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Theorem prmreclem2 16919
Description: Lemma for prmrec 16924. 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 7457 . . . 4 ({0, 1} ↑m (1...𝐾)) ∈ V
2 fveqeq2 6910 . . . . . . 7 (𝑥 = 𝑦 → ((𝑄𝑥) = 1 ↔ (𝑄𝑦) = 1))
32elrab 3681 . . . . . 6 (𝑦 ∈ {𝑥𝑀 ∣ (𝑄𝑥) = 1} ↔ (𝑦𝑀 ∧ (𝑄𝑦) = 1))
4 prmrec.4 . . . . . . . . . . . . . . . . . . . 20 𝑀 = {𝑛 ∈ (1...𝑁) ∣ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑛}
54ssrab3 4079 . . . . . . . . . . . . . . . . . . 19 𝑀 ⊆ (1...𝑁)
6 simprl 769 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) → 𝑦𝑀)
76ad2antrr 724 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑦𝑀)
85, 7sselid 3977 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑦 ∈ (1...𝑁))
9 elfznn 13584 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (1...𝑁) → 𝑦 ∈ ℕ)
108, 9syl 17 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑦 ∈ ℕ)
11 simpr 483 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑛 ∈ ℙ)
12 prmuz2 16697 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℙ → 𝑛 ∈ (ℤ‘2))
1311, 12syl 17 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑛 ∈ (ℤ‘2))
14 prmreclem2.5 . . . . . . . . . . . . . . . . . . 19 𝑄 = (𝑛 ∈ ℕ ↦ sup({𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑛}, ℝ, < ))
1514prmreclem1 16918 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ℕ → ((𝑄𝑦) ∈ ℕ ∧ ((𝑄𝑦)↑2) ∥ 𝑦 ∧ (𝑛 ∈ (ℤ‘2) → ¬ (𝑛↑2) ∥ (𝑦 / ((𝑄𝑦)↑2)))))
1615simp3d 1141 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ℕ → (𝑛 ∈ (ℤ‘2) → ¬ (𝑛↑2) ∥ (𝑦 / ((𝑄𝑦)↑2))))
1710, 13, 16sylc 65 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ¬ (𝑛↑2) ∥ (𝑦 / ((𝑄𝑦)↑2)))
18 simprr 771 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) → (𝑄𝑦) = 1)
1918ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑄𝑦) = 1)
2019oveq1d 7439 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑄𝑦)↑2) = (1↑2))
21 sq1 14213 . . . . . . . . . . . . . . . . . . . . 21 (1↑2) = 1
2220, 21eqtrdi 2782 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑄𝑦)↑2) = 1)
2322oveq2d 7440 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑦 / ((𝑄𝑦)↑2)) = (𝑦 / 1))
2410nncnd 12280 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑦 ∈ ℂ)
2524div1d 12033 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑦 / 1) = 𝑦)
2623, 25eqtrd 2766 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑦 / ((𝑄𝑦)↑2)) = 𝑦)
2726breq2d 5165 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑛↑2) ∥ (𝑦 / ((𝑄𝑦)↑2)) ↔ (𝑛↑2) ∥ 𝑦))
2810nnzd 12637 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑦 ∈ ℤ)
29 2nn0 12541 . . . . . . . . . . . . . . . . . . 19 2 ∈ ℕ0
3029a1i 11 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 2 ∈ ℕ0)
31 pcdvdsb 16871 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ ℙ ∧ 𝑦 ∈ ℤ ∧ 2 ∈ ℕ0) → (2 ≤ (𝑛 pCnt 𝑦) ↔ (𝑛↑2) ∥ 𝑦))
3211, 28, 30, 31syl3anc 1368 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (2 ≤ (𝑛 pCnt 𝑦) ↔ (𝑛↑2) ∥ 𝑦))
3327, 32bitr4d 281 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑛↑2) ∥ (𝑦 / ((𝑄𝑦)↑2)) ↔ 2 ≤ (𝑛 pCnt 𝑦)))
3417, 33mtbid 323 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ¬ 2 ≤ (𝑛 pCnt 𝑦))
3511, 10pccld 16852 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈ ℕ0)
3635nn0red 12585 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈ ℝ)
37 2re 12338 . . . . . . . . . . . . . . . 16 2 ∈ ℝ
38 ltnle 11343 . . . . . . . . . . . . . . . 16 (((𝑛 pCnt 𝑦) ∈ ℝ ∧ 2 ∈ ℝ) → ((𝑛 pCnt 𝑦) < 2 ↔ ¬ 2 ≤ (𝑛 pCnt 𝑦)))
3936, 37, 38sylancl 584 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑛 pCnt 𝑦) < 2 ↔ ¬ 2 ≤ (𝑛 pCnt 𝑦)))
4034, 39mpbird 256 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) < 2)
41 df-2 12327 . . . . . . . . . . . . . 14 2 = (1 + 1)
4240, 41breqtrdi 5194 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) < (1 + 1))
4335nn0zd 12636 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈ ℤ)
44 1z 12644 . . . . . . . . . . . . . 14 1 ∈ ℤ
45 zleltp1 12665 . . . . . . . . . . . . . 14 (((𝑛 pCnt 𝑦) ∈ ℤ ∧ 1 ∈ ℤ) → ((𝑛 pCnt 𝑦) ≤ 1 ↔ (𝑛 pCnt 𝑦) < (1 + 1)))
4643, 44, 45sylancl 584 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑛 pCnt 𝑦) ≤ 1 ↔ (𝑛 pCnt 𝑦) < (1 + 1)))
4742, 46mpbird 256 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ≤ 1)
48 nn0uz 12916 . . . . . . . . . . . . . 14 0 = (ℤ‘0)
4935, 48eleqtrdi 2836 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈ (ℤ‘0))
50 elfz5 13547 . . . . . . . . . . . . 13 (((𝑛 pCnt 𝑦) ∈ (ℤ‘0) ∧ 1 ∈ ℤ) → ((𝑛 pCnt 𝑦) ∈ (0...1) ↔ (𝑛 pCnt 𝑦) ≤ 1))
5149, 44, 50sylancl 584 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑛 pCnt 𝑦) ∈ (0...1) ↔ (𝑛 pCnt 𝑦) ≤ 1))
5247, 51mpbird 256 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈ (0...1))
53 0z 12621 . . . . . . . . . . . . 13 0 ∈ ℤ
54 fzpr 13610 . . . . . . . . . . . . 13 (0 ∈ ℤ → (0...(0 + 1)) = {0, (0 + 1)})
5553, 54ax-mp 5 . . . . . . . . . . . 12 (0...(0 + 1)) = {0, (0 + 1)}
56 1e0p1 12771 . . . . . . . . . . . . 13 1 = (0 + 1)
5756oveq2i 7435 . . . . . . . . . . . 12 (0...1) = (0...(0 + 1))
5856preq2i 4746 . . . . . . . . . . . 12 {0, 1} = {0, (0 + 1)}
5955, 57, 583eqtr4i 2764 . . . . . . . . . . 11 (0...1) = {0, 1}
6052, 59eleqtrdi 2836 . . . . . . . . . 10 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈ {0, 1})
61 c0ex 11258 . . . . . . . . . . . 12 0 ∈ V
6261prid1 4771 . . . . . . . . . . 11 0 ∈ {0, 1}
6362a1i 11 . . . . . . . . . 10 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ ¬ 𝑛 ∈ ℙ) → 0 ∈ {0, 1})
6460, 63ifclda 4568 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) → if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0) ∈ {0, 1})
6564fmpttd 7129 . . . . . . . 8 ((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) → (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)):(1...𝐾)⟶{0, 1})
66 prex 5438 . . . . . . . . 9 {0, 1} ∈ V
67 ovex 7457 . . . . . . . . 9 (1...𝐾) ∈ V
6866, 67elmap 8900 . . . . . . . 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})
6965, 68sylibr 233 . . . . . . 7 ((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) → (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) ∈ ({0, 1} ↑m (1...𝐾)))
7069ex 411 . . . . . 6 (𝜑 → ((𝑦𝑀 ∧ (𝑄𝑦) = 1) → (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) ∈ ({0, 1} ↑m (1...𝐾))))
713, 70biimtrid 241 . . . . 5 (𝜑 → (𝑦 ∈ {𝑥𝑀 ∣ (𝑄𝑥) = 1} → (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) ∈ ({0, 1} ↑m (1...𝐾))))
72 fveqeq2 6910 . . . . . . . 8 (𝑥 = 𝑧 → ((𝑄𝑥) = 1 ↔ (𝑄𝑧) = 1))
7372elrab 3681 . . . . . . 7 (𝑧 ∈ {𝑥𝑀 ∣ (𝑄𝑥) = 1} ↔ (𝑧𝑀 ∧ (𝑄𝑧) = 1))
743, 73anbi12i 626 . . . . . 6 ((𝑦 ∈ {𝑥𝑀 ∣ (𝑄𝑥) = 1} ∧ 𝑧 ∈ {𝑥𝑀 ∣ (𝑄𝑥) = 1}) ↔ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1)))
