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Theorem prmreclem2 15615
Description: Lemma for prmrec 15620. There are at most 2↑𝐾 squarefree numbers which divide no primes larger than 𝐾. (We could strengthen this to 2↑#(ℙ ∩ (1...𝐾)) but there's no reason to.) We establish the inequality by showing that the prime counts of the number up to 𝐾 completely determine it because all higher prime counts are zero, and they are all at most 1 because no square divides the number, so there are at most 2↑𝐾 possibilities. (Contributed by Mario Carneiro, 5-Aug-2014.)
Hypotheses
Ref Expression
prmrec.1 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (1 / 𝑛), 0))
prmrec.2 (𝜑𝐾 ∈ ℕ)
prmrec.3 (𝜑𝑁 ∈ ℕ)
prmrec.4 𝑀 = {𝑛 ∈ (1...𝑁) ∣ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑛}
prmreclem2.5 𝑄 = (𝑛 ∈ ℕ ↦ sup({𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑛}, ℝ, < ))
Assertion
Ref Expression
prmreclem2 (𝜑 → (#‘{𝑥𝑀 ∣ (𝑄𝑥) = 1}) ≤ (2↑𝐾))
Distinct variable groups:   𝑛,𝑝,𝑟,𝑥,𝐹   𝑛,𝐾,𝑝,𝑥   𝑛,𝑀,𝑝,𝑥   𝜑,𝑛,𝑝,𝑥   𝑄,𝑛,𝑝,𝑟,𝑥   𝑛,𝑁,𝑝,𝑥
Allowed substitution hints:   𝜑(𝑟)   𝐾(𝑟)   𝑀(𝑟)   𝑁(𝑟)

Proof of Theorem prmreclem2
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 6675 . . . 4 ({0, 1} ↑𝑚 (1...𝐾)) ∈ V
2 fveq2 6189 . . . . . . . 8 (𝑥 = 𝑦 → (𝑄𝑥) = (𝑄𝑦))
32eqeq1d 2623 . . . . . . 7 (𝑥 = 𝑦 → ((𝑄𝑥) = 1 ↔ (𝑄𝑦) = 1))
43elrab 3361 . . . . . 6 (𝑦 ∈ {𝑥𝑀 ∣ (𝑄𝑥) = 1} ↔ (𝑦𝑀 ∧ (𝑄𝑦) = 1))
5 prmrec.4 . . . . . . . . . . . . . . . . . . . 20 𝑀 = {𝑛 ∈ (1...𝑁) ∣ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑛}
6 ssrab2 3685 . . . . . . . . . . . . . . . . . . . 20 {𝑛 ∈ (1...𝑁) ∣ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑛} ⊆ (1...𝑁)
75, 6eqsstri 3633 . . . . . . . . . . . . . . . . . . 19 𝑀 ⊆ (1...𝑁)
8 simprl 794 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) → 𝑦𝑀)
98ad2antrr 762 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑦𝑀)
107, 9sseldi 3599 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑦 ∈ (1...𝑁))
11 elfznn 12367 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (1...𝑁) → 𝑦 ∈ ℕ)
1210, 11syl 17 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑦 ∈ ℕ)
13 simpr 477 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑛 ∈ ℙ)
14 prmuz2 15402 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℙ → 𝑛 ∈ (ℤ‘2))
1513, 14syl 17 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑛 ∈ (ℤ‘2))
16 prmreclem2.5 . . . . . . . . . . . . . . . . . . 19 𝑄 = (𝑛 ∈ ℕ ↦ sup({𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑛}, ℝ, < ))
1716prmreclem1 15614 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ℕ → ((𝑄𝑦) ∈ ℕ ∧ ((𝑄𝑦)↑2) ∥ 𝑦 ∧ (𝑛 ∈ (ℤ‘2) → ¬ (𝑛↑2) ∥ (𝑦 / ((𝑄𝑦)↑2)))))
1817simp3d 1074 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ℕ → (𝑛 ∈ (ℤ‘2) → ¬ (𝑛↑2) ∥ (𝑦 / ((𝑄𝑦)↑2))))
1912, 15, 18sylc 65 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ¬ (𝑛↑2) ∥ (𝑦 / ((𝑄𝑦)↑2)))
20 simprr 796 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) → (𝑄𝑦) = 1)
2120ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑄𝑦) = 1)
2221oveq1d 6662 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑄𝑦)↑2) = (1↑2))
23 sq1 12953 . . . . . . . . . . . . . . . . . . . . 21 (1↑2) = 1
