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Theorem prmreclem2 16950
Description: Lemma for prmrec 16955. 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 7463 . . . 4 ({0, 1} ↑m (1...𝐾)) ∈ V
2 fveqeq2 6915 . . . . . . 7 (𝑥 = 𝑦 → ((𝑄𝑥) = 1 ↔ (𝑄𝑦) = 1))
32elrab 3694 . . . . . 6 (𝑦 ∈ {𝑥𝑀 ∣ (𝑄𝑥) = 1} ↔ (𝑦𝑀 ∧ (𝑄𝑦) = 1))
4 prmrec.4 . . . . . . . . . . . . . . . . . . . 20 𝑀 = {𝑛 ∈ (1...𝑁) ∣ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑛}
54ssrab3 4091 . . . . . . . . . . . . . . . . . . 19 𝑀 ⊆ (1...𝑁)
6 simprl 771 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) → 𝑦𝑀)
76ad2antrr 726 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑦𝑀)
85, 7sselid 3992 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑦 ∈ (1...𝑁))
9 elfznn 13589 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (1...𝑁) → 𝑦 ∈ ℕ)
108, 9syl 17 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑦 ∈ ℕ)
11 simpr 484 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑛 ∈ ℙ)
12 prmuz2 16729 . . . . . . . . . . . . . . . . . 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 16949 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ℕ → ((𝑄𝑦) ∈ ℕ ∧ ((𝑄𝑦)↑2) ∥ 𝑦 ∧ (𝑛 ∈ (ℤ‘2) → ¬ (𝑛↑2) ∥ (𝑦 / ((𝑄𝑦)↑2)))))
1615simp3d 1143 . . . . . . . . . . . . . . . . 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 773 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) → (𝑄𝑦) = 1)
1918ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑄𝑦) = 1)
2019oveq1d 7445 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑄𝑦)↑2) = (1↑2))
21 sq1 14230 . . . . . . . . . . . . . . . . . . . . 21 (1↑2) = 1
2220, 21eqtrdi 2790 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑄𝑦)↑2) = 1)
2322oveq2d 7446 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑦 / ((𝑄𝑦)↑2)) = (𝑦 / 1))
2410nncnd 12279 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑦 ∈ ℂ)
2524div1d 12032 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑦 / 1) = 𝑦)
2623, 25eqtrd 2774 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑦 / ((𝑄𝑦)↑2)) = 𝑦)
2726breq2d 5159 . . . . . . . . . . . . . . . . 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 12540 . . . . . . . . . . . . . . . . . . 19 2 ∈ ℕ0
3029a1i 11 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 2 ∈ ℕ0)
31 pcdvdsb 16902 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ ℙ ∧ 𝑦 ∈ ℤ ∧ 2 ∈ ℕ0) → (2 ≤ (𝑛 pCnt 𝑦) ↔ (𝑛↑2) ∥ 𝑦))
3211, 28, 30, 31syl3anc 1370 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (2 ≤ (𝑛 pCnt 𝑦) ↔ (𝑛↑2) ∥ 𝑦))
3327, 32bitr4d 282 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑛↑2) ∥ (𝑦 / ((𝑄𝑦)↑2)) ↔ 2 ≤ (𝑛 pCnt 𝑦)))
3417, 33mtbid 324 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ¬ 2 ≤ (𝑛 pCnt 𝑦))
3511, 10pccld 16883 . . . . . . . . . . . . . . . . 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 12337 . . . . . . . . . . . . . . . 16 2 ∈ ℝ
38 ltnle 11337 . . . . . . . . . . . . . . . 16 (((𝑛 pCnt 𝑦) ∈ ℝ ∧ 2 ∈ ℝ) → ((𝑛 pCnt 𝑦) < 2 ↔ ¬ 2 ≤ (𝑛 pCnt 𝑦)))
3936, 37, 38sylancl 586 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑛 pCnt 𝑦) < 2 ↔ ¬ 2 ≤ (𝑛 pCnt 𝑦)))
4034, 39mpbird 257 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) < 2)
41 df-2 12326 . . . . . . . . . . . . . 14 2 = (1 + 1)
4240, 41breqtrdi 5188 . . . . . . . . . . . . 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 586 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑛 pCnt 𝑦) ≤ 1 ↔ (𝑛 pCnt 𝑦) < (1 + 1)))
