| Step | Hyp | Ref
| Expression |
| 1 | | ovex 7464 |
. . . 4
⊢ ({0, 1}
↑m (1...𝐾))
∈ V |
| 2 | | fveqeq2 6915 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → ((𝑄‘𝑥) = 1 ↔ (𝑄‘𝑦) = 1)) |
| 3 | 2 | elrab 3692 |
. . . . . 6
⊢ (𝑦 ∈ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ↔ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) |
| 4 | | prmrec.4 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 𝑀 = {𝑛 ∈ (1...𝑁) ∣ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑛} |
| 5 | 4 | ssrab3 4082 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝑀 ⊆ (1...𝑁) |
| 6 | | simprl 771 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) → 𝑦 ∈ 𝑀) |
| 7 | 6 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑦 ∈ 𝑀) |
| 8 | 5, 7 | sselid 3981 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑦 ∈ (1...𝑁)) |
| 9 | | elfznn 13593 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ (1...𝑁) → 𝑦 ∈ ℕ) |
| 10 | 8, 9 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑦 ∈ ℕ) |
| 11 | | simpr 484 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑛 ∈ ℙ) |
| 12 | | prmuz2 16733 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ ℙ → 𝑛 ∈
(ℤ≥‘2)) |
| 13 | 11, 12 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑛 ∈
(ℤ≥‘2)) |
| 14 | | prmreclem2.5 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝑄 = (𝑛 ∈ ℕ ↦ sup({𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑛}, ℝ, <
)) |
| 15 | 14 | prmreclem1 16954 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ ℕ → ((𝑄‘𝑦) ∈ ℕ ∧ ((𝑄‘𝑦)↑2) ∥ 𝑦 ∧ (𝑛 ∈ (ℤ≥‘2)
→ ¬ (𝑛↑2)
∥ (𝑦 / ((𝑄‘𝑦)↑2))))) |
| 16 | 15 | simp3d 1145 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ ℕ → (𝑛 ∈
(ℤ≥‘2) → ¬ (𝑛↑2) ∥ (𝑦 / ((𝑄‘𝑦)↑2)))) |
| 17 | 10, 13, 16 | sylc 65 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ¬ (𝑛↑2) ∥ (𝑦 / ((𝑄‘𝑦)↑2))) |
| 18 | | simprr 773 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) → (𝑄‘𝑦) = 1) |
| 19 | 18 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑄‘𝑦) = 1) |
| 20 | 19 | oveq1d 7446 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑄‘𝑦)↑2) = (1↑2)) |
| 21 | | sq1 14234 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(1↑2) = 1 |
| 22 | 20, 21 | eqtrdi 2793 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑄‘𝑦)↑2) = 1) |
| 23 | 22 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑦 / ((𝑄‘𝑦)↑2)) = (𝑦 / 1)) |
| 24 | 10 | nncnd 12282 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑦 ∈ ℂ) |
| 25 | 24 | div1d 12035 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑦 / 1) = 𝑦) |
| 26 | 23, 25 | eqtrd 2777 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑦 / ((𝑄‘𝑦)↑2)) = 𝑦) |
| 27 | 26 | breq2d 5155 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑛↑2) ∥ (𝑦 / ((𝑄‘𝑦)↑2)) ↔ (𝑛↑2) ∥ 𝑦)) |
| 28 | 10 | nnzd 12640 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑦 ∈ ℤ) |
| 29 | | 2nn0 12543 |
. . . . . . . . . . . . . . . . . . 19
⊢ 2 ∈
ℕ0 |
| 30 | 29 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 2 ∈
ℕ0) |
| 31 | | pcdvdsb 16907 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑛 ∈ ℙ ∧ 𝑦 ∈ ℤ ∧ 2 ∈
ℕ0) → (2 ≤ (𝑛 pCnt 𝑦) ↔ (𝑛↑2) ∥ 𝑦)) |
| 32 | 11, 28, 30, 31 | syl3anc 1373 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (2 ≤ (𝑛 pCnt 𝑦) ↔ (𝑛↑2) ∥ 𝑦)) |
| 33 | 27, 32 | bitr4d 282 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑛↑2) ∥ (𝑦 / ((𝑄‘𝑦)↑2)) ↔ 2 ≤ (𝑛 pCnt 𝑦))) |
| 34 | 17, 33 | mtbid 324 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ¬ 2 ≤ (𝑛 pCnt 𝑦)) |
| 35 | 11, 10 | pccld 16888 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈
ℕ0) |
| 36 | 35 | nn0red 12588 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈ ℝ) |
| 37 | | 2re 12340 |
. . . . . . . . . . . . . . . 16
⊢ 2 ∈
ℝ |
| 38 | | ltnle 11340 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑛 pCnt 𝑦) ∈ ℝ ∧ 2 ∈ ℝ)
→ ((𝑛 pCnt 𝑦) < 2 ↔ ¬ 2 ≤
(𝑛 pCnt 𝑦))) |
| 39 | 36, 37, 38 | sylancl 586 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑛 pCnt 𝑦) < 2 ↔ ¬ 2 ≤ (𝑛 pCnt 𝑦))) |
| 40 | 34, 39 | mpbird 257 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) < 2) |
| 41 | | df-2 12329 |
. . . . . . . . . . . . . 14
⊢ 2 = (1 +
1) |
| 42 | 40, 41 | breqtrdi 5184 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) < (1 + 1)) |
| 43 | 35 | nn0zd 12639 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈ ℤ) |
| 44 | | 1z 12647 |
. . . . . . . . . . . . . 14
⊢ 1 ∈
ℤ |
| 45 | | zleltp1 12668 |
. . . . . . . . . . . . . 14
⊢ (((𝑛 pCnt 𝑦) ∈ ℤ ∧ 1 ∈ ℤ)
→ ((𝑛 pCnt 𝑦) ≤ 1 ↔ (𝑛 pCnt 𝑦) < (1 + 1))) |
| 46 | 43, 44, 45 | sylancl 586 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑛 pCnt 𝑦) ≤ 1 ↔ (𝑛 pCnt 𝑦) < (1 + 1))) |
| 47 | 42, 46 | mpbird 257 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ≤ 1) |
| 48 | | nn0uz 12920 |
. . . . . . . . . . . . . 14
⊢
ℕ0 = (ℤ≥‘0) |
| 49 | 35, 48 | eleqtrdi 2851 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈
(ℤ≥‘0)) |
| 50 | | elfz5 13556 |
. . . . . . . . . . . . 13
⊢ (((𝑛 pCnt 𝑦) ∈ (ℤ≥‘0)
∧ 1 ∈ ℤ) → ((𝑛 pCnt 𝑦) ∈ (0...1) ↔ (𝑛 pCnt 𝑦) ≤ 1)) |
| 51 | 49, 44, 50 | sylancl 586 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑛 pCnt 𝑦) ∈ (0...1) ↔ (𝑛 pCnt 𝑦) ≤ 1)) |
| 52 | 47, 51 | mpbird 257 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈ (0...1)) |
| 53 | | 0z 12624 |
. . . . . . . . . . . . 13
⊢ 0 ∈
ℤ |
| 54 | | fzpr 13619 |
. . . . . . . . . . . . 13
⊢ (0 ∈
ℤ → (0...(0 + 1)) = {0, (0 + 1)}) |
| 55 | 53, 54 | ax-mp 5 |
. . . . . . . . . . . 12
⊢ (0...(0 +
1)) = {0, (0 + 1)} |
| 56 | | 1e0p1 12775 |
. . . . . . . . . . . . 13
⊢ 1 = (0 +
1) |
| 57 | 56 | oveq2i 7442 |
. . . . . . . . . . . 12
⊢ (0...1) =
(0...(0 + 1)) |
| 58 | 56 | preq2i 4737 |
. . . . . . . . . . . 12
⊢ {0, 1} =
{0, (0 + 1)} |
| 59 | 55, 57, 58 | 3eqtr4i 2775 |
. . . . . . . . . . 11
⊢ (0...1) =
{0, 1} |
| 60 | 52, 59 | eleqtrdi 2851 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈ {0, 1}) |
| 61 | | c0ex 11255 |
. . . . . . . . . . . 12
⊢ 0 ∈
V |
| 62 | 61 | prid1 4762 |
. . . . . . . . . . 11
⊢ 0 ∈
{0, 1} |
| 63 | 62 | a1i 11 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ ¬ 𝑛 ∈ ℙ) → 0 ∈ {0,
1}) |
| 64 | 60, 63 | ifclda 4561 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) → if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0) ∈ {0, 1}) |
| 65 | 64 | fmpttd 7135 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) → (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)):(1...𝐾)⟶{0, 1}) |
| 66 | | prex 5437 |
. . . . . . . . 9
⊢ {0, 1}
∈ V |
| 67 | | ovex 7464 |
. . . . . . . . 9
⊢
(1...𝐾) ∈
V |
| 68 | 66, 67 | elmap 8911 |
. . . . . . . 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}) |
| 69 | 65, 68 | sylibr 234 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) → (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) ∈ ({0, 1} ↑m
(1...𝐾))) |
| 70 | 69 | ex 412 |
. . . . . 6
⊢ (𝜑 → ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) → (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) ∈ ({0, 1} ↑m
(1...𝐾)))) |
| 71 | 3, 70 | biimtrid 242 |
. . . . 5
⊢ (𝜑 → (𝑦 ∈ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} → (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) ∈ ({0, 1} ↑m
(1...𝐾)))) |
| 72 | | fveqeq2 6915 |
. . . . . . . 8
⊢ (𝑥 = 𝑧 → ((𝑄‘𝑥) = 1 ↔ (𝑄‘𝑧) = 1)) |
| 73 | 72 | elrab 3692 |