2422, 23syl6eq 2671 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑄𝑦)↑2) = 1)
2524oveq2d 6663 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑦 / ((𝑄𝑦)↑2)) = (𝑦 / 1))
2612nncnd 11033 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑦 ∈ ℂ)
2726div1d 10790 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑦 / 1) = 𝑦)
2825, 27eqtrd 2655 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑦 / ((𝑄𝑦)↑2)) = 𝑦)
2928breq2d 4663 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑛↑2) ∥ (𝑦 / ((𝑄𝑦)↑2)) ↔ (𝑛↑2) ∥ 𝑦))
3012nnzd 11478 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑦 ∈ ℤ)
31 2nn0 11306 . . . . . . . . . . . . . . . . . . 19 2 ∈ ℕ0
3231a1i 11 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 2 ∈ ℕ0)
33 pcdvdsb 15567 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ ℙ ∧ 𝑦 ∈ ℤ ∧ 2 ∈ ℕ0) → (2 ≤ (𝑛 pCnt 𝑦) ↔ (𝑛↑2) ∥ 𝑦))
3413, 30, 32, 33syl3anc 1325 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (2 ≤ (𝑛 pCnt 𝑦) ↔ (𝑛↑2) ∥ 𝑦))
3529, 34bitr4d 271 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑛↑2) ∥ (𝑦 / ((𝑄𝑦)↑2)) ↔ 2 ≤ (𝑛 pCnt 𝑦)))
3619, 35mtbid 314 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ¬ 2 ≤ (𝑛 pCnt 𝑦))
3713, 12pccld 15549 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈ ℕ0)
3837nn0red 11349 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈ ℝ)
39 2re 11087 . . . . . . . . . . . . . . . 16 2 ∈ ℝ
40 ltnle 10114 . . . . . . . . . . . . . . . 16 (((𝑛 pCnt 𝑦) ∈ ℝ ∧ 2 ∈ ℝ) → ((𝑛 pCnt 𝑦) < 2 ↔ ¬ 2 ≤ (𝑛 pCnt 𝑦)))
4138, 39, 40sylancl 694 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑛 pCnt 𝑦) < 2 ↔ ¬ 2 ≤ (𝑛 pCnt 𝑦)))
4236, 41mpbird 247 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) < 2)
43 df-2 11076 . . . . . . . . . . . . . 14 2 = (1 + 1)
4442, 43syl6breq 4692 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) < (1 + 1))
4537nn0zd 11477 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈ ℤ)
46 1z 11404 . . . . . . . . . . . . . 14 1 ∈ ℤ
47 zleltp1 11425 . . . . . . . . . . . . . 14 (((𝑛 pCnt 𝑦) ∈ ℤ ∧ 1 ∈ ℤ) → ((𝑛 pCnt 𝑦) ≤ 1 ↔ (𝑛 pCnt 𝑦) < (1 + 1)))
4845, 46, 47sylancl 694 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑛 pCnt 𝑦) ≤ 1 ↔ (𝑛 pCnt 𝑦) < (1 + 1)))
4944, 48mpbird 247 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ≤ 1)
50 nn0uz 11719 . . . . . . . . . . . . . 14 0 = (ℤ‘0)
5137, 50syl6eleq 2710 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈ (ℤ‘0))
52 elfz5 12331 . . . . . . . . . . . . 13 (((𝑛 pCnt 𝑦) ∈ (ℤ‘0) ∧ 1 ∈ ℤ) → ((𝑛 pCnt 𝑦) ∈ (0...1) ↔ (𝑛 pCnt 𝑦) ≤ 1))
5351, 46, 52sylancl 694 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑛 pCnt 𝑦) ∈ (0...1) ↔ (𝑛 pCnt 𝑦) ≤ 1))
5449, 53mpbird 247 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈ (0...1))
55 0z 11385 . . . . . . . . . . . . 13 0 ∈ ℤ
56 fzpr 12393 . . . . . . . . . . . . 13 (0 ∈ ℤ → (0...(0 + 1)) = {0, (0 + 1)})
5755, 56ax-mp 5 . . . . . . . . . . . 12 (0...(0 + 1)) = {0, (0 + 1)}
58 1e0p1 11549 . . . . . . . . . . . . 13 1 = (0 + 1)
5958oveq2i 6658 . . . . . . . . . . . 12 (0...1) = (0...(0 + 1))
6058preq2i 4270 . . . . . . . . . . . 12 {0, 1} = {0, (0 + 1)}
