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Theorem poimirlem28 36106
Description: Lemma for poimir 36111, 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 12473 . . . . 5 (𝜑𝑁 ∈ ℕ0)
31nnred 12168 . . . . . 6 (𝜑𝑁 ∈ ℝ)
43leidd 11721 . . . . 5 (𝜑𝑁𝑁)
52, 2, 43jca 1128 . . . 4 (𝜑 → (𝑁 ∈ ℕ0𝑁 ∈ ℕ0𝑁𝑁))
6 oveq2 7365 . . . . . . . . . . . . . . . 16 (𝑘 = 0 → (1...𝑘) = (1...0))
7 fz10 13462 . . . . . . . . . . . . . . . 16 (1...0) = ∅
86, 7eqtrdi 2792 . . . . . . . . . . . . . . 15 (𝑘 = 0 → (1...𝑘) = ∅)
98oveq2d 7373 . . . . . . . . . . . . . 14 (𝑘 = 0 → ((0..^𝐾) ↑m (1...𝑘)) = ((0..^𝐾) ↑m ∅))
10 fzofi 13879 . . . . . . . . . . . . . . . 16 (0..^𝐾) ∈ Fin
11 map0e 8820 . . . . . . . . . . . . . . . 16 ((0..^𝐾) ∈ Fin → ((0..^𝐾) ↑m ∅) = 1o)
1210, 11ax-mp 5 . . . . . . . . . . . . . . 15 ((0..^𝐾) ↑m ∅) = 1o
13 df1o2 8419 . . . . . . . . . . . . . . 15 1o = {∅}
1412, 13eqtri 2764 . . . . . . . . . . . . . 14 ((0..^𝐾) ↑m ∅) = {∅}
159, 14eqtrdi 2792 . . . . . . . . . . . . 13 (𝑘 = 0 → ((0..^𝐾) ↑m (1...𝑘)) = {∅})
16 eqidd 2737 . . . . . . . . . . . . . . . . 17 (𝑘 = 0 → 𝑓 = 𝑓)
1716, 8, 8f1oeq123d 6778 . . . . . . . . . . . . . . . 16 (𝑘 = 0 → (𝑓:(1...𝑘)–1-1-onto→(1...𝑘) ↔ 𝑓:∅–1-1-onto→∅))
18 eqid 2736 . . . . . . . . . . . . . . . . 17 ∅ = ∅
19 f1o00 6819 . . . . . . . . . . . . . . . . 17 (𝑓:∅–1-1-onto→∅ ↔ (𝑓 = ∅ ∧ ∅ = ∅))
2018, 19mpbiran2 708 . . . . . . . . . . . . . . . 16 (𝑓:∅–1-1-onto→∅ ↔ 𝑓 = ∅)
2117, 20bitrdi 286 . . . . . . . . . . . . . . 15 (𝑘 = 0 → (𝑓:(1...𝑘)–1-1-onto→(1...𝑘) ↔ 𝑓 = ∅))
2221abbidv 2805 . . . . . . . . . . . . . 14 (𝑘 = 0 → {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)} = {𝑓𝑓 = ∅})
23 df-sn 4587 . . . . . . . . . . . . . 14 {∅} = {𝑓𝑓 = ∅}
2422, 23eqtr4di 2794 . . . . . . . . . . . . 13 (𝑘 = 0 → {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)} = {∅})
2515, 24xpeq12d 5664 . . . . . . . . . . . 12 (𝑘 = 0 → (((0..^𝐾) ↑m (1...𝑘)) × {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) = ({∅} × {∅}))
26 0ex 5264 . . . . . . . . . . . . 13 ∅ ∈ V
2726, 26xpsn 7087 . . . . . . . . . . . 12 ({∅} × {∅}) = {⟨∅, ∅⟩}
2825, 27eqtr2di 2793 . . . . . . . . . . 11 (𝑘 = 0 → {⟨∅, ∅⟩} = (((0..^𝐾) ↑m (1...𝑘)) × {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}))
29 elsni 4603 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ {⟨∅, ∅⟩} → 𝑠 = ⟨∅, ∅⟩)
3026, 26op1std 7931 . . . . . . . . . . . . . . . . . . 19 (𝑠 = ⟨∅, ∅⟩ → (1st𝑠) = ∅)
3129, 30syl 17 . . . . . . . . . . . . . . . . . 18 (𝑠 ∈ {⟨∅, ∅⟩} → (1st𝑠) = ∅)
3231oveq1d 7372 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ {⟨∅, ∅⟩} → ((1st𝑠) ∘f + ∅) = (∅ ∘f + ∅))
33 f0 6723 . . . . . . . . . . . . . . . . . . . 20 ∅:∅⟶∅
34 ffn 6668 . . . . . . . . . . . . . . . . . . . 20 (∅:∅⟶∅ → ∅ Fn ∅)
3533, 34mp1i 13 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ {⟨∅, ∅⟩} → ∅ Fn ∅)
3626a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ {⟨∅, ∅⟩} → ∅ ∈ V)
37 inidm 4178 . . . . . . . . . . . . . . . . . . 19 (∅ ∩ ∅) = ∅
38 0fv 6886 . . . . . . . . . . . . . . . . . . . 20 (∅‘𝑛) = ∅
3938a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝑠 ∈ {⟨∅, ∅⟩} ∧ 𝑛 ∈ ∅) → (∅‘𝑛) = ∅)
4035, 35, 36, 36, 37, 39, 39offval 7626 . . . . . . . . . . . . . . . . . 18 (𝑠 ∈ {⟨∅, ∅⟩} → (∅ ∘f + ∅) = (𝑛 ∈ ∅ ↦ (∅ + ∅)))
41 mpt0 6643 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ∅ ↦ (∅ + ∅)) = ∅
4240, 41eqtrdi 2792 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ {⟨∅, ∅⟩} → (∅ ∘f + ∅) = ∅)
4332, 42eqtrd 2776 . . . . . . . . . . . . . . . 16 (𝑠 ∈ {⟨∅, ∅⟩} → ((1st𝑠) ∘f + ∅) = ∅)
4443uneq1d 4122 . . . . . . . . . . . . . . 15 (𝑠 ∈ {⟨∅, ∅⟩} → (((1st𝑠) ∘f + ∅) ∪ ((1...𝑁) × {0})) = (∅ ∪ ((1...𝑁) × {0})))
45 uncom 4113 . . . . . . . . . . . . . . . 16 (∅ ∪ ((1...𝑁) × {0})) = (((1...𝑁) × {0}) ∪ ∅)
46 un0 4350 . . . . . . . . . . . . . . . 16 (((1...𝑁) × {0}) ∪ ∅) = ((1...𝑁) × {0})
4745, 46eqtri 2764 . . . . . . . . . . . . . . 15 (∅ ∪ ((1...𝑁) × {0})) = ((1...𝑁) × {0})
4844, 47eqtr2di 2793 . . . . . . . . . . . . . 14 (𝑠 ∈ {⟨∅, ∅⟩} → ((1...𝑁) × {0}) = (((1st𝑠) ∘f + ∅) ∪ ((1...𝑁) × {0})))
4948csbeq1d 3859 . . . . . . . . . . . . 13 (𝑠 ∈ {⟨∅, ∅⟩} → ((1...𝑁) × {0}) / 𝑝𝐵 = (((1st𝑠) ∘f + ∅) ∪ ((1...𝑁) × {0})) / 𝑝𝐵)
5049eqeq2d 2747 . . . . . . . . . . . 12 (𝑠 ∈ {⟨∅, ∅⟩} → (0 = ((1...𝑁) × {0}) / 𝑝𝐵 ↔ 0 = (((1st𝑠) ∘f + ∅) ∪ ((1...𝑁) × {0})) / 𝑝𝐵))
51 oveq2 7365 . . . . . . . . . . . . . . 15 (𝑘 = 0 → (0...𝑘) = (0...0))
52 0z 12510 . . . . . . . . . . . . . . . 16 0 ∈ ℤ
53 fzsn 13483 . . . . . . . . . . . . . . . 16 (0 ∈ ℤ → (0...0) = {0})
5452, 53ax-mp 5 . . . . . . . . . . . . . . 15 (0...0) = {0}
5551, 54eqtrdi 2792 . . . . . . . . . . . . . 14 (𝑘 = 0 → (0...𝑘) = {0})
56 oveq2 7365 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 0 → ((𝑗 + 1)...𝑘) = ((𝑗 + 1)...0))
5756imaeq2d 6013 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 0 → ((2nd𝑠) “ ((𝑗 + 1)...𝑘)) = ((2nd𝑠) “ ((𝑗 + 1)...0)))
5857xpeq1d 5662 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 0 → (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}) = (((2nd𝑠) “ ((𝑗 + 1)...0)) × {0}))
5958uneq2d 4123 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 0 → ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0})) = ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...0)) × {0})))
6059oveq2d 7373 . . . . . . . . . . . . . . . . . . 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 7364 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 0 → (𝑘 + 1) = (0 + 1))
62 0p1e1 12275 . . . . . . . . . . . . . . . . . . . . . 22 (0 + 1) = 1
6361, 62eqtrdi 2792 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 0 → (𝑘 + 1) = 1)
6463oveq1d 7372 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 0 → ((𝑘 + 1)...𝑁) = (1...𝑁))
6564xpeq1d 5662 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (((𝑘 + 1)...𝑁) × {0}) = ((1...𝑁) × {0}))
6660, 65uneq12d 4124 . . . . . . . . . . . . . . . . . 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 3859 . . . . . . . . . . . . . . . . 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 2747 . . . . . . . . . . . . . . . 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 3320 . . . . . . . . . . . . . . 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 11149 . . . . . . . . . . . . . . . 16 0 ∈ V
71 oveq2 7365 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 = 0 → (1...𝑗) = (1...0))
7271, 7eqtrdi 2792 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 = 0 → (1...𝑗) = ∅)
7372imaeq2d 6013 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 = 0 → ((2nd𝑠) “ (1...𝑗)) = ((2nd𝑠) “ ∅))
74 ima0 6029 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((2nd𝑠) “ ∅) = ∅
7573, 74eqtrdi 2792 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 = 0 → ((2nd𝑠) “ (1...𝑗)) = ∅)
7675xpeq1d 5662 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = 0 → (((2nd𝑠) “ (1...𝑗)) × {1}) = (∅ × {1}))
77 0xp 5730 . . . . . . . . . . . . . . . . . . . . . . 23 (∅ × {1}) = ∅
7876, 77eqtrdi 2792 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 0 → (((2nd𝑠) “ (1...𝑗)) × {1}) = ∅)
79 oveq1 7364 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 = 0 → (𝑗 + 1) = (0 + 1))
8079, 62eqtrdi 2792 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 = 0 → (𝑗 + 1) = 1)
8180oveq1d 7372 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 = 0 → ((𝑗 + 1)...0) = (1...0))
