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Theorem poimirlem28 33408
Description: Lemma for poimir 33413, 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 11336 . . . . 5 (𝜑𝑁 ∈ ℕ0)
31nnred 11020 . . . . . 6 (𝜑𝑁 ∈ ℝ)
43leidd 10579 . . . . 5 (𝜑𝑁𝑁)
52, 2, 43jca 1240 . . . 4 (𝜑 → (𝑁 ∈ ℕ0𝑁 ∈ ℕ0𝑁𝑁))
6 oveq2 6643 . . . . . . . . . . . . . . . 16 (𝑘 = 0 → (1...𝑘) = (1...0))
7 fz10 12347 . . . . . . . . . . . . . . . 16 (1...0) = ∅
86, 7syl6eq 2670 . . . . . . . . . . . . . . 15 (𝑘 = 0 → (1...𝑘) = ∅)
98oveq2d 6651 . . . . . . . . . . . . . 14 (𝑘 = 0 → ((0..^𝐾) ↑𝑚 (1...𝑘)) = ((0..^𝐾) ↑𝑚 ∅))
10 fzofi 12756 . . . . . . . . . . . . . . . 16 (0..^𝐾) ∈ Fin
11 map0e 7880 . . . . . . . . . . . . . . . 16 ((0..^𝐾) ∈ Fin → ((0..^𝐾) ↑𝑚 ∅) = 1𝑜)
1210, 11ax-mp 5 . . . . . . . . . . . . . . 15 ((0..^𝐾) ↑𝑚 ∅) = 1𝑜
13 df1o2 7557 . . . . . . . . . . . . . . 15 1𝑜 = {∅}
1412, 13eqtri 2642 . . . . . . . . . . . . . 14 ((0..^𝐾) ↑𝑚 ∅) = {∅}
159, 14syl6eq 2670 . . . . . . . . . . . . 13 (𝑘 = 0 → ((0..^𝐾) ↑𝑚 (1...𝑘)) = {∅})
16 eqidd 2621 . . . . . . . . . . . . . . . . 17 (𝑘 = 0 → 𝑓 = 𝑓)
1716, 8, 8f1oeq123d 6120 . . . . . . . . . . . . . . . 16 (𝑘 = 0 → (𝑓:(1...𝑘)–1-1-onto→(1...𝑘) ↔ 𝑓:∅–1-1-onto→∅))
18 eqid 2620 . . . . . . . . . . . . . . . . 17 ∅ = ∅
19 f1o00 6158 . . . . . . . . . . . . . . . . 17 (𝑓:∅–1-1-onto→∅ ↔ (𝑓 = ∅ ∧ ∅ = ∅))
2018, 19mpbiran2 953 . . . . . . . . . . . . . . . 16 (𝑓:∅–1-1-onto→∅ ↔ 𝑓 = ∅)
2117, 20syl6bb 276 . . . . . . . . . . . . . . 15 (𝑘 = 0 → (𝑓:(1...𝑘)–1-1-onto→(1...𝑘) ↔ 𝑓 = ∅))
2221abbidv 2739 . . . . . . . . . . . . . 14 (𝑘 = 0 → {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)} = {𝑓𝑓 = ∅})
23 df-sn 4169 . . . . . . . . . . . . . 14 {∅} = {𝑓𝑓 = ∅}
2422, 23syl6eqr 2672 . . . . . . . . . . . . 13 (𝑘 = 0 → {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)} = {∅})
2515, 24xpeq12d 5130 . . . . . . . . . . . 12 (𝑘 = 0 → (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) = ({∅} × {∅}))
26 0ex 4781 . . . . . . . . . . . . 13 ∅ ∈ V
2726, 26xpsn 6392 . . . . . . . . . . . 12 ({∅} × {∅}) = {⟨∅, ∅⟩}
2825, 27syl6req 2671 . . . . . . . . . . 11 (𝑘 = 0 → {⟨∅, ∅⟩} = (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}))
29 elsni 4185 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ {⟨∅, ∅⟩} → 𝑠 = ⟨∅, ∅⟩)
3026, 26op1std 7163 . . . . . . . . . . . . . . . . . . 19 (𝑠 = ⟨∅, ∅⟩ → (1st𝑠) = ∅)
3129, 30syl 17 . . . . . . . . . . . . . . . . . 18 (𝑠 ∈ {⟨∅, ∅⟩} → (1st𝑠) = ∅)
3231oveq1d 6650 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ {⟨∅, ∅⟩} → ((1st𝑠) ∘𝑓 + ∅) = (∅ ∘𝑓 + ∅))
33 f0 6073 . . . . . . . . . . . . . . . . . . . 20 ∅:∅⟶∅
34 ffn 6032 . . . . . . . . . . . . . . . . . . . 20 (∅:∅⟶∅ → ∅ Fn ∅)
3533, 34mp1i 13 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ {⟨∅, ∅⟩} → ∅ Fn ∅)
3626a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ {⟨∅, ∅⟩} → ∅ ∈ V)
37 inidm 3814 . . . . . . . . . . . . . . . . . . 19 (∅ ∩ ∅) = ∅
38 0fv 6214 . . . . . . . . . . . . . . . . . . . 20 (∅‘𝑛) = ∅
3938a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝑠 ∈ {⟨∅, ∅⟩} ∧ 𝑛 ∈ ∅) → (∅‘𝑛) = ∅)
4035, 35, 36, 36, 37, 39, 39offval 6889 . . . . . . . . . . . . . . . . . 18 (𝑠 ∈ {⟨∅, ∅⟩} → (∅ ∘𝑓 + ∅) = (𝑛 ∈ ∅ ↦ (∅ + ∅)))
41 mpt0 6008 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ∅ ↦ (∅ + ∅)) = ∅
4240, 41syl6eq 2670 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ {⟨∅, ∅⟩} → (∅ ∘𝑓 + ∅) = ∅)
4332, 42eqtrd 2654 . . . . . . . . . . . . . . . 16 (𝑠 ∈ {⟨∅, ∅⟩} → ((1st𝑠) ∘𝑓 + ∅) = ∅)
4443uneq1d 3758 . . . . . . . . . . . . . . 15 (𝑠 ∈ {⟨∅, ∅⟩} → (((1st𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) = (∅ ∪ ((1...𝑁) × {0})))
45 uncom 3749 . . . . . . . . . . . . . . . 16 (∅ ∪ ((1...𝑁) × {0})) = (((1...𝑁) × {0}) ∪ ∅)
46 un0 3958 . . . . . . . . . . . . . . . 16 (((1...𝑁) × {0}) ∪ ∅) = ((1...𝑁) × {0})
4745, 46eqtri 2642 . . . . . . . . . . . . . . 15 (∅ ∪ ((1...𝑁) × {0})) = ((1...𝑁) × {0})
4844, 47syl6req 2671 . . . . . . . . . . . . . 14 (𝑠 ∈ {⟨∅, ∅⟩} → ((1...𝑁) × {0}) = (((1st𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})))
4948csbeq1d 3533 . . . . . . . . . . . . 13 (𝑠 ∈ {⟨∅, ∅⟩} → ((1...𝑁) × {0}) / 𝑝𝐵 = (((1st𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝𝐵)
5049eqeq2d 2630 . . . . . . . . . . . 12 (𝑠 ∈ {⟨∅, ∅⟩} → (0 = ((1...𝑁) × {0}) / 𝑝𝐵 ↔ 0 = (((1st𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝𝐵))
51 oveq2 6643 . . . . . . . . . . . . . . 15 (𝑘 = 0 → (0...𝑘) = (0...0))
52 0z 11373 . . . . . . . . . . . . . . . 16 0 ∈ ℤ
53 fzsn 12368 . . . . . . . . . . . . . . . 16 (0 ∈ ℤ → (0...0) = {0})
5452, 53ax-mp 5 . . . . . . . . . . . . . . 15 (0...0) = {0}
5551, 54syl6eq 2670 . . . . . . . . . . . . . 14 (𝑘 = 0 → (0...𝑘) = {0})
56 oveq2 6643 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 0 → ((𝑗 + 1)...𝑘) = ((𝑗 + 1)...0))
5756imaeq2d 5454 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 0 → ((2nd𝑠) “ ((𝑗 + 1)...𝑘)) = ((2nd𝑠) “ ((𝑗 + 1)...0)))
5857xpeq1d 5128 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 0 → (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}) = (((2nd𝑠) “ ((𝑗 + 1)...0)) × {0}))
5958uneq2d 3759 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 0 → ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0})) = ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...0)) × {0})))
6059oveq2d 6651 . . . . . . . . . . . . . . . . . . 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 6642 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 0 → (𝑘 + 1) = (0 + 1))
62 0p1e1 11117 . . . . . . . . . . . . . . . . . . . . . 22 (0 + 1) = 1
6361, 62syl6eq 2670 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 0 → (𝑘 + 1) = 1)
6463oveq1d 6650 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 0 → ((𝑘 + 1)...𝑁) = (1...𝑁))
6564xpeq1d 5128 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (((𝑘 + 1)...𝑁) × {0}) = ((1...𝑁) × {0}))
6660, 65uneq12d 3760 . . . . . . . . . . . . . . . . . 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 3533 . . . . . . . . . . . . . . . . 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 2630 . . . . . . . . . . . . . . . 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 3148 . . . . . . . . . . . . . . 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 10019 . . . . . . . . . . . . . . . 16 0 ∈ V
71 oveq2 6643 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 = 0 → (1...𝑗) = (1...0))
7271, 7syl6eq 2670 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 = 0 → (1...𝑗) = ∅)
7372imaeq2d 5454 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 = 0 → ((2nd𝑠) “ (1...𝑗)) = ((2nd𝑠) “ ∅))
74 ima0 5469 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((2nd𝑠) “ ∅) = ∅
7573, 74syl6eq 2670 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 = 0 → ((2nd𝑠) “ (1...𝑗)) = ∅)
7675xpeq1d 5128 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = 0 → (((2nd𝑠) “ (1...𝑗)) × {1}) = (∅ × {1}))
77 0xp 5189 . . . . . . . . . . . . . . . . . . . . . . 23 (∅ × {1}) = ∅
7876, 77syl6eq 2670 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 0 → (((2nd𝑠) “ (1...𝑗)) × {1}) = ∅)
79 oveq1 6642 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 = 0 → (𝑗 + 1) = (0 + 1))
8079, 62syl6eq 2670 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 = 0 → (𝑗 + 1) = 1)
8180oveq1d 6650 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 = 0 → ((𝑗 + 1)...0) = (1...0))
8281, 7syl6eq 2670 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 = 0 → ((𝑗 + 1)...0) = ∅)
8382imaeq2d 5454 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 = 0 → ((2nd𝑠) “ ((𝑗 + 1)...0)) = ((2nd𝑠) “ ∅))
8483, 74syl6eq 2670 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 = 0 → ((2nd𝑠) “ ((𝑗 + 1)...0)) = ∅)