6157, 59, 603eqtr4i 2653 . . . . . . . . . . 11 (0...1) = {0, 1}
6254, 61syl6eleq 2710 . . . . . . . . . 10 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈ {0, 1})
63 c0ex 10031 . . . . . . . . . . . 12 0 ∈ V
6463prid1 4295 . . . . . . . . . . 11 0 ∈ {0, 1}
6564a1i 11 . . . . . . . . . 10 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ ¬ 𝑛 ∈ ℙ) → 0 ∈ {0, 1})
6662, 65ifclda 4118 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) → if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0) ∈ {0, 1})
67 eqid 2621 . . . . . . . . 9 (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) = (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0))
6866, 67fmptd 6383 . . . . . . . 8 ((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) → (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)):(1...𝐾)⟶{0, 1})
69 prex 4907 . . . . . . . . 9 {0, 1} ∈ V
70 ovex 6675 . . . . . . . . 9 (1...𝐾) ∈ V
7169, 70elmap 7883 . . . . . . . 8 ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) ∈ ({0, 1} ↑𝑚 (1...𝐾)) ↔ (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)):(1...𝐾)⟶{0, 1})
7268, 71sylibr 224 . . . . . . 7 ((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) → (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) ∈ ({0, 1} ↑𝑚 (1...𝐾)))
7372ex 450 . . . . . 6 (𝜑 → ((𝑦𝑀 ∧ (𝑄𝑦) = 1) → (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) ∈ ({0, 1} ↑𝑚 (1...𝐾))))
744, 73syl5bi 232 . . . . 5 (𝜑 → (𝑦 ∈ {𝑥𝑀 ∣ (𝑄𝑥) = 1} → (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) ∈ ({0, 1} ↑𝑚 (1...𝐾))))
75 fveq2 6189 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑄𝑥) = (𝑄𝑧))
7675eqeq1d 2623 . . . . . . . 8 (𝑥 = 𝑧 → ((𝑄𝑥) = 1 ↔ (𝑄𝑧) = 1))
7776elrab 3361 . . . . . . 7 (𝑧 ∈ {𝑥𝑀 ∣ (𝑄𝑥) = 1} ↔ (𝑧𝑀 ∧ (𝑄𝑧) = 1))
784, 77anbi12i 733 . . . . . 6 ((𝑦 ∈ {𝑥𝑀 ∣ (𝑄𝑥) = 1} ∧ 𝑧 ∈ {𝑥𝑀 ∣ (𝑄𝑥) = 1}) ↔ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1)))
79 ovex 6675 . . . . . . . . . . . 12 (𝑛 pCnt 𝑦) ∈ V
8079, 63ifex 4154 . . . . . . . . . . 11 if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0) ∈ V
8180, 67fnmpti 6020 . . . . . . . . . 10 (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) Fn (1...𝐾)
82 ovex 6675 . . . . . . . . . . . 12 (𝑛 pCnt 𝑧) ∈ V
8382, 63ifex 4154 . . . . . . . . . . 11 if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0) ∈ V
84 eqid 2621 . . . . . . . . . . 11 (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0)) = (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0))
8583, 84fnmpti 6020 . . . . . . . . . 10 (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0)) Fn (1...𝐾)
86 eqfnfv 6309 . . . . . . . . . 10 (((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) Fn (1...𝐾) ∧ (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0)) Fn (1...𝐾)) → ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) = (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0)) ↔ ∀𝑝 ∈ (1...𝐾)((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0))‘𝑝) = ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0))‘𝑝)))