8281, 7eqtrdi 2792 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 = 0 → ((𝑗 + 1)...0) = ∅)
8382imaeq2d 6013 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 = 0 → ((2nd𝑠) “ ((𝑗 + 1)...0)) = ((2nd𝑠) “ ∅))
8483, 74eqtrdi 2792 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 = 0 → ((2nd𝑠) “ ((𝑗 + 1)...0)) = ∅)
8584xpeq1d 5662 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = 0 → (((2nd𝑠) “ ((𝑗 + 1)...0)) × {0}) = (∅ × {0}))
86 0xp 5730 . . . . . . . . . . . . . . . . . . . . . . 23 (∅ × {0}) = ∅
8785, 86eqtrdi 2792 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 0 → (((2nd𝑠) “ ((𝑗 + 1)...0)) × {0}) = ∅)
8878, 87uneq12d 4124 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 0 → ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...0)) × {0})) = (∅ ∪ ∅))
89 un0 4350 . . . . . . . . . . . . . . . . . . . . 21 (∅ ∪ ∅) = ∅
9088, 89eqtrdi 2792 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 0 → ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...0)) × {0})) = ∅)
9190oveq2d 7373 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 0 → ((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...0)) × {0}))) = ((1st𝑠) ∘f + ∅))
9291uneq1d 4122 . . . . . . . . . . . . . . . . . 18 (𝑗 = 0 → (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪ ((1...𝑁) × {0})) = (((1st𝑠) ∘f + ∅) ∪ ((1...𝑁) × {0})))
9392csbeq1d 3859 . . . . . . . . . . . . . . . . 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 2747 . . . . . . . . . . . . . . . 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 4643 . . . . . . . . . . . . . . 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, 95bitrdi 286 . . . . . . . . . . . . . 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 3319 . . . . . . . . . . . . 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 2740 . . . . . . . . . . . . . 14 (𝑖 = 0 → (𝑖 = (((1st𝑠) ∘f + ∅) ∪ ((1...𝑁) × {0})) / 𝑝𝐵 ↔ 0 = (((1st𝑠) ∘f + ∅) ∪ ((1...𝑁) × {0})) / 𝑝𝐵))
9970, 98ralsn 4642 . . . . . . . . . . . . 13 (∀𝑖 ∈ {0}𝑖 = (((1st𝑠) ∘f + ∅) ∪ ((1...𝑁) × {0})) / 𝑝𝐵 ↔ 0 = (((1st𝑠) ∘f + ∅) ∪ ((1...𝑁) × {0})) / 𝑝𝐵)
10097, 99bitr2di 287 . . . . . . . . . . . 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 511 . . . . . . . . . . 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 3423 . . . . . . . . . 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 2742 . . . . . . . . 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 6846 . . . . . . . 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 5117 . . . . . . 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 317 . . . . . 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 340 . . . . 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 7365 . . . . . . . . . . . 12 (𝑘 = 𝑚 → (1...𝑘) = (1...𝑚))
109108oveq2d 7373 . . . . . . . . . . 11 (𝑘 = 𝑚 → ((0..^𝐾) ↑m (1...𝑘)) = ((0..^𝐾) ↑m (1...𝑚)))
110 eqidd 2737 . . . . . . . . . . . . 13 (𝑘 = 𝑚𝑓 = 𝑓)
111110, 108, 108f1oeq123d 6778 . . . . . . . . . . . 12 (𝑘 = 𝑚 → (𝑓:(1...𝑘)–1-1-onto→(1...𝑘) ↔ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)))
112111abbidv 2805 . . . . . . . . . . 11 (𝑘 = 𝑚 → {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)} = {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)})
113109, 112xpeq12d 5664 . . . . . . . . . 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 7365 . . . . . . . . . . 11 (𝑘 = 𝑚 → (0...𝑘) = (0...𝑚))
115 oveq2 7365 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑚 → ((𝑗 + 1)...𝑘) = ((𝑗 + 1)...𝑚))
116115imaeq2d 6013 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑚 → ((2nd𝑠) “ ((𝑗 + 1)...𝑘)) = ((2nd𝑠) “ ((𝑗 + 1)...𝑚)))
117116xpeq1d 5662 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚 → (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}) = (((2nd𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))
118117uneq2d 4123 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0})) = ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑚)) × {0})))
119118oveq2d 7373 . . . . . . . . . . . . . . 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 7364 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚 → (𝑘 + 1) = (𝑚 + 1))
121120oveq1d 7372 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → ((𝑘 + 1)...𝑁) = ((𝑚 + 1)...𝑁))
122121xpeq1d 5662 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚 → (((𝑘 + 1)...𝑁) × {0}) = (((𝑚 + 1)...𝑁) × {0}))
123119, 122uneq12d 4124 . . . . . . . . . . . . . 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 3859 . . . . . . . . . . . . 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 2747 . . . . . . . . . . . 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 3320 . . . . . . . . . . 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 3319 . . . . . . . . . 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 3424 . . . . . . . . 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 6846 . . . . . . . 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 5117 . . . . . . 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 317 . . . . . 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 340 . . . . 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 7365 . . . . . . . . . . . 12 (𝑘 = (𝑚 + 1) → (1...𝑘) = (1...(𝑚 + 1)))
134133oveq2d 7373 . . . . . . . . . . 11 (𝑘 = (𝑚 + 1) → ((0..^𝐾) ↑m (1...𝑘)) = ((0..^𝐾) ↑m (1...(𝑚 + 1))))
135 eqidd 2737 . . . . . . . . . . . . 13 (𝑘 = (𝑚 + 1) → 𝑓 = 𝑓)
136135, 133, 133f1oeq123d 6778 . . . . . . . . . . . 12 (𝑘 = (𝑚 + 1) → (𝑓:(1...𝑘)–1-1-onto→(1...𝑘) ↔ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))))
137136abbidv 2805 . . . . . . . . . . 11 (𝑘 = (𝑚 + 1) → {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)} = {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))})
138134, 137xpeq12d 5664 . . . . . . . . . 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 7365 . . . . . . . . . . 11 (𝑘 = (𝑚 + 1) → (0...𝑘) = (0...(𝑚 + 1)))
140 oveq2 7365 . . . . . . . . . . . . . . . . . . 19 (𝑘 = (𝑚 + 1) → ((𝑗 + 1)...𝑘) = ((𝑗 + 1)...(𝑚 + 1)))
141140imaeq2d 6013 . . . . . . . . . . . . . . . . . 18 (𝑘 = (𝑚 + 1) → ((2nd𝑠) “ ((𝑗 + 1)...𝑘)) = ((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))))
142141xpeq1d 5662 . . . . . . . . . . . . . . . . 17 (𝑘 = (𝑚 + 1) → (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}) = (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))
143142uneq2d 4123 . . . . . . . . . . . . . . . 16 (𝑘 = (𝑚 + 1) → ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0})) = ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0})))
144143oveq2d 7373 . . . . . . . . . . . . . . 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 7364 . . . . . . . . . . . . . . . . 17 (𝑘 = (𝑚 + 1) → (𝑘 + 1) = ((𝑚 + 1) + 1))
146145oveq1d 7372 . . . . . . . . . . . . . . . 16 (𝑘 = (𝑚 + 1) → ((𝑘 + 1)...𝑁) = (((𝑚 + 1) + 1)...𝑁))
147146xpeq1d 5662 . . . . . . . . . . . . . . 15 (𝑘 = (𝑚 + 1) → (((𝑘 + 1)...𝑁) × {0}) = ((((𝑚 + 1) + 1)...𝑁) × {0}))