8584xpeq1d 5128 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = 0 → (((2nd𝑠) “ ((𝑗 + 1)...0)) × {0}) = (∅ × {0}))
86 0xp 5189 . . . . . . . . . . . . . . . . . . . . . . 23 (∅ × {0}) = ∅
8785, 86syl6eq 2670 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 0 → (((2nd𝑠) “ ((𝑗 + 1)...0)) × {0}) = ∅)
8878, 87uneq12d 3760 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 0 → ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...0)) × {0})) = (∅ ∪ ∅))
89 un0 3958 . . . . . . . . . . . . . . . . . . . . 21 (∅ ∪ ∅) = ∅
9088, 89syl6eq 2670 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 0 → ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...0)) × {0})) = ∅)
9190oveq2d 6651 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 0 → ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...0)) × {0}))) = ((1st𝑠) ∘𝑓 + ∅))
9291uneq1d 3758 . . . . . . . . . . . . . . . . . 18 (𝑗 = 0 → (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪ ((1...𝑁) × {0})) = (((1st𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})))
9392csbeq1d 3533 . . . . . . . . . . . . . . . . 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 2630 . . . . . . . . . . . . . . . 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 4214 . . . . . . . . . . . . . . 15 (∃𝑗 ∈ {0}𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪ ((1...𝑁) × {0})) / 𝑝𝐵𝑖 = (((1st𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝𝐵)
9669, 95syl6bb 276 . . . . . . . . . . . . . 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 3147 . . . . . . . . . . . . 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 2624 . . . . . . . . . . . . . 14 (𝑖 = 0 → (𝑖 = (((1st𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝𝐵 ↔ 0 = (((1st𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝𝐵))
9970, 98ralsn 4213 . . . . . . . . . . . . 13 (∀𝑖 ∈ {0}𝑖 = (((1st𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝𝐵 ↔ 0 = (((1st𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝𝐵)
10097, 99syl6rbb 277 . . . . . . . . . . . 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 736 . . . . . . . . . . 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 3191 . . . . . . . . . 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 2626 . . . . . . . . 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 6182 . . . . . . . 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 4656 . . . . . . 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 308 . . . . . 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 330 . . . . 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 6643 . . . . . . . . . . . 12 (𝑘 = 𝑚 → (1...𝑘) = (1...𝑚))
109108oveq2d 6651 . . . . . . . . . . 11 (𝑘 = 𝑚 → ((0..^𝐾) ↑𝑚 (1...𝑘)) = ((0..^𝐾) ↑𝑚 (1...𝑚)))
110 eqidd 2621 . . . . . . . . . . . . 13 (𝑘 = 𝑚𝑓 = 𝑓)
111110, 108, 108f1oeq123d 6120 . . . . . . . . . . . 12 (𝑘 = 𝑚 → (𝑓:(1...𝑘)–1-1-onto→(1...𝑘) ↔ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)))
112111abbidv 2739 . . . . . . . . . . 11 (𝑘 = 𝑚 → {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)} = {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)})
113109, 112xpeq12d 5130 . . . . . . . . . 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 6643 . . . . . . . . . . 11 (𝑘 = 𝑚 → (0...𝑘) = (0...𝑚))
115 oveq2 6643 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑚 → ((𝑗 + 1)...𝑘) = ((𝑗 + 1)...𝑚))
116115imaeq2d 5454 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑚 → ((2nd𝑠) “ ((𝑗 + 1)...𝑘)) = ((2nd𝑠) “ ((𝑗 + 1)...𝑚)))
117116xpeq1d 5128 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚 → (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}) = (((2nd𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))
118117uneq2d 3759 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0})) = ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑚)) × {0})))
119118oveq2d 6651 . . . . . . . . . . . . . . 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 6642 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚 → (𝑘 + 1) = (𝑚 + 1))
121120oveq1d 6650 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → ((𝑘 + 1)...𝑁) = ((𝑚 + 1)...𝑁))
122121xpeq1d 5128 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚 → (((𝑘 + 1)...𝑁) × {0}) = (((𝑚 + 1)...𝑁) × {0}))
123119, 122uneq12d 3760 . . . . . . . . . . . . . 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 3533 . . . . . . . . . . . . 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 2630 . . . . . . . . . . . 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 3148 . . . . . . . . . . 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 3147 . . . . . . . . . 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 3190 . . . . . . . . 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 6182 . . . . . . . 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 4656 . . . . . . 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 308 . . . . . 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 330 . . . . 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 6643 . . . . . . . . . . . 12 (𝑘 = (𝑚 + 1) → (1...𝑘) = (1...(𝑚 + 1)))
134133oveq2d 6651 . . . . . . . . . . 11 (𝑘 = (𝑚 + 1) → ((0..^𝐾) ↑𝑚 (1...𝑘)) = ((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))))
135 eqidd 2621 . . . . . . . . . . . . 13 (𝑘 = (𝑚 + 1) → 𝑓 = 𝑓)
136135, 133, 133f1oeq123d 6120 . . . . . . . . . . . 12 (𝑘 = (𝑚 + 1) → (𝑓:(1...𝑘)–1-1-onto→(1...𝑘) ↔ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))))
137136abbidv 2739 . . . . . . . . . . 11 (𝑘 = (𝑚 + 1) → {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)} = {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))})
138134, 137xpeq12d 5130 . . . . . . . . . 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 6643 . . . . . . . . . . 11 (𝑘 = (𝑚 + 1) → (0...𝑘) = (0...(𝑚 + 1)))
140 oveq2 6643 . . . . . . . . . . . . . . . . . . 19 (𝑘 = (𝑚 + 1) → ((𝑗 + 1)...𝑘) = ((𝑗 + 1)...(𝑚 + 1)))
141140imaeq2d 5454 . . . . . . . . . . . . . . . . . 18 (𝑘 = (𝑚 + 1) → ((2nd𝑠) “ ((𝑗 + 1)...𝑘)) = ((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))))
142141xpeq1d 5128 . . . . . . . . . . . . . . . . 17 (𝑘 = (𝑚 + 1) → (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}) = (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))
143142uneq2d 3759 . . . . . . . . . . . . . . . 16 (𝑘 = (𝑚 + 1) → ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0})) = ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0})))
144143oveq2d 6651 . . . . . . . . . . . . . . 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 6642 . . . . . . . . . . . . . . . . 17 (𝑘 = (𝑚 + 1) → (𝑘 + 1) = ((𝑚 + 1) + 1))
146145oveq1d 6650 . . . . . . . . . . . . . . . 16 (𝑘 = (𝑚 + 1) → ((𝑘 + 1)...𝑁) = (((𝑚 + 1) + 1)...𝑁))
147146xpeq1d 5128 . . . . . . . . . . . . . . 15 (𝑘 = (𝑚 + 1) → (((𝑘 + 1)...𝑁) × {0}) = ((((𝑚 + 1) + 1)...𝑁) × {0}))
148144, 147uneq12d 3760 . . . . . . . . . . . . . 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 3533 . . . . . . . . . . . . 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 2630 . . . . . . . . . . . 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 3148 . . . . . . . . . . 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 3147 . . . . . . . . . 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 3190 . . . . . . . . 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 6182 . . . . . . . 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 4656 . . . . . . 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 308 . . . . . 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 330 . . . . 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 6643 . . . . . . . . . . . 12 (𝑘 = 𝑁 → (1...𝑘) = (1...𝑁))