. . . . . . 7
⊢ (𝑧 ∈ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ↔ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1)) |
| 74 | 3, 73 | anbi12i 628 |
. . . . . 6
⊢ ((𝑦 ∈ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ∧ 𝑧 ∈ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1}) ↔ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) |
| 75 | | ovex 7464 |
. . . . . . . . . . . 12
⊢ (𝑛 pCnt 𝑦) ∈ V |
| 76 | 75, 61 | ifex 4576 |
. . . . . . . . . . 11
⊢ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0) ∈ V |
| 77 | | eqid 2737 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) = (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) |
| 78 | 76, 77 | fnmpti 6711 |
. . . . . . . . . 10
⊢ (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) Fn (1...𝐾) |
| 79 | | ovex 7464 |
. . . . . . . . . . . 12
⊢ (𝑛 pCnt 𝑧) ∈ V |
| 80 | 79, 61 | ifex 4576 |
. . . . . . . . . . 11
⊢ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0) ∈ V |
| 81 | | eqid 2737 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0)) = (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0)) |
| 82 | 80, 81 | fnmpti 6711 |
. . . . . . . . . 10
⊢ (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0)) Fn (1...𝐾) |
| 83 | | eqfnfv 7051 |
. . . . . . . . . 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))) |
| 84 | 78, 82, 83 | mp2an 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 2824 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑝 → (𝑛 ∈ ℙ ↔ 𝑝 ∈ ℙ)) |
| 86 | | oveq1 7438 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑝 → (𝑛 pCnt 𝑦) = (𝑝 pCnt 𝑦)) |
| 87 | 85, 86 | ifbieq1d 4550 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑝 → if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0)) |
| 88 | | ovex 7464 |
. . . . . . . . . . . . 13
⊢ (𝑝 pCnt 𝑦) ∈ V |
| 89 | 88, 61 | ifex 4576 |
. . . . . . . . . . . 12
⊢ if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) ∈ V |
| 90 | 87, 77, 89 | fvmpt 7016 |
. . . . . . . . . . 11
⊢ (𝑝 ∈ (1...𝐾) → ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0))‘𝑝) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0)) |
| 91 | | oveq1 7438 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑝 → (𝑛 pCnt 𝑧) = (𝑝 pCnt 𝑧)) |
| 92 | 85, 91 | ifbieq1d 4550 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑝 → if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0)) |
| 93 | | ovex 7464 |
. . . . . . . . . . . . 13
⊢ (𝑝 pCnt 𝑧) ∈ V |
| 94 | 93, 61 | ifex 4576 |
. . . . . . . . . . . 12
⊢ if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ∈ V |
| 95 | 92, 81, 94 | fvmpt 7016 |
. . . . . . . . . . 11
⊢ (𝑝 ∈ (1...𝐾) → ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0))‘𝑝) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0)) |
| 96 | 90, 95 | eqeq12d 2753 |
. . . . . . . . . 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))) |
| 97 | 96 | ralbiia 3091 |
. . . . . . . . 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)) |
| 98 | 84, 97 | bitri 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 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → 𝑦 ∈ 𝑀) |
| 100 | | breq2 5147 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 = 𝑦 → (𝑝 ∥ 𝑛 ↔ 𝑝 ∥ 𝑦)) |
| 101 | 100 | notbid 318 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑦 → (¬ 𝑝 ∥ 𝑛 ↔ ¬ 𝑝 ∥ 𝑦)) |
| 102 | 101 | ralbidv 3178 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑦 → (∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑛 ↔ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑦)) |
| 103 | 102, 4 | elrab2 3695 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ 𝑀 ↔ (𝑦 ∈ (1...𝑁) ∧ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑦)) |