148144, 147uneq12d 4124 . . . . . . . . . . . . . 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 3859 . . . . . . . . . . . . 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 2747 . . . . . . . . . . . 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 3320 . . . . . . . . . . 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 3319 . . . . . . . . . 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 3424 . . . . . . . . 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 6846 . . . . . . . 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 5117 . . . . . . 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 317 . . . . . 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 340 . . . . 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 7365 . . . . . . . . . . . 12 (𝑘 = 𝑁 → (1...𝑘) = (1...𝑁))
159158oveq2d 7373 . . . . . . . . . . 11 (𝑘 = 𝑁 → ((0..^𝐾) ↑m (1...𝑘)) = ((0..^𝐾) ↑m (1...𝑁)))
160 eqidd 2737 . . . . . . . . . . . . 13 (𝑘 = 𝑁𝑓 = 𝑓)
161160, 158, 158f1oeq123d 6778 . . . . . . . . . . . 12 (𝑘 = 𝑁 → (𝑓:(1...𝑘)–1-1-onto→(1...𝑘) ↔ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)))
162161abbidv 2805 . . . . . . . . . . 11 (𝑘 = 𝑁 → {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)} = {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
163159, 162xpeq12d 5664 . . . . . . . . . 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 7365 . . . . . . . . . . 11 (𝑘 = 𝑁 → (0...𝑘) = (0...𝑁))
165 oveq2 7365 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑁 → ((𝑗 + 1)...𝑘) = ((𝑗 + 1)...𝑁))
166165imaeq2d 6013 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑁 → ((2nd𝑠) “ ((𝑗 + 1)...𝑘)) = ((2nd𝑠) “ ((𝑗 + 1)...𝑁)))
167166xpeq1d 5662 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑁 → (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}) = (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))
168167uneq2d 4123 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑁 → ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0})) = ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
169168oveq2d 7373 . . . . . . . . . . . . . . 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 7364 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑁 → (𝑘 + 1) = (𝑁 + 1))
171170oveq1d 7372 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑁 → ((𝑘 + 1)...𝑁) = ((𝑁 + 1)...𝑁))
172171xpeq1d 5662 . . . . . . . . . . . . . . 15 (𝑘 = 𝑁 → (((𝑘 + 1)...𝑁) × {0}) = (((𝑁 + 1)...𝑁) × {0}))
173169, 172uneq12d 4124 . . . . . . . . . . . . . 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 3859 . . . . . . . . . . . . 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 2747 . . . . . . . . . . . 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 3320 . . . . . . . . . . 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 3319 . . . . . . . . . 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 3424 . . . . . . . . 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 6846 . . . . . . . 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 5117 . . . . . . 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 317 . . . . . 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 340 . . . . 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 16250 . . . . . . 7 ¬ 2 ∥ 1
184 opex 5421 . . . . . . . . . 10 ⟨∅, ∅⟩ ∈ V
185 hashsng 14269 . . . . . . . . . 10 (⟨∅, ∅⟩ ∈ V → (♯‘{⟨∅, ∅⟩}) = 1)
186184, 185ax-mp 5 . . . . . . . . 9 (♯‘{⟨∅, ∅⟩}) = 1
187 nnuz 12806 . . . . . . . . . . . . . . . . 17 ℕ = (ℤ‘1)
1881, 187eleqtrdi 2848 . . . . . . . . . . . . . . . 16 (𝜑𝑁 ∈ (ℤ‘1))
189 eluzfz1 13448 . . . . . . . . . . . . . . . 16 (𝑁 ∈ (ℤ‘1) → 1 ∈ (1...𝑁))
190188, 189syl 17 . . . . . . . . . . . . . . 15 (𝜑 → 1 ∈ (1...𝑁))
191 poimirlem28.5 . . . . . . . . . . . . . . . . 17 (𝜑𝐾 ∈ ℕ)
192191nnnn0d 12473 . . . . . . . . . . . . . . . 16 (𝜑𝐾 ∈ ℕ0)
193 0elfz 13538 . . . . . . . . . . . . . . . 16 (𝐾 ∈ ℕ0 → 0 ∈ (0...𝐾))
194 fconst6g 6731 . . . . . . . . . . . . . . . 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 7153 . . . . . . . . . . . . . . . 16 (1 ∈ (1...𝑁) → (((1...𝑁) × {0})‘1) = 0)
197190, 196syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (((1...𝑁) × {0})‘1) = 0)
198190, 195, 1973jca 1128 . . . . . . . . . . . . . 14 (𝜑 → (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) = 0))
199 nfv 1917 . . . . . . . . . . . . . . . 16 𝑝(𝜑 ∧ (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) = 0))
200 nfcsb1v 3880 . . . . . . . . . . . . . . . . 17 𝑝((1...𝑁) × {0}) / 𝑝𝐵
201200nfeq1 2922 . . . . . . . . . . . . . . . 16 𝑝((1...𝑁) × {0}) / 𝑝𝐵 = 0
202199, 201nfim 1899 . . . . . . . . . . . . . . 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 7390 . . . . . . . . . . . . . . . 16 (1...𝑁) ∈ V
204 snex 5388 . . . . . . . . . . . . . . . 16 {0} ∈ V
205203, 204xpex 7687 . . . . . . . . . . . . . . 15 ((1...𝑁) × {0}) ∈ V
206 feq1 6649 . . . . . . . . . . . . . . . . . 18 (𝑝 = ((1...𝑁) × {0}) → (𝑝:(1...𝑁)⟶(0...𝐾) ↔ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾)))
207 fveq1 6841 . . . . . . . . . . . . . . . . . . 19 (𝑝 = ((1...𝑁) × {0}) → (𝑝‘1) = (((1...𝑁) × {0})‘1))
208207eqeq1d 2738 . . . . . . . . . . . . . . . . . 18 (𝑝 = ((1...𝑁) × {0}) → ((𝑝‘1) = 0 ↔ (((1...𝑁) × {0})‘1) = 0))
209206, 2083anbi23d 1439 . . . . . . . . . . . . . . . . 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 629 . . . . . . . . . . . . . . . 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 3869 . . . . . . . . . . . . . . . . 17 (𝑝 = ((1...𝑁) × {0}) → 𝐵 = ((1...𝑁) × {0}) / 𝑝𝐵)
212211eqeq1d 2738 . . . . . . . . . . . . . . . 16 (𝑝 = ((1...𝑁) × {0}) → (𝐵 = 0 ↔ ((1...𝑁) × {0}) / 𝑝𝐵 = 0))
213210, 212imbi12d 344 . . . . . . . . . . . . . . 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 11151 . . . . . . . . . . . . . . . . 17 1 ∈ V
215 eleq1 2825 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 1 → (𝑛 ∈ (1...𝑁) ↔ 1 ∈ (1...𝑁)))
216 fveqeq2 6851 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 1 → ((𝑝𝑛) = 0 ↔ (𝑝‘1) = 0))
217215, 2163anbi13d 1438 . . . . . . . . . . . . . . . . . . 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 629 . . . . . . . . . . . . . . . . . 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 5109 . . . . . . . . . . . . . . . . . 18 (𝑛 = 1 → (𝐵 < 𝑛𝐵 < 1))
220218, 219imbi12d 344 . . . . . . . . . . . . . . . . 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 3518 . . . . . . . . . . . . . . . 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 13534 . . . . . . . . . . . . . . . . . 18 (𝐵 ∈ (0...𝑁) → 𝐵 ∈ ℕ0)
225 nn0lt10b 12565 . . . . . . . . . . . . . . . . . 18 (𝐵 ∈ ℕ0 → (𝐵 < 1 ↔ 𝐵 = 0))
226223, 224, 2253syl 18 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝:(1...𝑁)⟶(0...𝐾)) → (𝐵 < 1 ↔ 𝐵 = 0))
2272263ad2antr2 1189 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)) → (𝐵 < 1 ↔ 𝐵 = 0))
228222, 227mpbid 231 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)) → 𝐵 = 0)
229202, 205, 213, 228vtoclf 3516 . . . . . . . . . . . . . 14 ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) = 0)) → ((1...𝑁) × {0}) / 𝑝𝐵 = 0)
230198, 229mpdan 685 . . . . . . . . . . . . 13 (𝜑((1...𝑁) × {0}) / 𝑝𝐵 = 0)
231230eqcomd 2742 . . . . . . . . . . . 12 (𝜑 → 0 = ((1...𝑁) × {0}) / 𝑝𝐵)
232231ralrimivw 3147 . . . . . . . . . . 11 (𝜑 → ∀𝑠 ∈ {⟨∅, ∅⟩}0 = ((1...𝑁) × {0}) / 𝑝𝐵)
233 rabid2 3436 . . . . . . . . . . 11 ({⟨∅, ∅⟩} = {𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ((1...𝑁) × {0}) / 𝑝𝐵} ↔ ∀𝑠 ∈ {⟨∅, ∅⟩}0 = ((1...𝑁) × {0}) / 𝑝𝐵)