159158oveq2d 6651 . . . . . . . . . . 11 (𝑘 = 𝑁 → ((0..^𝐾) ↑𝑚 (1...𝑘)) = ((0..^𝐾) ↑𝑚 (1...𝑁)))
160 eqidd 2621 . . . . . . . . . . . . 13 (𝑘 = 𝑁𝑓 = 𝑓)
161160, 158, 158f1oeq123d 6120 . . . . . . . . . . . 12 (𝑘 = 𝑁 → (𝑓:(1...𝑘)–1-1-onto→(1...𝑘) ↔ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)))
162161abbidv 2739 . . . . . . . . . . 11 (𝑘 = 𝑁 → {𝑓𝑓:(1...𝑘)–1-1-onto→(1...𝑘)} = {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
163159, 162xpeq12d 5130 . . . . . . . . . 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 6643 . . . . . . . . . . 11 (𝑘 = 𝑁 → (0...𝑘) = (0...𝑁))
165 oveq2 6643 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑁 → ((𝑗 + 1)...𝑘) = ((𝑗 + 1)...𝑁))
166165imaeq2d 5454 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑁 → ((2nd𝑠) “ ((𝑗 + 1)...𝑘)) = ((2nd𝑠) “ ((𝑗 + 1)...𝑁)))
167166xpeq1d 5128 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑁 → (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0}) = (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))
168167uneq2d 3759 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑁 → ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑘)) × {0})) = ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
169168oveq2d 6651 . . . . . . . . . . . . . . 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 6642 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑁 → (𝑘 + 1) = (𝑁 + 1))
171170oveq1d 6650 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑁 → ((𝑘 + 1)...𝑁) = ((𝑁 + 1)...𝑁))
172171xpeq1d 5128 . . . . . . . . . . . . . . 15 (𝑘 = 𝑁 → (((𝑘 + 1)...𝑁) × {0}) = (((𝑁 + 1)...𝑁) × {0}))
173169, 172uneq12d 3760 . . . . . . . . . . . . . 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 3533 . . . . . . . . . . . . 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 2630 . . . . . . . . . . . 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 3148 . . . . . . . . . . 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 3147 . . . . . . . . . 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 3190 . . . . . . . . 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 6182 . . . . . . . 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 4656 . . . . . . 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 308 . . . . . 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 330 . . . . 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 15085 . . . . . . 7 ¬ 2 ∥ 1
184 opex 4923 . . . . . . . . . 10 ⟨∅, ∅⟩ ∈ V
185 hashsng 13142 . . . . . . . . . 10 (⟨∅, ∅⟩ ∈ V → (#‘{⟨∅, ∅⟩}) = 1)
186184, 185ax-mp 5 . . . . . . . . 9 (#‘{⟨∅, ∅⟩}) = 1
187 nnuz 11708 . . . . . . . . . . . . . . . . 17 ℕ = (ℤ‘1)
1881, 187syl6eleq 2709 . . . . . . . . . . . . . . . 16 (𝜑𝑁 ∈ (ℤ‘1))
189 eluzfz1 12333 . . . . . . . . . . . . . . . 16 (𝑁 ∈ (ℤ‘1) → 1 ∈ (1...𝑁))
190188, 189syl 17 . . . . . . . . . . . . . . 15 (𝜑 → 1 ∈ (1...𝑁))
191 poimirlem28.5 . . . . . . . . . . . . . . . . 17 (𝜑𝐾 ∈ ℕ)
192191nnnn0d 11336 . . . . . . . . . . . . . . . 16 (𝜑𝐾 ∈ ℕ0)
193 0elfz 12420 . . . . . . . . . . . . . . . 16 (𝐾 ∈ ℕ0 → 0 ∈ (0...𝐾))
194 fconst6g 6081 . . . . . . . . . . . . . . . 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 6454 . . . . . . . . . . . . . . . 16 (1 ∈ (1...𝑁) → (((1...𝑁) × {0})‘1) = 0)
197190, 196syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (((1...𝑁) × {0})‘1) = 0)
198190, 195, 1973jca 1240 . . . . . . . . . . . . . 14 (𝜑 → (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) = 0))
199 nfv 1841 . . . . . . . . . . . . . . . 16 𝑝(𝜑 ∧ (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) = 0))
200 nfcsb1v 3542 . . . . . . . . . . . . . . . . 17 𝑝((1...𝑁) × {0}) / 𝑝𝐵
201200nfeq1 2775 . . . . . . . . . . . . . . . 16 𝑝((1...𝑁) × {0}) / 𝑝𝐵 = 0
202199, 201nfim 1823 . . . . . . . . . . . . . . 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 6663 . . . . . . . . . . . . . . . 16 (1...𝑁) ∈ V
204 snex 4899 . . . . . . . . . . . . . . . 16 {0} ∈ V
205203, 204xpex 6947 . . . . . . . . . . . . . . 15 ((1...𝑁) × {0}) ∈ V
206 feq1 6013 . . . . . . . . . . . . . . . . . 18 (𝑝 = ((1...𝑁) × {0}) → (𝑝:(1...𝑁)⟶(0...𝐾) ↔ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾)))
207 fveq1 6177 . . . . . . . . . . . . . . . . . . 19 (𝑝 = ((1...𝑁) × {0}) → (𝑝‘1) = (((1...𝑁) × {0})‘1))
208207eqeq1d 2622 . . . . . . . . . . . . . . . . . 18 (𝑝 = ((1...𝑁) × {0}) → ((𝑝‘1) = 0 ↔ (((1...𝑁) × {0})‘1) = 0))
209206, 2083anbi23d 1400 . . . . . . . . . . . . . . . . 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 739 . . . . . . . . . . . . . . . 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 3535 . . . . . . . . . . . . . . . . 17 (𝑝 = ((1...𝑁) × {0}) → 𝐵 = ((1...𝑁) × {0}) / 𝑝𝐵)
212211eqeq1d 2622 . . . . . . . . . . . . . . . 16 (𝑝 = ((1...𝑁) × {0}) → (𝐵 = 0 ↔ ((1...𝑁) × {0}) / 𝑝𝐵 = 0))
213210, 212imbi12d 334 . . . . . . . . . . . . . . 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 10020 . . . . . . . . . . . . . . . . 17 1 ∈ V
215 eleq1 2687 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 1 → (𝑛 ∈ (1...𝑁) ↔ 1 ∈ (1...𝑁)))
216 fveq2 6178 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 1 → (𝑝𝑛) = (𝑝‘1))
217216eqeq1d 2622 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 1 → ((𝑝𝑛) = 0 ↔ (𝑝‘1) = 0))
218215, 2173anbi13d 1399 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 1 → ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 0) ↔ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)))
219218anbi2d 739 . . . . . . . . . . . . . . . . . 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))))
220 breq2 4648 . . . . . . . . . . . . . . . . . 18 (𝑛 = 1 → (𝐵 < 𝑛𝐵 < 1))
221219, 220imbi12d 334 . . . . . . . . . . . . . . . . 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)))
222 poimirlem28.3 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 0)) → 𝐵 < 𝑛)
223214, 221, 222vtocl 3254 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)) → 𝐵 < 1)
224 poimirlem28.2 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁))
225 elfznn0 12417 . . . . . . . . . . . . . . . . . 18 (𝐵 ∈ (0...𝑁) → 𝐵 ∈ ℕ0)
226 nn0lt10b 11424 . . . . . . . . . . . . . . . . . 18 (𝐵 ∈ ℕ0 → (𝐵 < 1 ↔ 𝐵 = 0))
227224, 225, 2263syl 18 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝:(1...𝑁)⟶(0...𝐾)) → (𝐵 < 1 ↔ 𝐵 = 0))
2282273ad2antr2 1225 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)) → (𝐵 < 1 ↔ 𝐵 = 0))
229223, 228mpbid 222 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)) → 𝐵 = 0)
230202, 205, 213, 229vtoclf 3253 . . . . . . . . . . . . . 14 ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) = 0)) → ((1...𝑁) × {0}) / 𝑝𝐵 = 0)
231198, 230mpdan 701 . . . . . . . . . . . . 13 (𝜑((1...𝑁) × {0}) / 𝑝𝐵 = 0)
232231eqcomd 2626 . . . . . . . . . . . 12 (𝜑 → 0 = ((1...𝑁) × {0}) / 𝑝𝐵)
233232ralrimivw 2964 . . . . . . . . . . 11 (𝜑 → ∀𝑠 ∈ {⟨∅, ∅⟩}0 = ((1...𝑁) × {0}) / 𝑝𝐵)
234 rabid2 3113 . . . . . . . . . . 11 ({⟨∅, ∅⟩} = {𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ((1...𝑁) × {0}) / 𝑝𝐵} ↔ ∀𝑠 ∈ {⟨∅, ∅⟩}0 = ((1...𝑁) × {0}) / 𝑝𝐵)
235233, 234sylibr 224 . . . . . . . . . 10 (𝜑 → {⟨∅, ∅⟩} = {𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ((1...𝑁) × {0}) / 𝑝𝐵})
236235fveq2d 6182 . . . . . . . . 9 (𝜑 → (#‘{⟨∅, ∅⟩}) = (#‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ((1...𝑁) × {0}) / 𝑝𝐵}))
237186, 236syl5eqr 2668 . . . . . . . 8 (𝜑 → 1 = (#‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ((1...𝑁) × {0}) / 𝑝𝐵}))
238237breq2d 4656 . . . . . . 7 (𝜑 → (2 ∥ 1 ↔ 2 ∥ (#‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ((1...𝑁) × {0}) / 𝑝𝐵})))