75 ovex 7457 . . . . . . . . . . . 12 (𝑛 pCnt 𝑦) ∈ V
7675, 61ifex 4583 . . . . . . . . . . 11 if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0) ∈ V
77 eqid 2726 . . . . . . . . . . 11 (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) = (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0))
7876, 77fnmpti 6704 . . . . . . . . . 10 (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) Fn (1...𝐾)
79 ovex 7457 . . . . . . . . . . . 12 (𝑛 pCnt 𝑧) ∈ V
8079, 61ifex 4583 . . . . . . . . . . 11 if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0) ∈ V
81 eqid 2726 . . . . . . . . . . 11 (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0)) = (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0))
8280, 81fnmpti 6704 . . . . . . . . . 10 (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0)) Fn (1...𝐾)
83 eqfnfv 7044 . . . . . . . . . 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)))
8478, 82, 83mp2an 690 . . . . . . . . 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))
85 eleq1w 2809 . . . . . . . . . . . . 13 (𝑛 = 𝑝 → (𝑛 ∈ ℙ ↔ 𝑝 ∈ ℙ))
86 oveq1 7431 . . . . . . . . . . . . 13 (𝑛 = 𝑝 → (𝑛 pCnt 𝑦) = (𝑝 pCnt 𝑦))
8785, 86ifbieq1d 4557 . . . . . . . . . . . 12 (𝑛 = 𝑝 → if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0))
88 ovex 7457 . . . . . . . . . . . . 13 (𝑝 pCnt 𝑦) ∈ V
8988, 61ifex 4583 . . . . . . . . . . . 12 if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) ∈ V
9087, 77, 89fvmpt 7009 . . . . . . . . . . 11 (𝑝 ∈ (1...𝐾) → ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0))‘𝑝) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0))
91 oveq1 7431 . . . . . . . . . . . . 13 (𝑛 = 𝑝 → (𝑛 pCnt 𝑧) = (𝑝 pCnt 𝑧))
9285, 91ifbieq1d 4557 . . . . . . . . . . . 12 (𝑛 = 𝑝 → if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0))
93 ovex 7457 . . . . . . . . . . . . 13 (𝑝 pCnt 𝑧) ∈ V
9493, 61ifex 4583 . . . . . . . . . . . 12 if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ∈ V
9592, 81, 94fvmpt 7009 . . . . . . . . . . 11 (𝑝 ∈ (1...𝐾) → ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0))‘𝑝) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0))
9690, 95eqeq12d 2742 . . . . . . . . . 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)))
9796ralbiia 3081 . . . . . . . . 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))
9884, 97bitri 274 . . . . . . . 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))
99 simprll 777 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) → 𝑦𝑀)
100 breq2 5157 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑦 → (𝑝𝑛𝑝𝑦))
101100notbid 317 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑦 → (¬ 𝑝𝑛 ↔ ¬ 𝑝𝑦))
102101ralbidv 3168 . . . . . . . . . . . . . . 15 (𝑛 = 𝑦 → (∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑛 ↔ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑦))
103102, 4elrab2 3684 . . . . . . . . . . . . . 14 (𝑦𝑀 ↔ (𝑦 ∈ (1...𝑁) ∧ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑦))
104103simprbi 495 . . . . . . . . . . . . 13 (𝑦𝑀 → ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑦)
10599, 104syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) → ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑦)
106 simprrl 779 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) → 𝑧𝑀)