234232, 233sylibr 233 . . . . . . . . . 10 (𝜑 → {⟨∅, ∅⟩} = {𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ((1...𝑁) × {0}) / 𝑝𝐵})
235234fveq2d 6846 . . . . . . . . 9 (𝜑 → (♯‘{⟨∅, ∅⟩}) = (♯‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ((1...𝑁) × {0}) / 𝑝𝐵}))
236186, 235eqtr3id 2790 . . . . . . . 8 (𝜑 → 1 = (♯‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ((1...𝑁) × {0}) / 𝑝𝐵}))
237236breq2d 5117 . . . . . . 7 (𝜑 → (2 ∥ 1 ↔ 2 ∥ (♯‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ((1...𝑁) × {0}) / 𝑝𝐵})))
238183, 237mtbii 325 . . . . . 6 (𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ((1...𝑁) × {0}) / 𝑝𝐵}))
239238a1i 11 . . . . 5 (𝑁 ∈ ℕ0 → (𝜑 → ¬ 2 ∥ (♯‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ((1...𝑁) × {0}) / 𝑝𝐵})))
240 2z 12535 . . . . . . . . . . . . 13 2 ∈ ℤ
241 fzfi 13877 . . . . . . . . . . . . . . . . 17 (1...(𝑚 + 1)) ∈ Fin
242 mapfi 9292 . . . . . . . . . . . . . . . . 17 (((0..^𝐾) ∈ Fin ∧ (1...(𝑚 + 1)) ∈ Fin) → ((0..^𝐾) ↑m (1...(𝑚 + 1))) ∈ Fin)
24310, 241, 242mp2an 690 . . . . . . . . . . . . . . . 16 ((0..^𝐾) ↑m (1...(𝑚 + 1))) ∈ Fin
244 ovex 7390 . . . . . . . . . . . . . . . . . . 19 (1...(𝑚 + 1)) ∈ V
245244, 244mapval 8777 . . . . . . . . . . . . . . . . . 18 ((1...(𝑚 + 1)) ↑m (1...(𝑚 + 1))) = {𝑓𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))}
246 mapfi 9292 . . . . . . . . . . . . . . . . . . 19 (((1...(𝑚 + 1)) ∈ Fin ∧ (1...(𝑚 + 1)) ∈ Fin) → ((1...(𝑚 + 1)) ↑m (1...(𝑚 + 1))) ∈ Fin)
247241, 241, 246mp2an 690 . . . . . . . . . . . . . . . . . 18 ((1...(𝑚 + 1)) ↑m (1...(𝑚 + 1))) ∈ Fin
248245, 247eqeltrri 2835 . . . . . . . . . . . . . . . . 17 {𝑓𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))} ∈ Fin
249 f1of 6784 . . . . . . . . . . . . . . . . . 18 (𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1)) → 𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1)))
250249ss2abi 4023 . . . . . . . . . . . . . . . . 17 {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))} ⊆ {𝑓𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))}
251 ssfi 9117 . . . . . . . . . . . . . . . . 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 690 . . . . . . . . . . . . . . . 16 {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))} ∈ Fin
253 xpfi 9261 . . . . . . . . . . . . . . . 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 690 . . . . . . . . . . . . . . 15 (((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∈ Fin
255 rabfi 9213 . . . . . . . . . . . . . . 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 14256 . . . . . . . . . . . . . . 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 12528 . . . . . . . . . . . . 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 9213 . . . . . . . . . . . . . . 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 14256 . . . . . . . . . . . . . . 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 12528 . . . . . . . . . . . . 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 1339 . . . . . . . . . . . 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 12452 . . . . . . . . . . . . . . . 16 (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℕ)
265264ad2antrl 726 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → (𝑚 + 1) ∈ ℕ)
266 uneq1 4116 . . . . . . . . . . . . . . . 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 3859 . . . . . . . . . . . . . . 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 6728 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑚 + 1) + 1)...𝑁) × {0}):(((𝑚 + 1) + 1)...𝑁)⟶{0}
269268jctr 525 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑞:(1...(𝑚 + 1))⟶(0...𝐾) → (𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ ((((𝑚 + 1) + 1)...𝑁) × {0}):(((𝑚 + 1) + 1)...𝑁)⟶{0}))
270264nnred 12168 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℝ)
271270ltp1d 12085 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 ∈ ℕ0 → (𝑚 + 1) < ((𝑚 + 1) + 1))
272 fzdisj 13468 . . . . . . . . . . . . . . . . . . . . . . . 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 6704 . . . . . . . . . . . . . . . . . . . . . . 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 597 . . . . . . . . . . . . . . . . . . . . . 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 713 . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . 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 12170 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑚 ∈ ℕ0 → ((𝑚 + 1) + 1) ∈ ℕ)
279278, 187eleqtrdi 2848 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑚 ∈ ℕ0 → ((𝑚 + 1) + 1) ∈ (ℤ‘1))
280279ad2antrl 726 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → ((𝑚 + 1) + 1) ∈ (ℤ‘1))
281 nn0z 12524 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑚 ∈ ℕ0𝑚 ∈ ℤ)
2821nnzd 12526 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑𝑁 ∈ ℤ)
283 zltp1le 12553 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑚 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑚 < 𝑁 ↔ (𝑚 + 1) ≤ 𝑁))
284281, 282, 283syl2anr 597 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑚 ∈ ℕ0) → (𝑚 < 𝑁 ↔ (𝑚 + 1) ≤ 𝑁))
285284biimpa 477 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) → (𝑚 + 1) ≤ 𝑁)
286285anasss 467 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → (𝑚 + 1) ≤ 𝑁)
287281peano2zd 12610 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℤ)
288287adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑚 ∈ ℕ0𝑚 < 𝑁) → (𝑚 + 1) ∈ ℤ)
289 eluz 12777 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑚 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ‘(𝑚 + 1)) ↔ (𝑚 + 1) ≤ 𝑁))
290288, 282, 289syl2anr 597 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → (𝑁 ∈ (ℤ‘(𝑚 + 1)) ↔ (𝑚 + 1) ≤ 𝑁))
291286, 290mpbird 256 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → 𝑁 ∈ (ℤ‘(𝑚 + 1)))
292 fzsplit2 13466 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑚 + 1) + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ‘(𝑚 + 1))) → (1...𝑁) = ((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁)))
293280, 291, 292syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → (1...𝑁) = ((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁)))
294293eqcomd 2742 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → ((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁)) = (1...𝑁))
295192, 193syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → 0 ∈ (0...𝐾))
296295snssd 4769 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → {0} ⊆ (0...𝐾))
297 ssequn2 4143 . . . . . . . . . . . . . . . . . . . . . . . 24 ({0} ⊆ (0...𝐾) ↔ ((0...𝐾) ∪ {0}) = (0...𝐾))
298296, 297sylib 217 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((0...𝐾) ∪ {0}) = (0...𝐾))
299298adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → ((0...𝐾) ∪ {0}) = (0...𝐾))
300294, 299feq23d 6663 . . . . . . . . . . . . . . . . . . . . 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 715 . . . . . . . . . . . . . . . . . . . 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 231 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ((𝑚 ∈ ℕ0𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
303 nfv 1917 . . . . . . . . . . . . . . . . . . . . 21 𝑝(𝜑 ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
304 nfcsb1v 3880 . . . . . . . . . . . . . . . . . . . . . 22 𝑝(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵
305304nfel1 2923 . . . . . . . . . . . . . . . . . . . . 21 𝑝(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ (0...𝑁)
306303, 305nfim 1899 . . . . . . . . . . . . . . . . . . . 20 𝑝((𝜑 ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ (0...𝑁))
307 vex 3449 . . . . . . . . . . . . . . . . . . . . 21 𝑞 ∈ V
308 ovex 7390 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑚 + 1) + 1)...𝑁) ∈ V
309308, 204xpex 7687 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑚 + 1) + 1)...𝑁) × {0}) ∈ V
310307, 309unex 7680 . . . . . . . . . . . . . . . . . . . 20 (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) ∈ V
311 feq1 6649 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝑝:(1...𝑁)⟶(0...𝐾) ↔ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)))
312311anbi2d 629 . . . . . . . . . . . . . . . . . . . . 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 3869 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → 𝐵 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵)
314313eleq1d 2822 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝐵 ∈ (0...𝑁) ↔ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ (0...𝑁)))
315312, 314imbi12d 344 . . . . . . . . . . . . . . . . . . . 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 3516 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ (0...𝑁))
317302, 316syldan 591 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑚 ∈ ℕ0𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ (0...𝑁))
318317anassrs 468 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ (0...𝑁))
319 elfznn0 13534 . . . . . . . . . . . . . . . . 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 12473 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℕ0)
322321adantr 481 . . . . . . . . . . . . . . . . 17 ((𝑚 ∈ ℕ0𝑚 < 𝑁) → (𝑚 + 1) ∈ ℕ0)
323322ad2antlr 725 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑚 + 1) ∈ ℕ0)
324 leloe 11241 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑚 + 1) ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑚 + 1) ≤ 𝑁 ↔ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)))
325270, 3, 324syl2anr 597 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑚 ∈ ℕ0) → ((𝑚 + 1) ≤ 𝑁 ↔ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)))
326284, 325bitrd 278 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑚 ∈ ℕ0) → (𝑚 < 𝑁 ↔ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)))
327326biimpd 228 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ ℕ0) → (𝑚 < 𝑁 → ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)))
328327imdistani 569 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) → ((𝜑𝑚 ∈ ℕ0) ∧ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)))
329328anasss 467 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → ((𝜑𝑚 ∈ ℕ0) ∧ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)))
330 simplll 773 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → 𝜑)
331278nnge1d 12201 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 ∈ ℕ0 → 1 ≤ ((𝑚 + 1) + 1))
332331ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 1 ≤ ((𝑚 + 1) + 1))
333 zltp1le 12553 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑚 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑚 + 1) < 𝑁 ↔ ((𝑚 + 1) + 1) ≤ 𝑁))
334287, 282, 333syl2anr 597 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑚 ∈ ℕ0) → ((𝑚 + 1) < 𝑁 ↔ ((𝑚 + 1) + 1) ≤ 𝑁))
335334biimpa 477 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ≤ 𝑁)
336287peano2zd 12610 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑚 ∈ ℕ0 → ((𝑚 + 1) + 1) ∈ ℤ)
337 1z 12533 . . . . . . . . . . . . . . . . . . . . . . . . . 26 1 ∈ ℤ
338 elfz 13430 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝑚 + 1) + 1) ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((𝑚 + 1) + 1) ∈ (1...𝑁) ↔ (1 ≤ ((𝑚 + 1) + 1) ∧ ((𝑚 + 1) + 1) ≤ 𝑁)))
339337, 338mp3an2 1449 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑚 + 1) + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((𝑚 + 1) + 1) ∈ (1...𝑁) ↔ (1 ≤ ((𝑚 + 1) + 1) ∧ ((𝑚 + 1) + 1) ≤ 𝑁)))
340336, 282, 339syl2anr 597 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑚 ∈ ℕ0) → (((𝑚 + 1) + 1) ∈ (1...𝑁) ↔ (1 ≤ ((𝑚 + 1) + 1) ∧ ((𝑚 + 1) + 1) ≤ 𝑁)))
341340adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → (((𝑚 + 1) + 1) ∈ (1...𝑁) ↔ (1 ≤ ((𝑚 + 1) + 1) ∧ ((𝑚 + 1) + 1) ≤ 𝑁)))
342332, 335, 341mpbir2and 711 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ∈ (1...𝑁))
343342adantlr 713 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ∈ (1...𝑁))
344 nn0re 12422 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑚 ∈ ℕ0𝑚 ∈ ℝ)
345344ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑚 ∈ ℝ)
346270ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → (𝑚 + 1) ∈ ℝ)
3473ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑁 ∈ ℝ)
348344ltp1d 12085 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑚 ∈ ℕ0𝑚 < (𝑚 + 1))
349348ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑚 < (𝑚 + 1))
350 simpr 485 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → (𝑚 + 1) < 𝑁)
351345, 346, 347, 349, 350lttrd 11316 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑚 < 𝑁)
352351adantlr 713 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → 𝑚 < 𝑁)
353 anass 469 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) ↔ (𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)))
354302anassrs 468 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
355353, 354sylanb 581 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
356355an32s 650 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ 𝑚 < 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
357352, 356syldan 591 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
358 ffn 6668 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑞:(1...(𝑚 + 1))⟶(0...𝐾) → 𝑞 Fn (1...(𝑚 + 1)))
359358ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → 𝑞 Fn (1...(𝑚 + 1)))
360273ad3antlr 729 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅)
361 eluz 12777 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑚 + 1) + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ‘((𝑚 + 1) + 1)) ↔ ((𝑚 + 1) + 1) ≤ 𝑁))
362336, 282, 361syl2anr 597 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑚 ∈ ℕ0) → (𝑁 ∈ (ℤ‘((𝑚 + 1) + 1)) ↔ ((𝑚 + 1) + 1) ≤ 𝑁))
363362adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → (𝑁 ∈ (ℤ‘((𝑚 + 1) + 1)) ↔ ((𝑚 + 1) + 1) ≤ 𝑁))
364335, 363mpbird 256 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑁 ∈ (ℤ‘((𝑚 + 1) + 1)))
365 eluzfz1 13448 . . . . . . . . . . . . . . . . . . . . . . . . 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 713 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁))
368 fnconstg 6730 . . . . . . . . . . . . . . . . . . . . . . . . 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 6933 . . . . . . . . . . . . . . . . . . . . . . . 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 1449 . . . . . . . . . . . . . . . . . . . . . . 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 835 . . . . . . . . . . . . . . . . . . . . . 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 7153 . . . . . . . . . . . . . . . . . . . . . . 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 2776 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0)