8781, 85, 86mp2an 708 . . . . . . . . 9 ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) = (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0)) ↔ ∀𝑝 ∈ (1...𝐾)((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0))‘𝑝) = ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0))‘𝑝))
88 eleq1 2688 . . . . . . . . . . . . 13 (𝑛 = 𝑝 → (𝑛 ∈ ℙ ↔ 𝑝 ∈ ℙ))
89 oveq1 6654 . . . . . . . . . . . . 13 (𝑛 = 𝑝 → (𝑛 pCnt 𝑦) = (𝑝 pCnt 𝑦))
9088, 89ifbieq1d 4107 . . . . . . . . . . . 12 (𝑛 = 𝑝 → if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0))
91 ovex 6675 . . . . . . . . . . . . 13 (𝑝 pCnt 𝑦) ∈ V
9291, 63ifex 4154 . . . . . . . . . . . 12 if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) ∈ V
9390, 67, 92fvmpt 6280 . . . . . . . . . . 11 (𝑝 ∈ (1...𝐾) → ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0))‘𝑝) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0))
94 oveq1 6654 . . . . . . . . . . . . 13 (𝑛 = 𝑝 → (𝑛 pCnt 𝑧) = (𝑝 pCnt 𝑧))
9588, 94ifbieq1d 4107 . . . . . . . . . . . 12 (𝑛 = 𝑝 → if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0))
96 ovex 6675 . . . . . . . . . . . . 13 (𝑝 pCnt 𝑧) ∈ V
9796, 63ifex 4154 . . . . . . . . . . . 12 if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ∈ V
9895, 84, 97fvmpt 6280 . . . . . . . . . . 11 (𝑝 ∈ (1...𝐾) → ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0))‘𝑝) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0))
9993, 98eqeq12d 2636 . . . . . . . . . 10 (𝑝 ∈ (1...𝐾) → (((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0))‘𝑝) = ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0))‘𝑝) ↔ if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0)))
10099ralbiia 2978 . . . . . . . . 9 (∀𝑝 ∈ (1...𝐾)((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0))‘𝑝) = ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0))‘𝑝) ↔ ∀𝑝 ∈ (1...𝐾)if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0))
10187, 100bitri 264 . . . . . . . 8 ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) = (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0)) ↔ ∀𝑝 ∈ (1...𝐾)if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0))
102 simprll 802 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) → 𝑦𝑀)
103 breq2 4655 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑦 → (𝑝𝑛𝑝𝑦))
104103notbid 308 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑦 → (¬ 𝑝𝑛 ↔ ¬ 𝑝𝑦))
105104ralbidv 2985 . . . . . . . . . . . . . . 15 (𝑛 = 𝑦 → (∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑛 ↔ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑦))
106105, 5elrab2 3364 . . . . . . . . . . . . . 14 (𝑦𝑀 ↔ (𝑦 ∈ (1...𝑁) ∧ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑦))
107106simprbi 480 . . . . . . . . . . . . 13 (𝑦𝑀 → ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑦)
108102, 107syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) → ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑦)
109 simprrl 804 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) → 𝑧𝑀)
110 breq2 4655 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑧 → (𝑝𝑛𝑝𝑧))
111110notbid 308 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑧 → (¬ 𝑝𝑛 ↔ ¬ 𝑝𝑧))
112111ralbidv 2985 . . . . . . . . . . . . . . 15 (𝑛 = 𝑧 → (∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑛 ↔ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑧))