107 breq2 5157 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑧 → (𝑝𝑛𝑝𝑧))
108107notbid 317 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑧 → (¬ 𝑝𝑛 ↔ ¬ 𝑝𝑧))
109108ralbidv 3168 . . . . . . . . . . . . . . 15 (𝑛 = 𝑧 → (∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑛 ↔ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑧))
110109, 4elrab2 3684 . . . . . . . . . . . . . 14 (𝑧𝑀 ↔ (𝑧 ∈ (1...𝑁) ∧ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑧))
111110simprbi 495 . . . . . . . . . . . . 13 (𝑧𝑀 → ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑧)
112106, 111syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) → ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑧)
113 r19.26 3101 . . . . . . . . . . . . 13 (∀𝑝 ∈ (ℙ ∖ (1...𝐾))(¬ 𝑝𝑦 ∧ ¬ 𝑝𝑧) ↔ (∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑦 ∧ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑧))
114 eldifi 4126 . . . . . . . . . . . . . . . . 17 (𝑝 ∈ (ℙ ∖ (1...𝐾)) → 𝑝 ∈ ℙ)
115 fz1ssnn 13586 . . . . . . . . . . . . . . . . . . 19 (1...𝑁) ⊆ ℕ
1165, 115sstri 3989 . . . . . . . . . . . . . . . . . 18 𝑀 ⊆ ℕ
117116, 99sselid 3977 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) → 𝑦 ∈ ℕ)
118 pceq0 16873 . . . . . . . . . . . . . . . . 17 ((𝑝 ∈ ℙ ∧ 𝑦 ∈ ℕ) → ((𝑝 pCnt 𝑦) = 0 ↔ ¬ 𝑝𝑦))
119114, 117, 118syl2anr 595 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) ∧ 𝑝 ∈ (ℙ ∖ (1...𝐾))) → ((𝑝 pCnt 𝑦) = 0 ↔ ¬ 𝑝𝑦))
120116, 106sselid 3977 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) → 𝑧 ∈ ℕ)
121 pceq0 16873 . . . . . . . . . . . . . . . . 17 ((𝑝 ∈ ℙ ∧ 𝑧 ∈ ℕ) → ((𝑝 pCnt 𝑧) = 0 ↔ ¬ 𝑝𝑧))
122114, 120, 121syl2anr 595 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) ∧ 𝑝 ∈ (ℙ ∖ (1...𝐾))) → ((𝑝 pCnt 𝑧) = 0 ↔ ¬ 𝑝𝑧))
123119, 122anbi12d 630 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) ∧ 𝑝 ∈ (ℙ ∖ (1...𝐾))) → (((𝑝 pCnt 𝑦) = 0 ∧ (𝑝 pCnt 𝑧) = 0) ↔ (¬ 𝑝𝑦 ∧ ¬ 𝑝𝑧)))
124 eqtr3 2752 . . . . . . . . . . . . . . 15 (((𝑝 pCnt 𝑦) = 0 ∧ (𝑝 pCnt 𝑧) = 0) → (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧))
125123, 124biimtrrdi 253 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) ∧ 𝑝 ∈ (ℙ ∖ (1...𝐾))) → ((¬ 𝑝𝑦 ∧ ¬ 𝑝𝑧) → (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧)))
126125ralimdva 3157 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) → (∀𝑝 ∈ (ℙ ∖ (1...𝐾))(¬ 𝑝𝑦 ∧ ¬ 𝑝𝑧) → ∀𝑝 ∈ (ℙ ∖ (1...𝐾))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧)))
127113, 126biimtrrid 242 . . . . . . . . . . . 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)))
128105, 112, 127mp2and 697 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) → ∀𝑝 ∈ (ℙ ∖ (1...𝐾))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧))
129128biantrud 530 . . . . . . . . . 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))))
130 incom 4202 . . . . . . . . . . . . . . 15 (ℙ ∩ (1...𝐾)) = ((1...𝐾) ∩ ℙ)
131130uneq1i 4159 . . . . . . . . . . . . . 14 ((ℙ ∩ (1...𝐾)) ∪ ((1...𝐾) ∖ ℙ)) = (((1...𝐾) ∩ ℙ) ∪ ((1...𝐾) ∖ ℙ))
132 inundif 4483 . . . . . . . . . . . . . 14 (((1...𝐾) ∩ ℙ) ∪ ((1...𝐾) ∖ ℙ)) = (1...𝐾)
133131, 132eqtri 2754 . . . . . . . . . . . . 13 ((ℙ ∩ (1...𝐾)) ∪ ((1...𝐾) ∖ ℙ)) = (1...𝐾)
134133raleqi 3313 . . . . . . . . . . . 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))