376 nfv 1917 . . . . . . . . . . . . . . . . . . . . . . 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 2907 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑝 <
378 nfcv 2907 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑝((𝑚 + 1) + 1)
379304, 377, 378nfbr 5152 . . . . . . . . . . . . . . . . . . . . . . 23 𝑝(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 < ((𝑚 + 1) + 1)
380376, 379nfim 1899 . . . . . . . . . . . . . . . . . . . . . 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 6841 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝑝‘((𝑚 + 1) + 1)) = ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)))
382381eqeq1d 2738 . . . . . . . . . . . . . . . . . . . . . . . . 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 1439 . . . . . . . . . . . . . . . . . . . . . . . 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 629 . . . . . . . . . . . . . . . . . . . . . . 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 5115 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝐵 < ((𝑚 + 1) + 1) ↔ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 < ((𝑚 + 1) + 1)))
386384, 385imbi12d 344 . . . . . . . . . . . . . . . . . . . . . 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 7390 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑚 + 1) + 1) ∈ V
388 eleq1 2825 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = ((𝑚 + 1) + 1) → (𝑛 ∈ (1...𝑁) ↔ ((𝑚 + 1) + 1) ∈ (1...𝑁)))
389 fveqeq2 6851 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = ((𝑚 + 1) + 1) → ((𝑝𝑛) = 0 ↔ (𝑝‘((𝑚 + 1) + 1)) = 0))
390388, 3893anbi13d 1438 . . . . . . . . . . . . . . . . . . . . . . . . 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 629 . . . . . . . . . . . . . . . . . . . . . . . 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 5109 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = ((𝑚 + 1) + 1) → (𝐵 < 𝑛𝐵 < ((𝑚 + 1) + 1)))
393391, 392imbi12d 344 . . . . . . . . . . . . . . . . . . . . . . 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 3518 . . . . . . . . . . . . . . . . . . . . . 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 3516 . . . . . . . . . . . . . . . . . . . . 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 1372 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 < ((𝑚 + 1) + 1))
397353, 318sylanb 581 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ (0...𝑁))
398397an32s 650 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ 𝑚 < 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ (0...𝑁))
399398elfzelzd 13442 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ 𝑚 < 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ ℤ)
400352, 399syldan 591 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ ℤ)
401287ad3antlr 729 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → (𝑚 + 1) ∈ ℤ)
402 zleltp1 12554 . . . . . . . . . . . . . . . . . . . . 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)))
403400, 401, 402syl2anc 584 . . . . . . . . . . . . . . . . . . . 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)))
404396, 403mpbird 256 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ≤ (𝑚 + 1))
405348ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → 𝑚 < (𝑚 + 1))
406 breq2 5109 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑚 + 1) = 𝑁 → (𝑚 < (𝑚 + 1) ↔ 𝑚 < 𝑁))
407406biimpac 479 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑚 < (𝑚 + 1) ∧ (𝑚 + 1) = 𝑁) → 𝑚 < 𝑁)
408405, 407sylan 580 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) = 𝑁) → 𝑚 < 𝑁)
409 elfzle2 13445 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ (0...𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵𝑁)
410398, 409syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ 𝑚 < 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵𝑁)
411408, 410syldan 591 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) = 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵𝑁)
412 simpr 485 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) = 𝑁) → (𝑚 + 1) = 𝑁)
413411, 412breqtrrd 5133 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) = 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ≤ (𝑚 + 1))
414404, 413jaodan 956 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ≤ (𝑚 + 1))
415414an32s 650 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑚 ∈ ℕ0) ∧ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ≤ (𝑚 + 1))
416329, 415sylan 580 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ≤ (𝑚 + 1))
417 elfz2nn0 13532 . . . . . . . . . . . . . . . 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)))
418320, 323, 416, 417syl3anbrc 1343 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ (0...(𝑚 + 1)))
419 fzss2 13481 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ (ℤ‘(𝑚 + 1)) → (1...(𝑚 + 1)) ⊆ (1...𝑁))
420291, 419syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → (1...(𝑚 + 1)) ⊆ (1...𝑁))
421420sselda 3944 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) ∧ 𝑛 ∈ (1...(𝑚 + 1))) → 𝑛 ∈ (1...𝑁))
4224213ad2antr1 1188 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞𝑛) = 0)) → 𝑛 ∈ (1...𝑁))
4233543ad2antr2 1189 . . . . . . . . . . . . . . . . 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))
424358ad2antll 727 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → 𝑞 Fn (1...(𝑚 + 1)))
425273ad2antlr 725 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → ((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅)
426 simprl 769 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → 𝑛 ∈ (1...(𝑚 + 1)))
427 fvun1 6932 . . . . . . . . . . . . . . . . . . . . . 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))
428369, 427mp3an2 1449 . . . . . . . . . . . . . . . . . . . . 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))
429424, 425, 426, 428syl12anc 835 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = (𝑞𝑛))
430429adantlrr 719 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = (𝑞𝑛))
4314303adantr3 1171 . . . . . . . . . . . . . . . . . 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))
432 simpr3 1196 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞𝑛) = 0)) → (𝑞𝑛) = 0)
433431, 432eqtrd 2776 . . . . . . . . . . . . . . . . 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)
434422, 423, 4333jca 1128 . . . . . . . . . . . . . . . 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))
435 nfv 1917 . . . . . . . . . . . . . . . . . . 19 𝑝(𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0))
436 nfcv 2907 . . . . . . . . . . . . . . . . . . . 20 𝑝𝑛
437304, 377, 436nfbr 5152 . . . . . . . . . . . . . . . . . . 19 𝑝(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 < 𝑛
438435, 437nfim 1899 . . . . . . . . . . . . . . . . . 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)
439 fveq1 6841 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝑝𝑛) = ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛))
440439eqeq1d 2738 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝑝𝑛) = 0 ↔ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0))
441311, 4403anbi23d 1439 . . . . . . . . . . . . . . . . . . . 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)))