4742, 46mpbird 257 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ≤ 1)
48 nn0uz 12917 . . . . . . . . . . . . . 14 0 = (ℤ‘0)
4935, 48eleqtrdi 2848 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈ (ℤ‘0))
50 elfz5 13552 . . . . . . . . . . . . 13 (((𝑛 pCnt 𝑦) ∈ (ℤ‘0) ∧ 1 ∈ ℤ) → ((𝑛 pCnt 𝑦) ∈ (0...1) ↔ (𝑛 pCnt 𝑦) ≤ 1))
5149, 44, 50sylancl 586 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑛 pCnt 𝑦) ∈ (0...1) ↔ (𝑛 pCnt 𝑦) ≤ 1))
5247, 51mpbird 257 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈ (0...1))
53 0z 12621 . . . . . . . . . . . . 13 0 ∈ ℤ
54 fzpr 13615 . . . . . . . . . . . . 13 (0 ∈ ℤ → (0...(0 + 1)) = {0, (0 + 1)})
5553, 54ax-mp 5 . . . . . . . . . . . 12 (0...(0 + 1)) = {0, (0 + 1)}
56 1e0p1 12772 . . . . . . . . . . . . 13 1 = (0 + 1)
5756oveq2i 7441 . . . . . . . . . . . 12 (0...1) = (0...(0 + 1))
5856preq2i 4741 . . . . . . . . . . . 12 {0, 1} = {0, (0 + 1)}
5955, 57, 583eqtr4i 2772 . . . . . . . . . . 11 (0...1) = {0, 1}
6052, 59eleqtrdi 2848 . . . . . . . . . 10 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈ {0, 1})
61 c0ex 11252 . . . . . . . . . . . 12 0 ∈ V
6261prid1 4766 . . . . . . . . . . 11 0 ∈ {0, 1}
6362a1i 11 . . . . . . . . . 10 ((((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ ¬ 𝑛 ∈ ℙ) → 0 ∈ {0, 1})
6460, 63ifclda 4565 . . . . . . . . 9 (((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) → if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0) ∈ {0, 1})
6564fmpttd 7134 . . . . . . . 8 ((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) → (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)):(1...𝐾)⟶{0, 1})
66 prex 5442 . . . . . . . . 9 {0, 1} ∈ V
67 ovex 7463 . . . . . . . . 9 (1...𝐾) ∈ V
6866, 67elmap 8909 . . . . . . . 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 234 . . . . . . 7 ((𝜑 ∧ (𝑦𝑀 ∧ (𝑄𝑦) = 1)) → (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) ∈ ({0, 1} ↑m (1...𝐾)))
7069ex 412 . . . . . 6 (𝜑 → ((𝑦𝑀 ∧ (𝑄𝑦) = 1) → (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) ∈ ({0, 1} ↑m (1...𝐾))))
713, 70biimtrid 242 . . . . 5 (𝜑 → (𝑦 ∈ {𝑥𝑀 ∣ (𝑄𝑥) = 1} → (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) ∈ ({0, 1} ↑m (1...𝐾))))
72 fveqeq2 6915 . . . . . . . 8 (𝑥 = 𝑧 → ((𝑄𝑥) = 1 ↔ (𝑄𝑧) = 1))
7372elrab 3694 . . . . . . 7 (𝑧 ∈ {𝑥𝑀 ∣ (𝑄𝑥) = 1} ↔ (𝑧𝑀 ∧ (𝑄𝑧) = 1))
743, 73anbi12i 628 . . . . . 6 ((𝑦 ∈ {𝑥𝑀 ∣ (𝑄𝑥) = 1} ∧ 𝑧 ∈ {𝑥𝑀 ∣ (𝑄𝑥) = 1}) ↔ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1)))
75 ovex 7463 . . . . . . . . . . . 12 (𝑛 pCnt 𝑦) ∈ V
7675, 61ifex 4580 . . . . . . . . . . 11 if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0) ∈ V
77 eqid 2734 . . . . . . . . . . 11 (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) = (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0))
7876, 77fnmpti 6711 . . . . . . . . . 10 (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) Fn (1...𝐾)
79 ovex 7463 . . . . . . . . . . . 12 (𝑛 pCnt 𝑧) ∈ V
8079, 61ifex 4580 . . . . . . . . . . 11 if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0) ∈ V
81 eqid 2734 . . . . . . . . . . 11 (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0)) = (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0))
8280, 81fnmpti 6711 . . . . . . . . . 10 (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0)) Fn (1...𝐾)
83 eqfnfv 7050 . . . . . . . . . 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 692 . . . . . . . . 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 2821 . . . . . . . . . . . . 13 (𝑛 = 𝑝 → (𝑛 ∈ ℙ ↔ 𝑝 ∈ ℙ))