239183, 238mtbii 316 . . . . . 6 (𝜑 → ¬ 2 ∥ (#‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ((1...𝑁) × {0}) / 𝑝𝐵}))
240239a1i 11 . . . . 5 (𝑁 ∈ ℕ0 → (𝜑 → ¬ 2 ∥ (#‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ((1...𝑁) × {0}) / 𝑝𝐵})))
241 2z 11394 . . . . . . . . . . . . 13 2 ∈ ℤ
242 fzfi 12754 . . . . . . . . . . . . . . . . 17 (1...(𝑚 + 1)) ∈ Fin
243 mapfi 8247 . . . . . . . . . . . . . . . . 17 (((0..^𝐾) ∈ Fin ∧ (1...(𝑚 + 1)) ∈ Fin) → ((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) ∈ Fin)
24410, 242, 243mp2an 707 . . . . . . . . . . . . . . . 16 ((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) ∈ Fin
245 ovex 6663 . . . . . . . . . . . . . . . . . . 19 (1...(𝑚 + 1)) ∈ V
246245, 245mapval 7854 . . . . . . . . . . . . . . . . . 18 ((1...(𝑚 + 1)) ↑𝑚 (1...(𝑚 + 1))) = {𝑓𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))}
247 mapfi 8247 . . . . . . . . . . . . . . . . . . 19 (((1...(𝑚 + 1)) ∈ Fin ∧ (1...(𝑚 + 1)) ∈ Fin) → ((1...(𝑚 + 1)) ↑𝑚 (1...(𝑚 + 1))) ∈ Fin)
248242, 242, 247mp2an 707 . . . . . . . . . . . . . . . . . 18 ((1...(𝑚 + 1)) ↑𝑚 (1...(𝑚 + 1))) ∈ Fin
249246, 248eqeltrri 2696 . . . . . . . . . . . . . . . . 17 {𝑓𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))} ∈ Fin
250 f1of 6124 . . . . . . . . . . . . . . . . . 18 (𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1)) → 𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1)))
251250ss2abi 3666 . . . . . . . . . . . . . . . . 17 {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))} ⊆ {𝑓𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))}
252 ssfi 8165 . . . . . . . . . . . . . . . . 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)
253249, 251, 252mp2an 707 . . . . . . . . . . . . . . . 16 {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))} ∈ Fin
254 xpfi 8216 . . . . . . . . . . . . . . . 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)
255244, 253, 254mp2an 707 . . . . . . . . . . . . . . 15 (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∈ Fin
256 rabfi 8170 . . . . . . . . . . . . . . 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)
257 hashcl 13130 . . . . . . . . . . . . . . 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)
258255, 256, 257mp2b 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
259258nn0zi 11387 . . . . . . . . . . . . 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
260 rabfi 8170 . . . . . . . . . . . . . . 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)
261 hashcl 13130 . . . . . . . . . . . . . . 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)
262255, 260, 261mp2b 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
263262nn0zi 11387 . . . . . . . . . . . . 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
264241, 259, 2633pm3.2i 1237 . . . . . . . . . . . 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)
265 nn0p1nn 11317 . . . . . . . . . . . . . . . 16 (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℕ)
266265ad2antrl 763 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → (𝑚 + 1) ∈ ℕ)
267 uneq1 3752 . . . . . . . . . . . . . . . 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})))
268267csbeq1d 3533 . . . . . . . . . . . . . . 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)
26970fconst 6078 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑚 + 1) + 1)...𝑁) × {0}):(((𝑚 + 1) + 1)...𝑁)⟶{0}
270269jctr 564 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑞:(1...(𝑚 + 1))⟶(0...𝐾) → (𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ ((((𝑚 + 1) + 1)...𝑁) × {0}):(((𝑚 + 1) + 1)...𝑁)⟶{0}))
271265nnred 11020 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℝ)
272271ltp1d 10939 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 ∈ ℕ0 → (𝑚 + 1) < ((𝑚 + 1) + 1))
273 fzdisj 12353 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑚 + 1) < ((𝑚 + 1) + 1) → ((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅)
274272, 273syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 ∈ ℕ0 → ((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅)
275 fun 6053 . . . . . . . . . . . . . . . . . . . . . . 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}))
276270, 274, 275syl2anr 495 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑚 ∈ ℕ0𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁))⟶((0...𝐾) ∪ {0}))
277276adantlr 750 . . . . . . . . . . . . . . . . . . . . 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}))
278277adantl 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}))
279265peano2nnd 11022 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑚 ∈ ℕ0 → ((𝑚 + 1) + 1) ∈ ℕ)
280279, 187syl6eleq 2709 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑚 ∈ ℕ0 → ((𝑚 + 1) + 1) ∈ (ℤ‘1))
281280ad2antrl 763 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → ((𝑚 + 1) + 1) ∈ (ℤ‘1))
282 nn0z 11385 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑚 ∈ ℕ0𝑚 ∈ ℤ)
2831nnzd 11466 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑𝑁 ∈ ℤ)
284 zltp1le 11412 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑚 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑚 < 𝑁 ↔ (𝑚 + 1) ≤ 𝑁))
285282, 283, 284syl2anr 495 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑚 ∈ ℕ0) → (𝑚 < 𝑁 ↔ (𝑚 + 1) ≤ 𝑁))
286285biimpa 501 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) → (𝑚 + 1) ≤ 𝑁)
287286anasss 678 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → (𝑚 + 1) ≤ 𝑁)
288282peano2zd 11470 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℤ)
289288adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑚 ∈ ℕ0𝑚 < 𝑁) → (𝑚 + 1) ∈ ℤ)
290 eluz 11686 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑚 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ‘(𝑚 + 1)) ↔ (𝑚 + 1) ≤ 𝑁))
291289, 283, 290syl2anr 495 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → (𝑁 ∈ (ℤ‘(𝑚 + 1)) ↔ (𝑚 + 1) ≤ 𝑁))
292287, 291mpbird 247 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → 𝑁 ∈ (ℤ‘(𝑚 + 1)))
293 fzsplit2 12351 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑚 + 1) + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ‘(𝑚 + 1))) → (1...𝑁) = ((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁)))
294281, 292, 293syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → (1...𝑁) = ((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁)))
295294eqcomd 2626 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → ((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁)) = (1...𝑁))
296192, 193syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → 0 ∈ (0...𝐾))
297296snssd 4331 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → {0} ⊆ (0...𝐾))
298 ssequn2 3778 . . . . . . . . . . . . . . . . . . . . . . . 24 ({0} ⊆ (0...𝐾) ↔ ((0...𝐾) ∪ {0}) = (0...𝐾))
299297, 298sylib 208 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((0...𝐾) ∪ {0}) = (0...𝐾))
300299adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → ((0...𝐾) ∪ {0}) = (0...𝐾))
301295, 300feq23d 6027 . . . . . . . . . . . . . . . . . . . . 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)))
302301adantrr 752 . . . . . . . . . . . . . . . . . . . 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)))
303278, 302mpbid 222 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ((𝑚 ∈ ℕ0𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
304 nfv 1841 . . . . . . . . . . . . . . . . . . . . 21 𝑝(𝜑 ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
305 nfcsb1v 3542 . . . . . . . . . . . . . . . . . . . . . 22 𝑝(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵
306305nfel1 2776 . . . . . . . . . . . . . . . . . . . . 21 𝑝(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ (0...𝑁)