135 ralunb 4192 . . . . . . . . . . . 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)))
136134, 135bitr3i 276 . . . . . . . . . . 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)))
137 eldifn 4127 . . . . . . . . . . . . . . 15 (𝑝 ∈ ((1...𝐾) ∖ ℙ) → ¬ 𝑝 ∈ ℙ)
138 iffalse 4542 . . . . . . . . . . . . . . . 16 𝑝 ∈ ℙ → if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = 0)
139 iffalse 4542 . . . . . . . . . . . . . . . 16 𝑝 ∈ ℙ → if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) = 0)
140138, 139eqtr4d 2769 . . . . . . . . . . . . . . 15 𝑝 ∈ ℙ → if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0))
141137, 140syl 17 . . . . . . . . . . . . . 14 (𝑝 ∈ ((1...𝐾) ∖ ℙ) → if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0))
142141rgen 3053 . . . . . . . . . . . . 13 𝑝 ∈ ((1...𝐾) ∖ ℙ)if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0)
143142biantru 528 . . . . . . . . . . . 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)))
144 elinel1 4196 . . . . . . . . . . . . . 14 (𝑝 ∈ (ℙ ∩ (1...𝐾)) → 𝑝 ∈ ℙ)
145 iftrue 4539 . . . . . . . . . . . . . . 15 (𝑝 ∈ ℙ → if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = (𝑝 pCnt 𝑦))
146 iftrue 4539 . . . . . . . . . . . . . . 15 (𝑝 ∈ ℙ → if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) = (𝑝 pCnt 𝑧))
147145, 146eqeq12d 2742 . . . . . . . . . . . . . 14 (𝑝 ∈ ℙ → (if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ↔ (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧)))
148144, 147syl 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)))
149148ralbiia 3081 . . . . . . . . . . . 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))
150143, 149bitr3i 276 . . . . . . . . . . 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))
151136, 150bitri 274 . . . . . . . . . 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))
152 inundif 4483 . . . . . . . . . . . 12 ((ℙ ∩ (1...𝐾)) ∪ (ℙ ∖ (1...𝐾))) = ℙ
153152raleqi 3313 . . . . . . . . . . 11 (∀𝑝 ∈ ((ℙ ∩ (1...𝐾)) ∪ (ℙ ∖ (1...𝐾)))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧) ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧))
154 ralunb 4192 . . . . . . . . . . 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)))
155153, 154bitr3i 276 . . . . . . . . . 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)))
156129, 151, 1553bitr4g 313 . . . . . . . . 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)))
157117nnnn0d 12584 . . . . . . . . . 10 ((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) → 𝑦 ∈ ℕ0)
158120nnnn0d 12584 . . . . . . . . . 10 ((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) → 𝑧 ∈ ℕ0)
159 pc11 16882 . . . . . . . . . 10 ((𝑦 ∈ ℕ0𝑧 ∈ ℕ0) → (𝑦 = 𝑧 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧)))
160157, 158, 159syl2anc 582 . . . . . . . . 9 ((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) → (𝑦 = 𝑧 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧)))
161156, 160bitr4d 281 . . . . . . . 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))
16298, 161bitrid 282 . . . . . . 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))
163162ex 411 . . . . . 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)))
16474, 163biimtrid 241 . . . . 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)))
16571, 164dom2d 9024 . . . 4 (𝜑 → (({0, 1} ↑m (1...𝐾)) ∈ V → {𝑥𝑀 ∣ (𝑄𝑥) = 1} ≼ ({0, 1} ↑m (1...𝐾))))