86 oveq1 7437 . . . . . . . . . . . . 13 (𝑛 = 𝑝 → (𝑛 pCnt 𝑦) = (𝑝 pCnt 𝑦))
8785, 86ifbieq1d 4554 . . . . . . . . . . . 12 (𝑛 = 𝑝 → if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0))
88 ovex 7463 . . . . . . . . . . . . 13 (𝑝 pCnt 𝑦) ∈ V
8988, 61ifex 4580 . . . . . . . . . . . 12 if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) ∈ V
9087, 77, 89fvmpt 7015 . . . . . . . . . . 11 (𝑝 ∈ (1...𝐾) → ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0))‘𝑝) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0))
91 oveq1 7437 . . . . . . . . . . . . 13 (𝑛 = 𝑝 → (𝑛 pCnt 𝑧) = (𝑝 pCnt 𝑧))
9285, 91ifbieq1d 4554 . . . . . . . . . . . 12 (𝑛 = 𝑝 → if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0))
93 ovex 7463 . . . . . . . . . . . . 13 (𝑝 pCnt 𝑧) ∈ V
9493, 61ifex 4580 . . . . . . . . . . . 12 if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ∈ V
9592, 81, 94fvmpt 7015 . . . . . . . . . . 11 (𝑝 ∈ (1...𝐾) → ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0))‘𝑝) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0))
9690, 95eqeq12d 2750 . . . . . . . . . 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 3088 . . . . . . . . 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 275 . . . . . . . 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 779 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) → 𝑦𝑀)
100 breq2 5151 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑦 → (𝑝𝑛𝑝𝑦))
101100notbid 318 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑦 → (¬ 𝑝𝑛 ↔ ¬ 𝑝𝑦))
102101ralbidv 3175 . . . . . . . . . . . . . . 15 (𝑛 = 𝑦 → (∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑛 ↔ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑦))
103102, 4elrab2 3697 . . . . . . . . . . . . . 14 (𝑦𝑀 ↔ (𝑦 ∈ (1...𝑁) ∧ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑦))
104103simprbi 496 . . . . . . . . . . . . 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 781 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) → 𝑧𝑀)
107 breq2 5151 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑧 → (𝑝𝑛𝑝𝑧))
108107notbid 318 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑧 → (¬ 𝑝𝑛 ↔ ¬ 𝑝𝑧))
109108ralbidv 3175 . . . . . . . . . . . . . . 15 (𝑛 = 𝑧 → (∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑛 ↔ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑧))
110109, 4elrab2 3697 . . . . . . . . . . . . . 14 (𝑧𝑀 ↔ (𝑧 ∈ (1...𝑁) ∧ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑧))
111110simprbi 496 . . . . . . . . . . . . 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 3108 . . . . . . . . . . . . 13 (∀𝑝 ∈ (ℙ ∖ (1...𝐾))(¬ 𝑝𝑦 ∧ ¬ 𝑝𝑧) ↔ (∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑦 ∧ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑧))
114 eldifi 4140 . . . . . . . . . . . . . . . . 17 (𝑝 ∈ (ℙ ∖ (1...𝐾)) → 𝑝 ∈ ℙ)
115 fz1ssnn 13591 . . . . . . . . . . . . . . . . . . 19 (1...𝑁) ⊆ ℕ
1165, 115sstri 4004 . . . . . . . . . . . . . . . . . 18 𝑀 ⊆ ℕ
117116, 99sselid 3992 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) → 𝑦 ∈ ℕ)
118 pceq0 16904 . . . . . . . . . . . . . . . . 17 ((𝑝 ∈ ℙ ∧ 𝑦 ∈ ℕ) → ((𝑝 pCnt 𝑦) = 0 ↔ ¬ 𝑝𝑦))
119114, 117, 118syl2anr 597 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) ∧ 𝑝 ∈ (ℙ ∖ (1...𝐾))) → ((𝑝 pCnt 𝑦) = 0 ↔ ¬ 𝑝𝑦))
