . . . . . . . 8 (2 ∈ ℤ → 2 ∥ 0)
544240, 543ax-mp 5 . . . . . . 7 2 ∥ 0
545 hash0 13374 . . . . . . 7 (♯‘∅) = 0
546544, 545breqtrri 4869 . . . . . 6 2 ∥ (♯‘∅)
547 fveq2 6406 . . . . . 6 ({𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} = ∅ → (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}) = (♯‘∅))
548546, 547syl5breqr 4880 . . . . 5 ({𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} = ∅ → 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}))
5493ltp1d 11237 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 < (𝑁 + 1))
550282peano2zd 11749 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑁 + 1) ∈ ℤ)
551 fzn 12578 . . . . . . . . . . . . . . . . . . 19 (((𝑁 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅))
552550, 282, 551syl2anc 575 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅))
553549, 552mpbid 223 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑁 + 1)...𝑁) = ∅)
554553xpeq1d 5337 . . . . . . . . . . . . . . . 16 (𝜑 → (((𝑁 + 1)...𝑁) × {0}) = (∅ × {0}))
555554, 86syl6eq 2854 . . . . . . . . . . . . . . 15 (𝜑 → (((𝑁 + 1)...𝑁) × {0}) = ∅)
556555uneq2d 3964 . . . . . . . . . . . . . 14 (𝜑 → (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ ∅))
557 un0 4163 . . . . . . . . . . . . . 14 (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ ∅) = ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
558556, 557syl6eq 2854 . . . . . . . . . . . . 13 (𝜑 → (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) = ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))))
559558csbeq1d 3733 . . . . . . . . . . . 12 (𝜑(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝𝐵 = ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝𝐵)
560 ovex 6904 . . . . . . . . . . . . 13 ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V
561 poimirlem28.1 . . . . . . . . . . . . 13 (𝑝 = ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶)
562560, 561csbie 3752 . . . . . . . . . . . 12 ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝𝐵 = 𝐶
563559, 562syl6eq 2854 . . . . . . . . . . 11 (𝜑(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝𝐵 = 𝐶)
564563eqeq2d 2814 . . . . . . . . . 10 (𝜑 → (𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝𝐵𝑖 = 𝐶))
565564rexbidv 3238 . . . . . . . . 9 (𝜑 → (∃𝑗 ∈ (0...𝑁)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝𝐵 ↔ ∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶))
566565ralbidv 3172 . . . . . . . 8 (𝜑 → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝𝐵 ↔ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶))
567566rabbidv 3377 . . . . . . 7 (𝜑 → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝𝐵} = {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})
568567fveq2d 6410 . . . . . 6 (𝜑 → (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝𝐵}) = (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}))
569568breq2d 4854 . . . . 5 (𝜑 → (2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝𝐵}) ↔ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})))
570548, 569syl5ibr 237 . . . 4 (𝜑 → ({𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} = ∅ → 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝𝐵})))
571570necon3bd 2990 . . 3 (𝜑 → (¬ 2 ∥ (♯‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝𝐵}) → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} ≠ ∅))
572542, 571mpd 15 . 2 (𝜑 → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} ≠ ∅)
573 rabn0 4156 . 2 ({𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} ≠ ∅ ↔ ∃𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)
574572, 573sylib 209 1 (𝜑 → ∃𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 865  w3a 1100   = wceq 1637  wcel 2156  {cab 2790  wne 2976  wral 3094  wrex 3095  {crab 3098  Vcvv 3389  csb 3726  cdif 3764  cun 3765  cin 3766  wss 3767  c0 4114  {csn 4368  cop 4374   class class class wbr 4842  cmpt 4921   × cxp 5307  cima 5312   Fn wfn 6094  wf 6095  1-1-ontowf1o 6098  cfv 6099  (class class class)co 6872  𝑓 cof 7123  1st c1st 7394  2nd c2nd 7395  1𝑜c1o 7787  𝑚 cmap 8090  cen 8187  Fincfn 8190  cc 10217  cr 10218  0cc0 10219  1c1 10220   + caddc 10222   < clt 10357  cle 10358  cmin 10549  cn 11303  2c2 11354  0cn0 11557  cz 11641  cuz 11902  ...cfz 12547  ..^cfzo 12687  chash 13335  cdvds 15201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2782  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5094  ax-un 7177  ax-inf2 8783  ax-cnex 10275  ax-resscn 10276  ax-1cn 10277  ax-icn 10278  ax-addcl 10279  ax-addrcl 10280  ax-mulcl 10281  ax-mulrcl 10282  ax-mulcom 10283  ax-addass 10284  ax-mulass 10285  ax-distr 10286  ax-i2m1 10287  ax-1ne0 10288  ax-1rid 10289  ax-rnegex 10290  ax-rrecex 10291  ax-cnre 10292  ax-pre-lttri 10293  ax-pre-lttrn 10294  ax-pre-ltadd 10295  ax-pre-mulgt0 10296  ax-pre-sup 10297
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2791  df-cleq 2797  df-clel 2800  df-nfc 2935  df-ne 2977  df-nel 3080  df-ral 3099  df-rex 3100  df-reu 3101  df-rmo 3102  df-rab 3103  df-v 3391  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4115  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-disj 4811  df-br 4843  df-opab 4905  df-mpt 4922  df-tr 4945  df-id 5217  df-eprel 5222  df-po 5230  df-so 5231  df-fr 5268  df-se 5269  df-we 5270  df-xp 5315  df-rel 5316  df-cnv 5317  df-co 5318  df-dm 5319  df-rn 5320  df-res 5321  df-ima 5322  df-pred 5891  df-ord 5937  df-on 5938  df-lim 5939  df-suc 5940  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-isom 6108  df-riota 6833  df-ov 6875  df-oprab 6876  df-mpt2 6877  df-of 7125  df-om 7294  df-1st 7396  df-2nd 7397  df-wrecs 7640  df-recs 7702  df-rdg 7740  df-1o 7794  df-2o 7795  df-oadd 7798  df-er 7977  df-map 8092  df-pm 8093  df-en 8191  df-dom 8192  df-sdom 8193  df-fin 8194  df-sup 8585  df-oi 8652  df-card 9046  df-cda 9273  df-pnf 10359  df-mnf 10360  df-xr 10361  df-ltxr 10362  df-le 10363  df-sub 10551  df-neg 10552  df-div 10968  df-nn 11304  df-2 11362  df-3 11363  df-n0 11558  df-xnn0 11628  df-z 11642  df-uz 11903  df-rp 12045  df-fz 12548  df-fzo 12688  df-seq 13023  df-exp 13082  df-fac 13279  df-bc 13308  df-hash 13336  df-cj 14060  df-re 14061  df-im 14062  df-sqrt 14196  df-abs 14197  df-clim 14440  df-sum 14638  df-dvds 15202
This theorem is referenced by:  poimirlem32  33752
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