| 104 | 103 | simprbi 496 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ 𝑀 → ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑦) |
| 105 | 99, 104 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑦) |
| 106 | | simprrl 781 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → 𝑧 ∈ 𝑀) |
| 107 | | breq2 5147 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 = 𝑧 → (𝑝 ∥ 𝑛 ↔ 𝑝 ∥ 𝑧)) |
| 108 | 107 | notbid 318 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑧 → (¬ 𝑝 ∥ 𝑛 ↔ ¬ 𝑝 ∥ 𝑧)) |
| 109 | 108 | ralbidv 3178 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑧 → (∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑛 ↔ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑧)) |
| 110 | 109, 4 | elrab2 3695 |
. . . . . . . . . . . . . 14
⊢ (𝑧 ∈ 𝑀 ↔ (𝑧 ∈ (1...𝑁) ∧ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑧)) |
| 111 | 110 | simprbi 496 |
. . . . . . . . . . . . 13
⊢ (𝑧 ∈ 𝑀 → ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑧) |
| 112 | 106, 111 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑧) |
| 113 | | r19.26 3111 |
. . . . . . . . . . . . 13
⊢
(∀𝑝 ∈
(ℙ ∖ (1...𝐾))(¬ 𝑝 ∥ 𝑦 ∧ ¬ 𝑝 ∥ 𝑧) ↔ (∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑦 ∧ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑧)) |
| 114 | | eldifi 4131 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑝 ∈ (ℙ ∖
(1...𝐾)) → 𝑝 ∈
ℙ) |
| 115 | | fz1ssnn 13595 |
. . . . . . . . . . . . . . . . . . 19
⊢
(1...𝑁) ⊆
ℕ |
| 116 | 5, 115 | sstri 3993 |
. . . . . . . . . . . . . . . . . 18
⊢ 𝑀 ⊆
ℕ |
| 117 | 116, 99 | sselid 3981 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → 𝑦 ∈ ℕ) |
| 118 | | pceq0 16909 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑝 ∈ ℙ ∧ 𝑦 ∈ ℕ) → ((𝑝 pCnt 𝑦) = 0 ↔ ¬ 𝑝 ∥ 𝑦)) |
| 119 | 114, 117,
118 | syl2anr 597 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) ∧ 𝑝 ∈ (ℙ ∖ (1...𝐾))) → ((𝑝 pCnt 𝑦) = 0 ↔ ¬ 𝑝 ∥ 𝑦)) |
| 120 | 116, 106 | sselid 3981 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → 𝑧 ∈ ℕ) |
| 121 | | pceq0 16909 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑝 ∈ ℙ ∧ 𝑧 ∈ ℕ) → ((𝑝 pCnt 𝑧) = 0 ↔ ¬ 𝑝 ∥ 𝑧)) |
| 122 | 114, 120,
121 | syl2anr 597 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) ∧ 𝑝 ∈ (ℙ ∖ (1...𝐾))) → ((𝑝 pCnt 𝑧) = 0 ↔ ¬ 𝑝 ∥ 𝑧)) |
| 123 | 119, 122 | anbi12d 632 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) ∧ 𝑝 ∈ (ℙ ∖ (1...𝐾))) → (((𝑝 pCnt 𝑦) = 0 ∧ (𝑝 pCnt 𝑧) = 0) ↔ (¬ 𝑝 ∥ 𝑦 ∧ ¬ 𝑝 ∥ 𝑧))) |
| 124 | | eqtr3 2763 |
. . . . . . . . . . . . . . 15
⊢ (((𝑝 pCnt 𝑦) = 0 ∧ (𝑝 pCnt 𝑧) = 0) → (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧)) |
| 125 | 123, 124 | biimtrrdi 254 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) ∧ 𝑝 ∈ (ℙ ∖ (1...𝐾))) → ((¬ 𝑝 ∥ 𝑦 ∧ ¬ 𝑝 ∥ 𝑧) → (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧))) |
| 126 | 125 | ralimdva 3167 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → (∀𝑝 ∈ (ℙ ∖ (1...𝐾))(¬ 𝑝 ∥ 𝑦 ∧ ¬ 𝑝 ∥ 𝑧) → ∀𝑝 ∈ (ℙ ∖ (1...𝐾))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧))) |
| 127 | 113, 126 | biimtrrid 243 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → ((∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑦 ∧ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑧) → ∀𝑝 ∈ (ℙ ∖ (1...𝐾))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧))) |
| 128 | 105, 112,
127 | mp2and 699 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → ∀𝑝 ∈ (ℙ ∖ (1...𝐾))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧)) |
| 129 | 128 | biantrud 531 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 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 4209 |