113112, 5elrab2 3364 . . . . . . . . . . . . . 14 (𝑧𝑀 ↔ (𝑧 ∈ (1...𝑁) ∧ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑧))
114113simprbi 480 . . . . . . . . . . . . 13 (𝑧𝑀 → ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑧)
115109, 114syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) → ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑧)
116 r19.26 3062 . . . . . . . . . . . . 13 (∀𝑝 ∈ (ℙ ∖ (1...𝐾))(¬ 𝑝𝑦 ∧ ¬ 𝑝𝑧) ↔ (∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑦 ∧ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑧))
117 eldifi 3730 . . . . . . . . . . . . . . . . 17 (𝑝 ∈ (ℙ ∖ (1...𝐾)) → 𝑝 ∈ ℙ)
118 fz1ssnn 12369 . . . . . . . . . . . . . . . . . . 19 (1...𝑁) ⊆ ℕ
1197, 118sstri 3610 . . . . . . . . . . . . . . . . . 18 𝑀 ⊆ ℕ
120119, 102sseldi 3599 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) → 𝑦 ∈ ℕ)
121 pceq0 15569 . . . . . . . . . . . . . . . . 17 ((𝑝 ∈ ℙ ∧ 𝑦 ∈ ℕ) → ((𝑝 pCnt 𝑦) = 0 ↔ ¬ 𝑝𝑦))
122117, 120, 121syl2anr 495 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) ∧ 𝑝 ∈ (ℙ ∖ (1...𝐾))) → ((𝑝 pCnt 𝑦) = 0 ↔ ¬ 𝑝𝑦))
123119, 109sseldi 3599 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) → 𝑧 ∈ ℕ)
124 pceq0 15569 . . . . . . . . . . . . . . . . 17 ((𝑝 ∈ ℙ ∧ 𝑧 ∈ ℕ) → ((𝑝 pCnt 𝑧) = 0 ↔ ¬ 𝑝𝑧))
125117, 123, 124syl2anr 495 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) ∧ 𝑝 ∈ (ℙ ∖ (1...𝐾))) → ((𝑝 pCnt 𝑧) = 0 ↔ ¬ 𝑝𝑧))
126122, 125anbi12d 747 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) ∧ 𝑝 ∈ (ℙ ∖ (1...𝐾))) → (((𝑝 pCnt 𝑦) = 0 ∧ (𝑝 pCnt 𝑧) = 0) ↔ (¬ 𝑝𝑦 ∧ ¬ 𝑝𝑧)))
127 eqtr3 2642 . . . . . . . . . . . . . . 15 (((𝑝 pCnt 𝑦) = 0 ∧ (𝑝 pCnt 𝑧) = 0) → (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧))
128126, 127syl6bir 244 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) ∧ 𝑝 ∈ (ℙ ∖ (1...𝐾))) → ((¬ 𝑝𝑦 ∧ ¬ 𝑝𝑧) → (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧)))
129128ralimdva 2961 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) → (∀𝑝 ∈ (ℙ ∖ (1...𝐾))(¬ 𝑝𝑦 ∧ ¬ 𝑝𝑧) → ∀𝑝 ∈ (ℙ ∖ (1...𝐾))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧)))
130116, 129syl5bir 233 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) → ((∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑦 ∧ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑧) → ∀𝑝 ∈ (ℙ ∖ (1...𝐾))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧)))
131108, 115, 130mp2and 715 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) → ∀𝑝 ∈ (ℙ ∖ (1...𝐾))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧))
132131biantrud 528 . . . . . . . . . 10 ((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) → (∀𝑝 ∈ (ℙ ∩ (1...𝐾))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧) ↔ (∀𝑝 ∈ (ℙ ∩ (1...𝐾))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧) ∧ ∀𝑝 ∈ (ℙ ∖ (1...𝐾))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧))))
133 incom 3803 . . . . . . . . . . . . . . 15 (ℙ ∩ (1...𝐾)) = ((1...𝐾) ∩ ℙ)
134133uneq1i 3761 . . . . . . . . . . . . . 14 ((ℙ ∩ (1...𝐾)) ∪ ((1...𝐾) ∖ ℙ)) = (((1...𝐾) ∩ ℙ) ∪ ((1...𝐾) ∖ ℙ))
135 inundif 4044 . . . . . . . . . . . . . 14 (((1...𝐾) ∩ ℙ) ∪ ((1...𝐾) ∖ ℙ)) = (1...𝐾)