307304, 306nfim 1823 . . . . . . . . . . . . . . . . . . . 20 𝑝((𝜑 ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ (0...𝑁))
308 vex 3198 . . . . . . . . . . . . . . . . . . . . 21 𝑞 ∈ V
309 ovex 6663 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑚 + 1) + 1)...𝑁) ∈ V
310309, 204xpex 6947 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑚 + 1) + 1)...𝑁) × {0}) ∈ V
311308, 310unex 6941 . . . . . . . . . . . . . . . . . . . 20 (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) ∈ V
312 feq1 6013 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝑝:(1...𝑁)⟶(0...𝐾) ↔ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)))
313312anbi2d 739 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝜑𝑝:(1...𝑁)⟶(0...𝐾)) ↔ (𝜑 ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))))
314 csbeq1a 3535 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → 𝐵 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵)
315314eleq1d 2684 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝐵 ∈ (0...𝑁) ↔ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ (0...𝑁)))
316313, 315imbi12d 334 . . . . . . . . . . . . . . . . . . . 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))))
317307, 311, 316, 224vtoclf 3253 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ (0...𝑁))
318303, 317syldan 487 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑚 ∈ ℕ0𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ (0...𝑁))
319318anassrs 679 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ (0...𝑁))
320 elfznn0 12417 . . . . . . . . . . . . . . . . 17 ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ (0...𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ ℕ0)
321319, 320syl 17 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ ℕ0)
322265nnnn0d 11336 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℕ0)
323322adantr 481 . . . . . . . . . . . . . . . . 17 ((𝑚 ∈ ℕ0𝑚 < 𝑁) → (𝑚 + 1) ∈ ℕ0)
324323ad2antlr 762 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑚 + 1) ∈ ℕ0)
325 leloe 10109 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑚 + 1) ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑚 + 1) ≤ 𝑁 ↔ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)))
326271, 3, 325syl2anr 495 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑚 ∈ ℕ0) → ((𝑚 + 1) ≤ 𝑁 ↔ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)))
327285, 326bitrd 268 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑚 ∈ ℕ0) → (𝑚 < 𝑁 ↔ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)))
328327biimpd 219 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚 ∈ ℕ0) → (𝑚 < 𝑁 → ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)))
329328imdistani 725 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) → ((𝜑𝑚 ∈ ℕ0) ∧ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)))
330329anasss 678 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → ((𝜑𝑚 ∈ ℕ0) ∧ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)))
331 simplll 797 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → 𝜑)
332279nnge1d 11048 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 ∈ ℕ0 → 1 ≤ ((𝑚 + 1) + 1))
333332ad2antlr 762 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 1 ≤ ((𝑚 + 1) + 1))
334 zltp1le 11412 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑚 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑚 + 1) < 𝑁 ↔ ((𝑚 + 1) + 1) ≤ 𝑁))
335288, 283, 334syl2anr 495 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑚 ∈ ℕ0) → ((𝑚 + 1) < 𝑁 ↔ ((𝑚 + 1) + 1) ≤ 𝑁))
336335biimpa 501 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ≤ 𝑁)
337288peano2zd 11470 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑚 ∈ ℕ0 → ((𝑚 + 1) + 1) ∈ ℤ)
338 1z 11392 . . . . . . . . . . . . . . . . . . . . . . . . . 26 1 ∈ ℤ
339 elfz 12317 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝑚 + 1) + 1) ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((𝑚 + 1) + 1) ∈ (1...𝑁) ↔ (1 ≤ ((𝑚 + 1) + 1) ∧ ((𝑚 + 1) + 1) ≤ 𝑁)))
340338, 339mp3an2 1410 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑚 + 1) + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((𝑚 + 1) + 1) ∈ (1...𝑁) ↔ (1 ≤ ((𝑚 + 1) + 1) ∧ ((𝑚 + 1) + 1) ≤ 𝑁)))
341337, 283, 340syl2anr 495 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑚 ∈ ℕ0) → (((𝑚 + 1) + 1) ∈ (1...𝑁) ↔ (1 ≤ ((𝑚 + 1) + 1) ∧ ((𝑚 + 1) + 1) ≤ 𝑁)))
342341adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → (((𝑚 + 1) + 1) ∈ (1...𝑁) ↔ (1 ≤ ((𝑚 + 1) + 1) ∧ ((𝑚 + 1) + 1) ≤ 𝑁)))
343333, 336, 342mpbir2and 956 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ∈ (1...𝑁))
344343adantlr 750 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ∈ (1...𝑁))
345 nn0re 11286 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑚 ∈ ℕ0𝑚 ∈ ℝ)
346345ad2antlr 762 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑚 ∈ ℝ)
347271ad2antlr 762 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → (𝑚 + 1) ∈ ℝ)
3483ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑁 ∈ ℝ)
349345ltp1d 10939 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑚 ∈ ℕ0𝑚 < (𝑚 + 1))
350349ad2antlr 762 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑚 < (𝑚 + 1))
351 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → (𝑚 + 1) < 𝑁)
352346, 347, 348, 350, 351lttrd 10183 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑚 < 𝑁)
353352adantlr 750 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → 𝑚 < 𝑁)
354 anass 680 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) ↔ (𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)))
355303anassrs 679 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
356354, 355sylanb 489 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
357356an32s 845 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ 𝑚 < 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
358353, 357syldan 487 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
359 ffn 6032 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑞:(1...(𝑚 + 1))⟶(0...𝐾) → 𝑞 Fn (1...(𝑚 + 1)))
360359ad2antlr 762 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → 𝑞 Fn (1...(𝑚 + 1)))
361274ad3antlr 766 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅)
362 eluz 11686 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑚 + 1) + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ‘((𝑚 + 1) + 1)) ↔ ((𝑚 + 1) + 1) ≤ 𝑁))
363337, 283, 362syl2anr 495 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑚 ∈ ℕ0) → (𝑁 ∈ (ℤ‘((𝑚 + 1) + 1)) ↔ ((𝑚 + 1) + 1) ≤ 𝑁))
364363adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → (𝑁 ∈ (ℤ‘((𝑚 + 1) + 1)) ↔ ((𝑚 + 1) + 1) ≤ 𝑁))
365336, 364mpbird 247 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑁 ∈ (ℤ‘((𝑚 + 1) + 1)))
366 eluzfz1 12333 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ (ℤ‘((𝑚 + 1) + 1)) → ((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁))
367365, 366syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁))
368367adantlr 750 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁))
369 fnconstg 6080 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0 ∈ V → ((((𝑚 + 1) + 1)...𝑁) × {0}) Fn (((𝑚 + 1) + 1)...𝑁))
37070, 369ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑚 + 1) + 1)...𝑁) × {0}) Fn (((𝑚 + 1) + 1)...𝑁)
371 fvun2 6257 . . . . . . . . . . . . . . . . . . . . . . . 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)))
372370, 371mp3an2 1410 . . . . . . . . . . . . . . . . . . . . . . 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)))
373360, 361, 368, 372syl12anc 1322 . . . . . . . . . . . . . . . . . . . . . 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)))