1661, 165mpi 20 . . 3 (𝜑 → {𝑥𝑀 ∣ (𝑄𝑥) = 1} ≼ ({0, 1} ↑m (1...𝐾)))
167 fzfi 13992 . . . . . . 7 (1...𝑁) ∈ Fin
168 ssrab2 4076 . . . . . . 7 {𝑛 ∈ (1...𝑁) ∣ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑛} ⊆ (1...𝑁)
169 ssfi 9211 . . . . . . 7 (((1...𝑁) ∈ Fin ∧ {𝑛 ∈ (1...𝑁) ∣ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑛} ⊆ (1...𝑁)) → {𝑛 ∈ (1...𝑁) ∣ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑛} ∈ Fin)
170167, 168, 169mp2an 690 . . . . . 6 {𝑛 ∈ (1...𝑁) ∣ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑛} ∈ Fin
1714, 170eqeltri 2822 . . . . 5 𝑀 ∈ Fin
172 ssrab2 4076 . . . . 5 {𝑥𝑀 ∣ (𝑄𝑥) = 1} ⊆ 𝑀
173 ssfi 9211 . . . . 5 ((𝑀 ∈ Fin ∧ {𝑥𝑀 ∣ (𝑄𝑥) = 1} ⊆ 𝑀) → {𝑥𝑀 ∣ (𝑄𝑥) = 1} ∈ Fin)
174171, 172, 173mp2an 690 . . . 4 {𝑥𝑀 ∣ (𝑄𝑥) = 1} ∈ Fin
175 prfi 9365 . . . . 5 {0, 1} ∈ Fin
176 fzfid 13993 . . . . 5 (𝜑 → (1...𝐾) ∈ Fin)
177 mapfi 9392 . . . . 5 (({0, 1} ∈ Fin ∧ (1...𝐾) ∈ Fin) → ({0, 1} ↑m (1...𝐾)) ∈ Fin)
178175, 176, 177sylancr 585 . . . 4 (𝜑 → ({0, 1} ↑m (1...𝐾)) ∈ Fin)
179 hashdom 14396 . . . 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))))
180174, 178, 179sylancr 585 . . 3 (𝜑 → ((♯‘{𝑥𝑀 ∣ (𝑄𝑥) = 1}) ≤ (♯‘({0, 1} ↑m (1...𝐾))) ↔ {𝑥𝑀 ∣ (𝑄𝑥) = 1} ≼ ({0, 1} ↑m (1...𝐾))))
181166, 180mpbird 256 . 2 (𝜑 → (♯‘{𝑥𝑀 ∣ (𝑄𝑥) = 1}) ≤ (♯‘({0, 1} ↑m (1...𝐾))))
182 hashmap 14452 . . . 4 (({0, 1} ∈ Fin ∧ (1...𝐾) ∈ Fin) → (♯‘({0, 1} ↑m (1...𝐾))) = ((♯‘{0, 1})↑(♯‘(1...𝐾))))
183175, 176, 182sylancr 585 . . 3 (𝜑 → (♯‘({0, 1} ↑m (1...𝐾))) = ((♯‘{0, 1})↑(♯‘(1...𝐾))))
184 prhash2ex 14416 . . . . 5 (♯‘{0, 1}) = 2
185184a1i 11 . . . 4 (𝜑 → (♯‘{0, 1}) = 2)
186 prmrec.2 . . . . . 6 (𝜑𝐾 ∈ ℕ)
187186nnnn0d 12584 . . . . 5 (𝜑𝐾 ∈ ℕ0)
188 hashfz1 14363 . . . . 5 (𝐾 ∈ ℕ0 → (♯‘(1...𝐾)) = 𝐾)
189187, 188syl 17 . . . 4 (𝜑 → (♯‘(1...𝐾)) = 𝐾)
190185, 189oveq12d 7442 . . 3 (𝜑 → ((♯‘{0, 1})↑(♯‘(1...𝐾))) = (2↑𝐾))
191183, 190eqtrd 2766 . 2 (𝜑 → (♯‘({0, 1} ↑m (1...𝐾))) = (2↑𝐾))
192181, 191breqtrd 5179 1 (𝜑 → (♯‘{𝑥𝑀 ∣ (𝑄𝑥) = 1}) ≤ (2↑𝐾))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  wral 3051  {crab 3419  Vcvv 3462  cdif 3944  cun 3945  cin 3946  wss 3947  ifcif 4533  {cpr 4635   class class class wbr 5153  cmpt 5236   Fn wfn 6549  wf 6550  cfv 6554  (class class class)co 7424  m cmap 8855  cdom 8972  Fincfn 8974  supcsup 9483  cr 11157  0cc0 11158  1c1 11159   + caddc 11161   < clt 11298  cle 11299   / cdiv 11921  cn 12264  2c2 12319  0cn0 12524  cz 12610  cuz 12874  ...cfz 13538  cexp 14081  chash 14347  cdvds 16256  cprime 16672   pCnt cpc 16838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235  ax-pre-sup 11236
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-2o 8497  df-oadd 8500  df-er 8734  df-map 8857  df-pm 8858  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-sup 9485  df-inf 9486  df-dju 9944  df-card 9982  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-div 11922  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-xnn0 12597  df-z 12611  df-uz 12875  df-q 12985  df-rp 13029  df-fz 13539  df-fl 13812  df-mod 13890  df-seq 14022  df-exp 14082  df-hash 14348  df-cj 15104  df-re 15105  df-im 15106  df-sqrt 15240  df-abs 15241  df-dvds 16257  df-gcd 16495  df-prm 16673  df-pc 16839
This theorem is referenced by:  prmreclem3  16920
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