136134, 135eqtri 2643 . . . . . . . . . . . . 13 ((ℙ ∩ (1...𝐾)) ∪ ((1...𝐾) ∖ ℙ)) = (1...𝐾)
137136raleqi 3140 . . . . . . . . . . . 12 (∀𝑝 ∈ ((ℙ ∩ (1...𝐾)) ∪ ((1...𝐾) ∖ ℙ))if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ↔ ∀𝑝 ∈ (1...𝐾)if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0))
138 ralunb 3792 . . . . . . . . . . . 12 (∀𝑝 ∈ ((ℙ ∩ (1...𝐾)) ∪ ((1...𝐾) ∖ ℙ))if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ↔ (∀𝑝 ∈ (ℙ ∩ (1...𝐾))if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ∧ ∀𝑝 ∈ ((1...𝐾) ∖ ℙ)if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0)))
139137, 138bitr3i 266 . . . . . . . . . . 11 (∀𝑝 ∈ (1...𝐾)if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ↔ (∀𝑝 ∈ (ℙ ∩ (1...𝐾))if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ∧ ∀𝑝 ∈ ((1...𝐾) ∖ ℙ)if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0)))
140 eldifn 3731 . . . . . . . . . . . . . . 15 (𝑝 ∈ ((1...𝐾) ∖ ℙ) → ¬ 𝑝 ∈ ℙ)
141 iffalse 4093 . . . . . . . . . . . . . . . 16 𝑝 ∈ ℙ → if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = 0)
142 iffalse 4093 . . . . . . . . . . . . . . . 16 𝑝 ∈ ℙ → if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) = 0)
143141, 142eqtr4d 2658 . . . . . . . . . . . . . . 15 𝑝 ∈ ℙ → if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0))
144140, 143syl 17 . . . . . . . . . . . . . 14 (𝑝 ∈ ((1...𝐾) ∖ ℙ) → if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0))
145144rgen 2921 . . . . . . . . . . . . 13 𝑝 ∈ ((1...𝐾) ∖ ℙ)if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0)
146145biantru 526 . . . . . . . . . . . 12 (∀𝑝 ∈ (ℙ ∩ (1...𝐾))if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ↔ (∀𝑝 ∈ (ℙ ∩ (1...𝐾))if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ∧ ∀𝑝 ∈ ((1...𝐾) ∖ ℙ)if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0)))
147 elinel1 3797 . . . . . . . . . . . . . 14 (𝑝 ∈ (ℙ ∩ (1...𝐾)) → 𝑝 ∈ ℙ)
148 iftrue 4090 . . . . . . . . . . . . . . 15 (𝑝 ∈ ℙ → if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = (𝑝 pCnt 𝑦))
149 iftrue 4090 . . . . . . . . . . . . . . 15 (𝑝 ∈ ℙ → if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) = (𝑝 pCnt 𝑧))
150148, 149eqeq12d 2636 . . . . . . . . . . . . . 14 (𝑝 ∈ ℙ → (if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ↔ (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧)))
151147, 150syl 17 . . . . . . . . . . . . 13 (𝑝 ∈ (ℙ ∩ (1...𝐾)) → (if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ↔ (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧)))
152151ralbiia 2978 . . . . . . . . . . . 12 (∀𝑝 ∈ (ℙ ∩ (1...𝐾))if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ↔ ∀𝑝 ∈ (ℙ ∩ (1...𝐾))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧))
153146, 152bitr3i 266 . . . . . . . . . . 11 ((∀𝑝 ∈ (ℙ ∩ (1...𝐾))if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ∧ ∀𝑝 ∈ ((1...𝐾) ∖ ℙ)if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0)) ↔ ∀𝑝 ∈ (ℙ ∩ (1...𝐾))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧))
154139, 153bitri 264 . . . . . . . . . 10 (∀𝑝 ∈ (1...𝐾)if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ↔ ∀𝑝 ∈ (ℙ ∩ (1...𝐾))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧))