120116, 106sselid 3992 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) → 𝑧 ∈ ℕ)
121 pceq0 16904 . . . . . . . . . . . . . . . . 17 ((𝑝 ∈ ℙ ∧ 𝑧 ∈ ℕ) → ((𝑝 pCnt 𝑧) = 0 ↔ ¬ 𝑝𝑧))
122114, 120, 121syl2anr 597 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) ∧ 𝑝 ∈ (ℙ ∖ (1...𝐾))) → ((𝑝 pCnt 𝑧) = 0 ↔ ¬ 𝑝𝑧))
123119, 122anbi12d 632 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) ∧ 𝑝 ∈ (ℙ ∖ (1...𝐾))) → (((𝑝 pCnt 𝑦) = 0 ∧ (𝑝 pCnt 𝑧) = 0) ↔ (¬ 𝑝𝑦 ∧ ¬ 𝑝𝑧)))
124 eqtr3 2760 . . . . . . . . . . . . . . 15 (((𝑝 pCnt 𝑦) = 0 ∧ (𝑝 pCnt 𝑧) = 0) → (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧))
125123, 124biimtrrdi 254 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) ∧ 𝑝 ∈ (ℙ ∖ (1...𝐾))) → ((¬ 𝑝𝑦 ∧ ¬ 𝑝𝑧) → (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧)))
126125ralimdva 3164 . . . . . . . . . . . . 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 243 . . . . . . . . . . . 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 699 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) → ∀𝑝 ∈ (ℙ ∖ (1...𝐾))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧))
129128biantrud 531 . . . . . . . . . 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 4216 . . . . . . . . . . . . . . 15 (ℙ ∩ (1...𝐾)) = ((1...𝐾) ∩ ℙ)
131130uneq1i 4173 . . . . . . . . . . . . . 14 ((ℙ ∩ (1...𝐾)) ∪ ((1...𝐾) ∖ ℙ)) = (((1...𝐾) ∩ ℙ) ∪ ((1...𝐾) ∖ ℙ))
132 inundif 4484 . . . . . . . . . . . . . 14 (((1...𝐾) ∩ ℙ) ∪ ((1...𝐾) ∖ ℙ)) = (1...𝐾)
133131, 132eqtri 2762 . . . . . . . . . . . . 13 ((ℙ ∩ (1...𝐾)) ∪ ((1...𝐾) ∖ ℙ)) = (1...𝐾)
134133raleqi 3321 . . . . . . . . . . . 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 4206 . . . . . . . . . . . 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 277 . . . . . . . . . . 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 4141 . . . . . . . . . . . . . . 15 (𝑝 ∈ ((1...𝐾) ∖ ℙ) → ¬ 𝑝 ∈ ℙ)
138 iffalse 4539 . . . . . . . . . . . . . . . 16 𝑝 ∈ ℙ → if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = 0)
139 iffalse 4539 . . . . . . . . . . . . . . . 16 𝑝 ∈ ℙ → if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) = 0)
140138, 139eqtr4d 2777 . . . . . . . . . . . . . . 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 3060 . . . . . . . . . . . . 13 𝑝 ∈ ((1...𝐾) ∖ ℙ)if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0)
143142biantru 529 . . . . . . . . . . . 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 4210 . . . . . . . . . . . . . 14 (𝑝 ∈ (ℙ ∩ (1...𝐾)) → 𝑝 ∈ ℙ)
145 iftrue 4536 . . . . . . . . . . . . . . 15 (𝑝 ∈ ℙ → if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = (𝑝 pCnt 𝑦))
146 iftrue 4536 . . . . . . . . . . . . . . 15 (𝑝 ∈ ℙ → if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) = (𝑝 pCnt 𝑧))
147145, 146eqeq12d 2750 . . . . . . . . . . . . . 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 3088 . . . . . . . . . . . 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 277 . . . . . . . . . . 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 275 . . . . . . . . . 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 4484 . . . . . . . . . . . 12 ((ℙ ∩ (1...𝐾)) ∪ (ℙ ∖ (1...𝐾))) = ℙ
153152raleqi 3321 . . . . . . . . . . 11 (∀𝑝 ∈ ((ℙ ∩ (1...𝐾)) ∪ (ℙ ∖ (1...𝐾)))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧) ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧))