37470fvconst2 6454 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁) → (((((𝑚 + 1) + 1)...𝑁) × {0})‘((𝑚 + 1) + 1)) = 0)
375368, 374syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → (((((𝑚 + 1) + 1)...𝑁) × {0})‘((𝑚 + 1) + 1)) = 0)
376373, 375eqtrd 2654 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0)
377 nfv 1841 . . . . . . . . . . . . . . . . . . . . . . 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))
378 nfcv 2762 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑝 <
379 nfcv 2762 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑝((𝑚 + 1) + 1)
380305, 378, 379nfbr 4690 . . . . . . . . . . . . . . . . . . . . . . 23 𝑝(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 < ((𝑚 + 1) + 1)
381377, 380nfim 1823 . . . . . . . . . . . . . . . . . . . . . 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))
382 fveq1 6177 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝑝‘((𝑚 + 1) + 1)) = ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)))
383382eqeq1d 2622 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝑝‘((𝑚 + 1) + 1)) = 0 ↔ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0))
384312, 3833anbi23d 1400 . . . . . . . . . . . . . . . . . . . . . . . 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)))
385384anbi2d 739 . . . . . . . . . . . . . . . . . . . . . . 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))))
386314breq1d 4654 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝐵 < ((𝑚 + 1) + 1) ↔ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 < ((𝑚 + 1) + 1)))
387385, 386imbi12d 334 . . . . . . . . . . . . . . . . . . . . . 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))))
388 ovex 6663 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑚 + 1) + 1) ∈ V
389 eleq1 2687 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = ((𝑚 + 1) + 1) → (𝑛 ∈ (1...𝑁) ↔ ((𝑚 + 1) + 1) ∈ (1...𝑁)))
390 fveq2 6178 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛 = ((𝑚 + 1) + 1) → (𝑝𝑛) = (𝑝‘((𝑚 + 1) + 1)))
391390eqeq1d 2622 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = ((𝑚 + 1) + 1) → ((𝑝𝑛) = 0 ↔ (𝑝‘((𝑚 + 1) + 1)) = 0))
392389, 3913anbi13d 1399 . . . . . . . . . . . . . . . . . . . . . . . . 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)))
393392anbi2d 739 . . . . . . . . . . . . . . . . . . . . . . . 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))))
394 breq2 4648 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = ((𝑚 + 1) + 1) → (𝐵 < 𝑛𝐵 < ((𝑚 + 1) + 1)))
395393, 394imbi12d 334 . . . . . . . . . . . . . . . . . . . . . . 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))))
396388, 395, 222vtocl 3254 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘((𝑚 + 1) + 1)) = 0)) → 𝐵 < ((𝑚 + 1) + 1))
397381, 311, 387, 396vtoclf 3253 . . . . . . . . . . . . . . . . . . . . 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))
398331, 344, 358, 376, 397syl13anc 1326 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 < ((𝑚 + 1) + 1))
399354, 319sylanb 489 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ (0...𝑁))
400399an32s 845 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ 𝑚 < 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ (0...𝑁))
401 elfzelz 12327 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ (0...𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ ℤ)
402400, 401syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ 𝑚 < 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ ℤ)
403353, 402syldan 487 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ ℤ)
404288ad3antlr 766 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → (𝑚 + 1) ∈ ℤ)
405 zleltp1 11413 . . . . . . . . . . . . . . . . . . . . 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)))
406403, 404, 405syl2anc 692 . . . . . . . . . . . . . . . . . . . 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)))
407398, 406mpbird 247 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ≤ (𝑚 + 1))
408349ad2antlr 762 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → 𝑚 < (𝑚 + 1))
409 breq2 4648 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑚 + 1) = 𝑁 → (𝑚 < (𝑚 + 1) ↔ 𝑚 < 𝑁))
410409biimpac 503 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑚 < (𝑚 + 1) ∧ (𝑚 + 1) = 𝑁) → 𝑚 < 𝑁)
411408, 410sylan 488 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) = 𝑁) → 𝑚 < 𝑁)
412 elfzle2 12330 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ (0...𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵𝑁)
413400, 412syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ 𝑚 < 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵𝑁)
414411, 413syldan 487 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) = 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵𝑁)
415 simpr 477 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) = 𝑁) → (𝑚 + 1) = 𝑁)
416414, 415breqtrrd 4672 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) = 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ≤ (𝑚 + 1))
417407, 416jaodan 825 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ≤ (𝑚 + 1))
418417an32s 845 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑚 ∈ ℕ0) ∧ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ≤ (𝑚 + 1))
419330, 418sylan 488 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ≤ (𝑚 + 1))
420 elfz2nn0 12415 . . . . . . . . . . . . . . . 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)))
421321, 324, 419, 420syl3anbrc 1244 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∈ (0...(𝑚 + 1)))
422 fzss2 12366 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ (ℤ‘(𝑚 + 1)) → (1...(𝑚 + 1)) ⊆ (1...𝑁))
423292, 422syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → (1...(𝑚 + 1)) ⊆ (1...𝑁))
424423sselda 3595 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) ∧ 𝑛 ∈ (1...(𝑚 + 1))) → 𝑛 ∈ (1...𝑁))
4254243ad2antr1 1224 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞𝑛) = 0)) → 𝑛 ∈ (1...𝑁))
4263553ad2antr2 1225 . . . . . . . . . . . . . . . . 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))
427359ad2antll 764 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → 𝑞 Fn (1...(𝑚 + 1)))
428274ad2antlr 762 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → ((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅)
429 simprl 793 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → 𝑛 ∈ (1...(𝑚 + 1)))
430 fvun1 6256 . . . . . . . . . . . . . . . . . . . . . 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))
431370, 430mp3an2 1410 . . . . . . . . . . . . . . . . . . . . 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))
432427, 428, 429, 431syl12anc 1322 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = (𝑞𝑛))
433432adantlrr 756 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = (𝑞𝑛))
4344333adantr3 1220 . . . . . . . . . . . . . . . . . 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))
435 simpr3 1067 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞𝑛) = 0)) → (𝑞𝑛) = 0)
436434, 435eqtrd 2654 . . . . . . . . . . . . . . . . 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)
437425, 426, 4363jca 1240 . . . . . . . . . . . . . . . 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))
438 nfv 1841 . . . . . . . . . . . . . . . . . . 19 𝑝(𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0))
439 nfcv 2762 . . . . . . . . . . . . . . . . . . . 20 𝑝𝑛
440305, 378, 439nfbr 4690 . . . . . . . . . . . . . . . . . . 19 𝑝(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 < 𝑛
441438, 440nfim 1823 . . . . . . . . . . . . . . . . . 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)
442 fveq1 6177 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝑝𝑛) = ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛))