155 inundif 4044 . . . . . . . . . . . 12 ((ℙ ∩ (1...𝐾)) ∪ (ℙ ∖ (1...𝐾))) = ℙ
156155raleqi 3140 . . . . . . . . . . 11 (∀𝑝 ∈ ((ℙ ∩ (1...𝐾)) ∪ (ℙ ∖ (1...𝐾)))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧) ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧))
157 ralunb 3792 . . . . . . . . . . 11 (∀𝑝 ∈ ((ℙ ∩ (1...𝐾)) ∪ (ℙ ∖ (1...𝐾)))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧) ↔ (∀𝑝 ∈ (ℙ ∩ (1...𝐾))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧) ∧ ∀𝑝 ∈ (ℙ ∖ (1...𝐾))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧)))
158156, 157bitr3i 266 . . . . . . . . . 10 (∀𝑝 ∈ ℙ (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧) ↔ (∀𝑝 ∈ (ℙ ∩ (1...𝐾))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧) ∧ ∀𝑝 ∈ (ℙ ∖ (1...𝐾))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧)))
159132, 154, 1583bitr4g 303 . . . . . . . . 9 ((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) → (∀𝑝 ∈ (1...𝐾)if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧)))
160120nnnn0d 11348 . . . . . . . . . 10 ((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) → 𝑦 ∈ ℕ0)
161123nnnn0d 11348 . . . . . . . . . 10 ((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) → 𝑧 ∈ ℕ0)
162 pc11 15578 . . . . . . . . . 10 ((𝑦 ∈ ℕ0𝑧 ∈ ℕ0) → (𝑦 = 𝑧 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧)))
163160, 161, 162syl2anc 693 . . . . . . . . 9 ((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) → (𝑦 = 𝑧 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧)))
164159, 163bitr4d 271 . . . . . . . 8 ((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) → (∀𝑝 ∈ (1...𝐾)if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ↔ 𝑦 = 𝑧))
165101, 164syl5bb 272 . . . . . . 7 ((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) → ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) = (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0)) ↔ 𝑦 = 𝑧))
166165ex 450 . . . . . 6 (𝜑 → (((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1)) → ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) = (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0)) ↔ 𝑦 = 𝑧)))
16778, 166syl5bi 232 . . . . 5 (𝜑 → ((𝑦 ∈ {𝑥𝑀 ∣ (𝑄𝑥) = 1} ∧ 𝑧 ∈ {𝑥𝑀 ∣ (𝑄𝑥) = 1}) → ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) = (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0)) ↔ 𝑦 = 𝑧)))
16874, 167dom2d 7993 . . . 4 (𝜑 → (({0, 1} ↑𝑚 (1...𝐾)) ∈ V → {𝑥𝑀 ∣ (𝑄𝑥) = 1} ≼ ({0, 1} ↑𝑚 (1...𝐾))))
1691, 168mpi 20 . . 3 (𝜑 → {𝑥𝑀 ∣ (𝑄𝑥) = 1} ≼ ({0, 1} ↑𝑚 (1...𝐾)))
170 fzfi 12766 . . . . . . 7 (1...𝑁) ∈ Fin
171 ssfi 8177 . . . . . . 7 (((1...𝑁) ∈ Fin ∧ {𝑛 ∈ (1...𝑁) ∣ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑛} ⊆ (1...𝑁)) → {𝑛 ∈ (1...𝑁) ∣ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑛} ∈ Fin)
172170, 6, 171mp2an 708 . . . . . 6 {𝑛 ∈ (1...𝑁) ∣ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑛} ∈ Fin
1735, 172eqeltri 2696 . . . . 5 𝑀 ∈ Fin
174 ssrab2 3685 . . . . 5 {𝑥𝑀 ∣ (𝑄𝑥) = 1} ⊆ 𝑀
175 ssfi 8177 . . . . 5 ((𝑀 ∈ Fin ∧ {𝑥𝑀 ∣ (𝑄𝑥) = 1} ⊆ 𝑀) → {𝑥𝑀 ∣ (𝑄𝑥) = 1} ∈ Fin)
176173, 174, 175mp2an 708 . . . 4 {𝑥𝑀 ∣ (𝑄𝑥) = 1} ∈ Fin
177 prfi 8232 . . . . 5 {0, 1} ∈ Fin