442441anbi2d 629 . . . . . . . . . . . . . . . . . . 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))))
443313breq1d 5115 . . . . . . . . . . . . . . . . . . 19 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝐵 < 𝑛(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 < 𝑛))
444442, 443imbi12d 344 . . . . . . . . . . . . . . . . . 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)))
445438, 310, 444, 221vtoclf 3516 . . . . . . . . . . . . . . . . 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)
446445adantlr 713 . . . . . . . . . . . . . . . 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)
447434, 446syldan 591 . . . . . . . . . . . . . . 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)
448 simp1 1136 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞𝑛) = 𝐾) → 𝑛 ∈ (1...(𝑚 + 1)))
449421anasss 467 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ((𝑚 ∈ ℕ0𝑚 < 𝑁) ∧ 𝑛 ∈ (1...(𝑚 + 1)))) → 𝑛 ∈ (1...𝑁))
450448, 449sylanr2 681 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑚 ∈ ℕ0𝑚 < 𝑁) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞𝑛) = 𝐾))) → 𝑛 ∈ (1...𝑁))
451 simp2 1137 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞𝑛) = 𝐾) → 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))
452451, 302sylanr2 681 . . . . . . . . . . . . . . . . . 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))
4534293adantr3 1171 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞𝑛) = 𝐾)) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = (𝑞𝑛))
454 simpr3 1196 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞𝑛) = 𝐾)) → (𝑞𝑛) = 𝐾)
455453, 454eqtrd 2776 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞𝑛) = 𝐾)) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)
456455anasss 467 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞𝑛) = 𝐾))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)
457456adantrlr 721 . . . . . . . . . . . . . . . . . 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)
458450, 452, 4573jca 1128 . . . . . . . . . . . . . . . . 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))
459 nfv 1917 . . . . . . . . . . . . . . . . . . 19 𝑝(𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾))
460 nfcv 2907 . . . . . . . . . . . . . . . . . . . 20 𝑝(𝑛 − 1)
461304, 460nfne 3045 . . . . . . . . . . . . . . . . . . 19 𝑝(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ≠ (𝑛 − 1)
462459, 461nfim 1899 . . . . . . . . . . . . . . . . . 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))
463439eqeq1d 2738 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝑝𝑛) = 𝐾 ↔ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾))
464311, 4633anbi23d 1439 . . . . . . . . . . . . . . . . . . . 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)))
465464anbi2d 629 . . . . . . . . . . . . . . . . . . 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))))
466313neeq1d 3003 . . . . . . . . . . . . . . . . . . 19 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝐵 ≠ (𝑛 − 1) ↔ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ≠ (𝑛 − 1)))
467465, 466imbi12d 344 . . . . . . . . . . . . . . . . . 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))))
468 poimirlem28.4 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 𝐾)) → 𝐵 ≠ (𝑛 − 1))
469462, 310, 467, 468vtoclf 3516 . . . . . . . . . . . . . . . . 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))
470458, 469syldan 591 . . . . . . . . . . . . . . . 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))
471470anassrs 468 . . . . . . . . . . . . . . 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))
472265, 267, 418, 447, 471poimirlem27 36105 . . . . . . . . . . . . . 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))})))
473265, 267, 418poimirlem26 36104 . . . . . . . . . . . . . 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})))
474 fzfi 13877 . . . . . . . . . . . . . . . . . . 19 (0...(𝑚 + 1)) ∈ Fin
475 xpfi 9261 . . . . . . . . . . . . . . . . . . 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)
476254, 474, 475mp2an 690 . . . . . . . . . . . . . . . . . 18 ((((0..^𝐾) ↑m (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∈ Fin
477 rabfi 9213 . . . . . . . . . . . . . . . . . 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)
478 hashcl 14256 . . . . . . . . . . . . . . . . . 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)
479476, 477, 478mp2b 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
480479nn0zi 12528 . . . . . . . . . . . . . . . 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
481 zsubcl 12545 . . . . . . . . . . . . . . . 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)
482480, 262, 481mp2an 690 . . . . . . . . . . . . . . 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
483 zsubcl 12545 . . . . . . . . . . . . . . . 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)
484480, 258, 483mp2an 690 . . . . . . . . . . . . . . 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
485 dvds2sub 16173 . . . . . . . . . . . . . . 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})))))
486240, 482, 484, 485mp3an 1461 . . . . . . . . . . . . . 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}))))
487472, 473, 486syl2anc 584 . . . . . . . . . . . . 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}))))
488479nn0cni 12425 . . . . . . . . . . . . . 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
489261nn0cni 12425 . . . . . . . . . . . . . 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
490257nn0cni 12425 . . . . . . . . . . . . . 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
491 nnncan1 11437 . . . . . . . . . . . . . 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))})))
492488, 489, 490, 491mp3an 1461 . . . . . . . . . . . . 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))}))
493487, 492breqtrdi 5146 . . . . . . . . . . . 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))})))
494 dvdssub2 16183 . . . . . . . . . . . 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))})))
495263, 493, 494sylancr 587 . . . . . . . . . . 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))})))
496 nn0cn 12423 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ ℕ0𝑚 ∈ ℂ)
497 pncan1 11579 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ ℂ → ((𝑚 + 1) − 1) = 𝑚)
498496, 497syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑚 ∈ ℕ0 → ((𝑚 + 1) − 1) = 𝑚)
499498oveq2d 7373 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ ℕ0 → (0...((𝑚 + 1) − 1)) = (0...𝑚))
500499rexeqdv 3314 . . . . . . . . . . . . . . . . . 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))
501499, 500raleqbidv 3319 . . . . . . . . . . . . . . . . 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))
5025013anbi1d 1440 . . . . . . . . . . . . . . . 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))))
503502rabbidv 3415 . . . . . . . . . . . . . . 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))})
504503fveq2d 6846 . . . . . . . . . . . . . 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))}))
505504ad2antrl 726 . . . . . . . . . . . . 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))}))
5061adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → 𝑁 ∈ ℕ)