443442eqeq1d 2622 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝑝𝑛) = 0 ↔ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0))
444312, 4433anbi23d 1400 . . . . . . . . . . . . . . . . . . . 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)))
445444anbi2d 739 . . . . . . . . . . . . . . . . . . 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))))
446314breq1d 4654 . . . . . . . . . . . . . . . . . . 19 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝐵 < 𝑛(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 < 𝑛))
447445, 446imbi12d 334 . . . . . . . . . . . . . . . . . 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)))
448441, 311, 447, 222vtoclf 3253 . . . . . . . . . . . . . . . . 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)
449448adantlr 750 . . . . . . . . . . . . . . . 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)
450437, 449syldan 487 . . . . . . . . . . . . . . 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)
451 simp1 1059 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞𝑛) = 𝐾) → 𝑛 ∈ (1...(𝑚 + 1)))
452424anasss 678 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ((𝑚 ∈ ℕ0𝑚 < 𝑁) ∧ 𝑛 ∈ (1...(𝑚 + 1)))) → 𝑛 ∈ (1...𝑁))
453451, 452sylanr2 684 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑚 ∈ ℕ0𝑚 < 𝑁) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞𝑛) = 𝐾))) → 𝑛 ∈ (1...𝑁))
454 simp2 1060 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞𝑛) = 𝐾) → 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))
455454, 303sylanr2 684 . . . . . . . . . . . . . . . . . 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))
4564323adantr3 1220 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞𝑛) = 𝐾)) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = (𝑞𝑛))
457 simpr3 1067 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞𝑛) = 𝐾)) → (𝑞𝑛) = 𝐾)
458456, 457eqtrd 2654 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞𝑛) = 𝐾)) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)
459458anasss 678 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞𝑛) = 𝐾))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)
460459adantrlr 758 . . . . . . . . . . . . . . . . . 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)
461453, 455, 4603jca 1240 . . . . . . . . . . . . . . . . 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))
462 nfv 1841 . . . . . . . . . . . . . . . . . . 19 𝑝(𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾))
463 nfcv 2762 . . . . . . . . . . . . . . . . . . . 20 𝑝(𝑛 − 1)
464305, 463nfne 2891 . . . . . . . . . . . . . . . . . . 19 𝑝(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ≠ (𝑛 − 1)
465462, 464nfim 1823 . . . . . . . . . . . . . . . . . 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))
466442eqeq1d 2622 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝑝𝑛) = 𝐾 ↔ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾))
467312, 4663anbi23d 1400 . . . . . . . . . . . . . . . . . . . 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)))
468467anbi2d 739 . . . . . . . . . . . . . . . . . . 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))))
469314neeq1d 2850 . . . . . . . . . . . . . . . . . . 19 (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝐵 ≠ (𝑛 − 1) ↔ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ≠ (𝑛 − 1)))
470468, 469imbi12d 334 . . . . . . . . . . . . . . . . . 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))))
471 poimirlem28.4 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝𝑛) = 𝐾)) → 𝐵 ≠ (𝑛 − 1))
472465, 311, 470, 471vtoclf 3253 . . . . . . . . . . . . . . . . 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))
473461, 472syldan 487 . . . . . . . . . . . . . . . 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))
474473anassrs 679 . . . . . . . . . . . . . . 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))
475266, 268, 421, 450, 474poimirlem27 33407 . . . . . . . . . . . . . 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))})))
476266, 268, 421poimirlem26 33406 . . . . . . . . . . . . . 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})))
477 fzfi 12754 . . . . . . . . . . . . . . . . . . 19 (0...(𝑚 + 1)) ∈ Fin
478 xpfi 8216 . . . . . . . . . . . . . . . . . . 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)
479255, 477, 478mp2an 707 . . . . . . . . . . . . . . . . . 18 ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∈ Fin
480 rabfi 8170 . . . . . . . . . . . . . . . . . 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)
481 hashcl 13130 . . . . . . . . . . . . . . . . . 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)
482479, 480, 481mp2b 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
483482nn0zi 11387 . . . . . . . . . . . . . . . 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
484 zsubcl 11404 . . . . . . . . . . . . . . . 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)
485483, 263, 484mp2an 707 . . . . . . . . . . . . . . 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
486 zsubcl 11404 . . . . . . . . . . . . . . . 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)
487483, 259, 486mp2an 707 . . . . . . . . . . . . . . 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
488 dvds2sub 14997 . . . . . . . . . . . . . . 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})))))
489241, 485, 487, 488mp3an 1422 . . . . . . . . . . . . . 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}))))
490475, 476, 489syl2anc 692 . . . . . . . . . . . . 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}))))
491482nn0cni 11289 . . . . . . . . . . . . . 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
492262nn0cni 11289 . . . . . . . . . . . . . 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
493258nn0cni 11289 . . . . . . . . . . . . . 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
494 nnncan1 10302 . . . . . . . . . . . . . 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))})))
495491, 492, 493, 494mp3an 1422 . . . . . . . . . . . . 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))}))
496490, 495syl6breq 4685 . . . . . . . . . . . 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))})))
497 dvdssub2 15004 . . . . . . . . . . . 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))})))
498264, 496, 497sylancr 694 . . . . . . . . . . 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))})))
499 nn0cn 11287 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ ℕ0𝑚 ∈ ℂ)
500 pncan1 10439 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ ℂ → ((𝑚 + 1) − 1) = 𝑚)
501499, 500syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑚 ∈ ℕ0 → ((𝑚 + 1) − 1) = 𝑚)
502501oveq2d 6651 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ ℕ0 → (0...((𝑚 + 1) − 1)) = (0...𝑚))
503502rexeqdv 3140 . . . . . . . . . . . . . . . . . 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))
504502, 503raleqbidv 3147 . . . . . . . . . . . . . . . . 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))
5055043anbi1d 1401 . . . . . . . . . . . . . . . 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))))
506505rabbidv 3184 . . . . . . . . . . . . . . 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))})
507506fveq2d 6182 . . . . . . . . . . . . . 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))}))
508507ad2antrl 763 . . . . . . . . . . . . 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))}))
5091adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → 𝑁 ∈ ℕ)
510191adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → 𝐾 ∈ ℕ)