178 fzfid 12767 . . . . 5 (𝜑 → (1...𝐾) ∈ Fin)
179 mapfi 8259 . . . . 5 (({0, 1} ∈ Fin ∧ (1...𝐾) ∈ Fin) → ({0, 1} ↑𝑚 (1...𝐾)) ∈ Fin)
180177, 178, 179sylancr 695 . . . 4 (𝜑 → ({0, 1} ↑𝑚 (1...𝐾)) ∈ Fin)
181 hashdom 13163 . . . 4 (({𝑥𝑀 ∣ (𝑄𝑥) = 1} ∈ Fin ∧ ({0, 1} ↑𝑚 (1...𝐾)) ∈ Fin) → ((#‘{𝑥𝑀 ∣ (𝑄𝑥) = 1}) ≤ (#‘({0, 1} ↑𝑚 (1...𝐾))) ↔ {𝑥𝑀 ∣ (𝑄𝑥) = 1} ≼ ({0, 1} ↑𝑚 (1...𝐾))))
182176, 180, 181sylancr 695 . . 3 (𝜑 → ((#‘{𝑥𝑀 ∣ (𝑄𝑥) = 1}) ≤ (#‘({0, 1} ↑𝑚 (1...𝐾))) ↔ {𝑥𝑀 ∣ (𝑄𝑥) = 1} ≼ ({0, 1} ↑𝑚 (1...𝐾))))
183169, 182mpbird 247 . 2 (𝜑 → (#‘{𝑥𝑀 ∣ (𝑄𝑥) = 1}) ≤ (#‘({0, 1} ↑𝑚 (1...𝐾))))
184 hashmap 13217 . . . 4 (({0, 1} ∈ Fin ∧ (1...𝐾) ∈ Fin) → (#‘({0, 1} ↑𝑚 (1...𝐾))) = ((#‘{0, 1})↑(#‘(1...𝐾))))
185177, 178, 184sylancr 695 . . 3 (𝜑 → (#‘({0, 1} ↑𝑚 (1...𝐾))) = ((#‘{0, 1})↑(#‘(1...𝐾))))
186 prhash2ex 13182 . . . . 5 (#‘{0, 1}) = 2
187186a1i 11 . . . 4 (𝜑 → (#‘{0, 1}) = 2)
188 prmrec.2 . . . . . 6 (𝜑𝐾 ∈ ℕ)
189188nnnn0d 11348 . . . . 5 (𝜑𝐾 ∈ ℕ0)
190 hashfz1 13129 . . . . 5 (𝐾 ∈ ℕ0 → (#‘(1...𝐾)) = 𝐾)
191189, 190syl 17 . . . 4 (𝜑 → (#‘(1...𝐾)) = 𝐾)
192187, 191oveq12d 6665 . . 3 (𝜑 → ((#‘{0, 1})↑(#‘(1...𝐾))) = (2↑𝐾))
193185, 192eqtrd 2655 . 2 (𝜑 → (#‘({0, 1} ↑𝑚 (1...𝐾))) = (2↑𝐾))
194183, 193breqtrd 4677 1 (𝜑 → (#‘{𝑥𝑀 ∣ (𝑄𝑥) = 1}) ≤ (2↑𝐾))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1482  wcel 1989  wral 2911  {crab 2915  Vcvv 3198  cdif 3569  cun 3570  cin 3571  wss 3572  ifcif 4084  {cpr 4177   class class class wbr 4651  cmpt 4727   Fn wfn 5881  wf 5882  cfv 5886  (class class class)co 6647  𝑚 cmap 7854  cdom 7950  Fincfn 7952  supcsup 8343  cr 9932  0cc0 9933  1c1 9934   + caddc 9936   < clt 10071  cle 10072   / cdiv 10681  cn 11017  2c2 11067  0cn0 11289  cz 11374  cuz 11684  ...cfz 12323  cexp 12855  #chash 13112  cdvds 14977  cprime 15379   pCnt cpc 15535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-cnex 9989  ax-resscn 9990  ax-1cn 9991  ax-icn 9992  ax-addcl 9993  ax-addrcl 9994  ax-mulcl 9995  ax-mulrcl 9996  ax-mulcom 9997  ax-addass 9998  ax-mulass 9999  ax-distr 10000  ax-i2m1 10001  ax-1ne0 10002  ax-1rid 10003  ax-rnegex 10004  ax-rrecex 10005  ax-cnre 10006  ax-pre-lttri 10007  ax-pre-lttrn 10008  ax-pre-ltadd 10009  ax-pre-mulgt0 10010  ax-pre-sup 10011
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-riota 6608  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-om 7063  df-1st 7165  df-2nd 7166  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-1o 7557  df-2o 7558  df-oadd 7561  df-er 7739  df-map 7856  df-pm 7857  df-en 7953  df-dom 7954  df-sdom 7955  df-fin 7956  df-sup 8345  df-inf 8346  df-card 8762  df-cda 8987  df-pnf 10073  df-mnf 10074  df-xr 10075  df-ltxr 10076  df-le 10077  df-sub 10265  df-neg 10266  df-div 10682  df-nn 11018  df-2 11076  df-3 11077  df-n0 11290  df-xnn0 11361  df-z 11375  df-uz 11685  df-q 11786  df-rp 11830  df-fz 12324  df-fl 12588  df-mod 12664  df-seq 12797  df-exp 12856  df-hash 13113  df-cj 13833  df-re 13834  df-im 13835  df-sqrt 13969  df-abs 13970  df-dvds 14978  df-gcd 15211  df-prm 15380  df-pc 15536
This theorem is referenced by:  prmreclem3  15616
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