154 ralunb 4206 . . . . . . . . . . 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 277 . . . . . . . . . 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 314 . . . . . . . . 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 16913 . . . . . . . . . 10 ((𝑦 ∈ ℕ0𝑧 ∈ ℕ0) → (𝑦 = 𝑧 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧)))
160157, 158, 159syl2anc 584 . . . . . . . . 9 ((𝜑 ∧ ((𝑦𝑀 ∧ (𝑄𝑦) = 1) ∧ (𝑧𝑀 ∧ (𝑄𝑧) = 1))) → (𝑦 = 𝑧 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧)))
161156, 160bitr4d 282 . . . . . . . 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 283 . . . . . . 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 412 . . . . . 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 242 . . . . 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 9031 . . . 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 14009 . . . . . . 7 (1...𝑁) ∈ Fin
168 ssrab2 4089 . . . . . . 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 692 . . . . . 6 {𝑛 ∈ (1...𝑁) ∣ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑛} ∈ Fin
1714, 170eqeltri 2834 . . . . 5 𝑀 ∈ Fin
172 ssrab2 4089 . . . . 5 {𝑥𝑀 ∣ (𝑄𝑥) = 1} ⊆ 𝑀
173 ssfi 9211 . . . . 5 ((𝑀 ∈ Fin ∧ {𝑥𝑀 ∣ (𝑄𝑥) = 1} ⊆ 𝑀) → {𝑥𝑀 ∣ (𝑄𝑥) = 1} ∈ Fin)
174171, 172, 173mp2an 692 . . . 4 {𝑥𝑀 ∣ (𝑄𝑥) = 1} ∈ Fin
175 prfi 9360 . . . . 5 {0, 1} ∈ Fin
176 fzfid 14010 . . . . 5 (𝜑 → (1...𝐾) ∈ Fin)
177 mapfi 9385 . . . . 5 (({0, 1} ∈ Fin ∧ (1...𝐾) ∈ Fin) → ({0, 1} ↑m (1...𝐾)) ∈ Fin)
178175, 176, 177sylancr 587 . . . 4 (𝜑 → ({0, 1} ↑m (1...𝐾)) ∈ Fin)
179 hashdom 14414 . . . 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 587 . . 3 (𝜑 → ((♯‘{𝑥𝑀 ∣ (𝑄𝑥) = 1}) ≤ (♯‘({0, 1} ↑m (1...𝐾))) ↔ {𝑥𝑀 ∣ (𝑄𝑥) = 1} ≼ ({0, 1} ↑m (1...𝐾))))
181166, 180mpbird 257 . 2 (𝜑 → (♯‘{𝑥𝑀 ∣ (𝑄𝑥) = 1}) ≤ (♯‘({0, 1} ↑m (1...𝐾))))
182 hashmap 14470 . . . 4 (({0, 1} ∈ Fin ∧ (1...𝐾) ∈ Fin) → (♯‘({0, 1} ↑m (1...𝐾))) = ((♯‘{0, 1})↑(♯‘(1...𝐾))))
183175, 176, 182sylancr 587 . . 3 (𝜑 → (♯‘({0, 1} ↑m (1...𝐾))) = ((♯‘{0, 1})↑(♯‘(1...𝐾))))
184 prhash2ex 14434 . . . . 5 (♯‘{0, 1}) = 2
185184a1i 11 . . . 4 (𝜑 → (♯‘{0, 1}) = 2)
186 prmrec.2 . . . . . 6 (𝜑𝐾 ∈ ℕ)
187186nnnn0d 12584 . . . . 5 (𝜑𝐾 ∈ ℕ0)
188 hashfz1 14381 . . . . 5 (𝐾 ∈ ℕ0 → (♯‘(1...𝐾)) = 𝐾)
189187, 188syl 17 . . . 4 (𝜑 → (♯‘(1...𝐾)) = 𝐾)
190185, 189oveq12d 7448 . . 3 (𝜑 → ((♯‘{0, 1})↑(♯‘(1...𝐾))) = (2↑𝐾))
191183, 190eqtrd 2774 . 2 (𝜑 → (♯‘({0, 1} ↑m (1...𝐾))) = (2↑𝐾))
192181, 191breqtrd 5173 1 (𝜑 → (♯‘{𝑥𝑀 ∣ (𝑄𝑥) = 1}) ≤ (2↑𝐾))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wral 3058  {crab 3432  Vcvv 3477  cdif 3959  cun 3960  cin 3961  wss 3962  ifcif 4530  {cpr 4632   class class class wbr 5147  cmpt 5230   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430  m cmap 8864  cdom 8981  Fincfn 8983  supcsup 9477  cr 11151  0cc0 11152  1c1 11153   + caddc 11155   < clt 11292  cle 11293   / cdiv 11917  cn 12263  2c2 12318  0cn0 12523  cz 12610  cuz 12875  ...cfz 13543  cexp 14098  chash 14365  cdvds 16286  cprime 16704   pCnt cpc 16869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-oadd 8508  df-er 8743  df-map 8866  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-inf 9480  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-xnn0 12597  df-z 12611  df-uz 12876  df-q 12988  df-rp 13032  df-fz 13544  df-fl 13828  df-mod 13906  df-seq 14039  df-exp 14099  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-dvds 16287  df-gcd 16528  df-prm 16705  df-pc 16870
This theorem is referenced by:  prmreclem3  16951
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