511 simprl 793 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → 𝑚 ∈ ℕ0)
512 simprr 795 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑚 ∈ ℕ0𝑚 < 𝑁)) → 𝑚 < 𝑁)
513509, 510, 511, 512poimirlem4 33384 . . . . . . . . . . . . . 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))})
514 fzfi 12754 . . . . . . . . . . . . . . . . . 18 (1...𝑚) ∈ Fin
515 mapfi 8247 . . . . . . . . . . . . . . . . . 18 (((0..^𝐾) ∈ Fin ∧ (1...𝑚) ∈ Fin) → ((0..^𝐾) ↑𝑚 (1...𝑚)) ∈ Fin)
51610, 514, 515mp2an 707 . . . . . . . . . . . . . . . . 17 ((0..^𝐾) ↑𝑚 (1...𝑚)) ∈ Fin
517 ovex 6663 . . . . . . . . . . . . . . . . . . . 20 (1...𝑚) ∈ V
518517, 517mapval 7854 . . . . . . . . . . . . . . . . . . 19 ((1...𝑚) ↑𝑚 (1...𝑚)) = {𝑓𝑓:(1...𝑚)⟶(1...𝑚)}
519 mapfi 8247 . . . . . . . . . . . . . . . . . . . 20 (((1...𝑚) ∈ Fin ∧ (1...𝑚) ∈ Fin) → ((1...𝑚) ↑𝑚 (1...𝑚)) ∈ Fin)
520514, 514, 519mp2an 707 . . . . . . . . . . . . . . . . . . 19 ((1...𝑚) ↑𝑚 (1...𝑚)) ∈ Fin
521518, 520eqeltrri 2696 . . . . . . . . . . . . . . . . . 18 {𝑓𝑓:(1...𝑚)⟶(1...𝑚)} ∈ Fin
522 f1of 6124 . . . . . . . . . . . . . . . . . . 19 (𝑓:(1...𝑚)–1-1-onto→(1...𝑚) → 𝑓:(1...𝑚)⟶(1...𝑚))
523522ss2abi 3666 . . . . . . . . . . . . . . . . . 18 {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)} ⊆ {𝑓𝑓:(1...𝑚)⟶(1...𝑚)}
524 ssfi 8165 . . . . . . . . . . . . . . . . . 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)
525521, 523, 524mp2an 707 . . . . . . . . . . . . . . . . 17 {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)} ∈ Fin
526 xpfi 8216 . . . . . . . . . . . . . . . . 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)
527516, 525, 526mp2an 707 . . . . . . . . . . . . . . . 16 (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∈ Fin
528 rabfi 8170 . . . . . . . . . . . . . . . 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)
529527, 528ax-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
530 rabfi 8170 . . . . . . . . . . . . . . . 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)
531255, 530ax-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
532 hashen 13118 . . . . . . . . . . . . . . 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))}))
533529, 531, 532mp2an 707 . . . . . . . . . . . . . 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))})
534513, 533sylibr 224 . . . . . . . . . . . . 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))}))
535508, 534eqtr4d 2657 . . . . . . . . . . . 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}))
536535breq2d 4656 . . . . . . . . . . 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})))
537498, 536bitrd 268 . . . . . . . . . 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})))
538537biimpd 219 . . . . . . . . 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})))
539538con3d 148 . . . . . . . 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})))
540539expcom 451 . . . . . . 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}))))
541540a2d 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}))))
5425413adant1 1077 . . . . 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}))))
543107, 132, 157, 182, 240, 542fnn0ind 11461 . . . 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})))
5445, 543mpcom 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}))
545 dvds0 14978 . . . . . . . 8 (2 ∈ ℤ → 2 ∥ 0)
546241, 545ax-mp 5 . . . . . . 7 2 ∥ 0
547 hash0 13141 . . . . . . 7 (#‘∅) = 0
548546, 547breqtrri 4671 . . . . . 6 2 ∥ (#‘∅)
549 fveq2 6178 . . . . . 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}) = (#‘∅))
550548, 549syl5breqr 4682 . . . . 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}))
5513ltp1d 10939 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 < (𝑁 + 1))
552283peano2zd 11470 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑁 + 1) ∈ ℤ)
553 fzn 12342 . . . . . . . . . . . . . . . . . . 19 (((𝑁 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅))
554552, 283, 553syl2anc 692 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅))
555551, 554mpbid 222 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑁 + 1)...𝑁) = ∅)
556555xpeq1d 5128 . . . . . . . . . . . . . . . 16 (𝜑 → (((𝑁 + 1)...𝑁) × {0}) = (∅ × {0}))
557556, 86syl6eq 2670 . . . . . . . . . . . . . . 15 (𝜑 → (((𝑁 + 1)...𝑁) × {0}) = ∅)
558557uneq2d 3759 . . . . . . . . . . . . . 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}))) ∪ ∅))
559 un0 3958 . . . . . . . . . . . . . 14 (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ ∅) = ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
560558, 559syl6eq 2670 . . . . . . . . . . . . 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}))))
561560csbeq1d 3533 . . . . . . . . . . . 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)
562 ovex 6663 . . . . . . . . . . . . 13 ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V
563 poimirlem28.1 . . . . . . . . . . . . 13 (𝑝 = ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶)
564562, 563csbie 3552 . . . . . . . . . . . 12 ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝𝐵 = 𝐶
565561, 564syl6eq 2670 . . . . . . . . . . 11 (𝜑(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝𝐵 = 𝐶)
566565eqeq2d 2630 . . . . . . . . . 10 (𝜑 → (𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝𝐵𝑖 = 𝐶))
567566rexbidv 3048 . . . . . . . . 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))
568567ralbidv 2983 . . . . . . . 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))
569568rabbidv 3184 . . . . . . 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})
570569fveq2d 6182 . . . . . 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}))
571570breq2d 4656 . . . . 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})))
572550, 571syl5ibr 236 . . . 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})))
573572necon3bd 2805 . . 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} ≠ ∅))
574544, 573mpd 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} ≠ ∅)
575 rabn0 3949 . 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)
576574, 575sylib 208 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 196  wo 383  wa 384  w3a 1036   = wceq 1481  wcel 1988  {cab 2606  wne 2791  wral 2909  wrex 2910  {crab 2913  Vcvv 3195  csb 3526  cdif 3564  cun 3565  cin 3566  wss 3567  c0 3907  {csn 4168  cop 4174   class class class wbr 4644  cmpt 4720   × cxp 5102  cima 5107   Fn wfn 5871  wf 5872  1-1-ontowf1o 5875  cfv 5876  (class class class)co 6635  𝑓 cof 6880  1st c1st 7151  2nd c2nd 7152  1𝑜c1o 7538  𝑚 cmap 7842  cen 7937  Fincfn 7940  cc 9919  cr 9920  0cc0 9921  1c1 9922   + caddc 9924   < clt 10059  cle 10060  cmin 10251  cn 11005  2c2 11055  0cn0 11277  cz 11362  cuz 11672  ...cfz 12311  ..^cfzo 12449  #chash 13100  cdvds 14964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998  ax-pre-sup 9999
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-disj 4612  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-of 6882  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-2o 7546  df-oadd 7549  df-er 7727  df-map 7844  df-pm 7845  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-sup 8333  df-oi 8400  df-card 8750  df-cda 8975  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-div 10670  df-nn 11006  df-2 11064  df-3 11065  df-n0 11278  df-xnn0 11349  df-z 11363  df-uz 11673  df-rp 11818  df-fz 12312  df-fzo 12450  df-seq 12785  df-exp 12844  df-fac 13044  df-bc 13073  df-hash 13101  df-cj 13820  df-re 13821  df-im 13822  df-sqrt 13956  df-abs 13957  df-clim 14200  df-sum 14398  df-dvds 14965
This theorem is referenced by:  poimirlem32  33412
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