507191adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → 𝐾 ∈ ℕ)
508 simprl 769 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → 𝑚 ∈ ℕ0)
509 simprr 771 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → 𝑚 < 𝑁)
510506, 507, 508, 509poimirlem4 36082 . . . . . . . . . . . . . 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))})
511 fzfi 13877 . . . . . . . . . . . . . . . . . 18 (1...𝑚) ∈ Fin
512 mapfi 9292 . . . . . . . . . . . . . . . . . 18 (((0..^𝐾) ∈ Fin ∧ (1...𝑚) ∈ Fin) → ((0..^𝐾) ↑m (1...𝑚)) ∈ Fin)
51310, 511, 512mp2an 690 . . . . . . . . . . . . . . . . 17 ((0..^𝐾) ↑m (1...𝑚)) ∈ Fin
514 ovex 7390 . . . . . . . . . . . . . . . . . . . 20 (1...𝑚) ∈ V
515514, 514mapval 8777 . . . . . . . . . . . . . . . . . . 19 ((1...𝑚) ↑m (1...𝑚)) = {𝑓𝑓:(1...𝑚)⟶(1...𝑚)}
516 mapfi 9292 . . . . . . . . . . . . . . . . . . . 20 (((1...𝑚) ∈ Fin ∧ (1...𝑚) ∈ Fin) → ((1...𝑚) ↑m (1...𝑚)) ∈ Fin)
517511, 511, 516mp2an 690 . . . . . . . . . . . . . . . . . . 19 ((1...𝑚) ↑m (1...𝑚)) ∈ Fin
518515, 517eqeltrri 2835 . . . . . . . . . . . . . . . . . 18 {𝑓𝑓:(1...𝑚)⟶(1...𝑚)} ∈ Fin
519 f1of 6784 . . . . . . . . . . . . . . . . . . 19 (𝑓:(1...𝑚)–1-1-onto→(1...𝑚) → 𝑓:(1...𝑚)⟶(1...𝑚))
520519ss2abi 4023 . . . . . . . . . . . . . . . . . 18 {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)} ⊆ {𝑓𝑓:(1...𝑚)⟶(1...𝑚)}
521 ssfi 9117 . . . . . . . . . . . . . . . . . 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)
522518, 520, 521mp2an 690 . . . . . . . . . . . . . . . . 17 {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)} ∈ Fin
523 xpfi 9261 . . . . . . . . . . . . . . . . 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)
524513, 522, 523mp2an 690 . . . . . . . . . . . . . . . 16 (((0..^𝐾) ↑m (1...𝑚)) × {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∈ Fin
525 rabfi 9213 . . . . . . . . . . . . . . . 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)
526524, 525ax-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
527 rabfi 9213 . . . . . . . . . . . . . . . 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)
528254, 527ax-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
529 hashen 14247 . . . . . . . . . . . . . . 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))}))
530526, 528, 529mp2an 690 . . . . . . . . . . . . . 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))})
531510, 530sylibr 233 . . . . . . . . . . . . 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))}))
532505, 531eqtr4d 2779 . . . . . . . . . . . 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}))
533532breq2d 5117 . . . . . . . . . . 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})))
534495, 533bitrd 278 . . . . . . . . . 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})))
535534biimpd 228 . . . . . . . . 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})))
536535con3d 152 . . . . . . . 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})))
537536expcom 414 . . . . . . 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}))))
538537a2d 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}))))
5395383adant1 1130 . . . . 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}))))
540107, 132, 157, 182, 239, 539fnn0ind 12602 . . . 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})))
5415, 540mpcom 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}))
542 dvds0 16154 . . . . . . . 8 (2 ∈ ℤ → 2 ∥ 0)
543240, 542ax-mp 5 . . . . . . 7 2 ∥ 0
544 hash0 14267 . . . . . . 7 (♯‘∅) = 0
545543, 544breqtrri 5132 . . . . . 6 2 ∥ (♯‘∅)
546 fveq2 6842 . . . . . 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}) = (♯‘∅))
547545, 546breqtrrid 5143 . . . . 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}))
5483ltp1d 12085 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 < (𝑁 + 1))
549282peano2zd 12610 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑁 + 1) ∈ ℤ)
550 fzn 13457 . . . . . . . . . . . . . . . . . . 19 (((𝑁 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅))
551549, 282, 550syl2anc 584 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅))
552548, 551mpbid 231 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑁 + 1)...𝑁) = ∅)
553552xpeq1d 5662 . . . . . . . . . . . . . . . 16 (𝜑 → (((𝑁 + 1)...𝑁) × {0}) = (∅ × {0}))
554553, 86eqtrdi 2792 . . . . . . . . . . . . . . 15 (𝜑 → (((𝑁 + 1)...𝑁) × {0}) = ∅)
555554uneq2d 4123 . . . . . . . . . . . . . 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}))) ∪ ∅))
556 un0 4350 . . . . . . . . . . . . . 14 (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ ∅) = ((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
557555, 556eqtrdi 2792 . . . . . . . . . . . . 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}))))
558557csbeq1d 3859 . . . . . . . . . . . 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)
559 ovex 7390 . . . . . . . . . . . . 13 ((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V
560 poimirlem28.1 . . . . . . . . . . . . 13 (𝑝 = ((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶)
561559, 560csbie 3891 . . . . . . . . . . . 12 ((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝𝐵 = 𝐶
562558, 561eqtrdi 2792 . . . . . . . . . . 11 (𝜑(((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝𝐵 = 𝐶)
563562eqeq2d 2747 . . . . . . . . . 10 (𝜑 → (𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝𝐵𝑖 = 𝐶))
564563rexbidv 3175 . . . . . . . . 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))
565564ralbidv 3174 . . . . . . . 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))
566565rabbidv 3415 . . . . . . 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})
567566fveq2d 6846 . . . . . 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}))
568567breq2d 5117 . . . . 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})))
569547, 568syl5ibr 245 . . . 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})))
570569necon3bd 2957 . . 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} ≠ ∅))
571541, 570mpd 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} ≠ ∅)
572 rabn0 4345 . 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)
573571, 572sylib 217 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 205  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  {cab 2713  wne 2943  wral 3064  wrex 3073  {crab 3407  Vcvv 3445  csb 3855  cdif 3907  cun 3908  cin 3909  wss 3910  c0 4282  {csn 4586  cop 4592   class class class wbr 5105  cmpt 5188   × cxp 5631  cima 5636   Fn wfn 6491  wf 6492  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  f cof 7615  1st c1st 7919  2nd c2nd 7920  1oc1o 8405  m cmap 8765  cen 8880  Fincfn 8883  cc 11049  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   < clt 11189  cle 11190  cmin 11385  cn 12153  2c2 12208  0cn0 12413  cz 12499  cuz 12763  ...cfz 13424  ..^cfzo 13567  chash 14230  cdvds 16136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-disj 5071  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-oi 9446  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-seq 13907  df-exp 13968  df-fac 14174  df-bc 14203  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-sum 15571  df-dvds 16137
This theorem is referenced by:  poimirlem32  36110
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