. . . . . . . . . . . . . . 15
⊢ (ℙ
∩ (1...𝐾)) =
((1...𝐾) ∩
ℙ) |
| 131 | 130 | uneq1i 4164 |
. . . . . . . . . . . . . 14
⊢ ((ℙ
∩ (1...𝐾)) ∪
((1...𝐾) ∖ ℙ))
= (((1...𝐾) ∩ ℙ)
∪ ((1...𝐾) ∖
ℙ)) |
| 132 | | inundif 4479 |
. . . . . . . . . . . . . 14
⊢
(((1...𝐾) ∩
ℙ) ∪ ((1...𝐾)
∖ ℙ)) = (1...𝐾) |
| 133 | 131, 132 | eqtri 2765 |
. . . . . . . . . . . . 13
⊢ ((ℙ
∩ (1...𝐾)) ∪
((1...𝐾) ∖ ℙ))
= (1...𝐾) |
| 134 | 133 | raleqi 3324 |
. . . . . . . . . . . 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 4197 |
. . . . . . . . . . . 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))) |
| 136 | 134, 135 | bitr3i 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 4132 |
. . . . . . . . . . . . . . 15
⊢ (𝑝 ∈ ((1...𝐾) ∖ ℙ) → ¬ 𝑝 ∈
ℙ) |
| 138 | | iffalse 4534 |
. . . . . . . . . . . . . . . 16
⊢ (¬
𝑝 ∈ ℙ →
if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = 0) |
| 139 | | iffalse 4534 |
. . . . . . . . . . . . . . . 16
⊢ (¬
𝑝 ∈ ℙ →
if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) = 0) |
| 140 | 138, 139 | eqtr4d 2780 |
. . . . . . . . . . . . . . 15
⊢ (¬
𝑝 ∈ ℙ →
if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0)) |
| 141 | 137, 140 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑝 ∈ ((1...𝐾) ∖ ℙ) → if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0)) |
| 142 | 141 | rgen 3063 |
. . . . . . . . . . . . 13
⊢
∀𝑝 ∈
((1...𝐾) ∖
ℙ)if(𝑝 ∈
ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) |
| 143 | 142 | biantru 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 4201 |
. . . . . . . . . . . . . 14
⊢ (𝑝 ∈ (ℙ ∩
(1...𝐾)) → 𝑝 ∈
ℙ) |
| 145 | | iftrue 4531 |
. . . . . . . . . . . . . . 15
⊢ (𝑝 ∈ ℙ → if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = (𝑝 pCnt 𝑦)) |
| 146 | | iftrue 4531 |
. . . . . . . . . . . . . . 15
⊢ (𝑝 ∈ ℙ → if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) = (𝑝 pCnt 𝑧)) |
| 147 | 145, 146 | eqeq12d 2753 |
. . . . . . . . . . . . . 14
⊢ (𝑝 ∈ ℙ → (if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ↔ (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧))) |
| 148 | 144, 147 | syl 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))) |
| 149 | 148 | ralbiia 3091 |
. . . . . . . . . . . 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)) |
| 150 | 143, 149 | bitr3i 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)) |
| 151 | 136, 150 | bitri 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 4479 |
. . . . . . . . . . . 12
⊢ ((ℙ
∩ (1...𝐾)) ∪
(ℙ ∖ (1...𝐾)))
= ℙ |
| 153 | 152 | raleqi 3324 |
. . . . . . . . . . 11
⊢
(∀𝑝 ∈
((ℙ ∩ (1...𝐾))
∪ (ℙ ∖ (1...𝐾)))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧) ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧)) |
| 154 | | ralunb 4197 |
. . . . . . . . . . 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))) |
| 155 | 153, 154 | bitr3i 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))) |
| 156 | 129, 151,
155 | 3bitr4g 314 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 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))) |
| 157 | 117 | nnnn0d 12587 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → 𝑦 ∈ ℕ0) |
| 158 | 120 | nnnn0d 12587 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → 𝑧 ∈ ℕ0) |
| 159 | | pc11 16918 |
. . . . . . . . . 10
⊢ ((𝑦 ∈ ℕ0
∧ 𝑧 ∈
ℕ0) → (𝑦 = 𝑧 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧))) |
| 160 | 157, 158,
159 | syl2anc 584 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → (𝑦 = 𝑧 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧))) |
| 161 | 156, 160 | bitr4d 282 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → (∀𝑝 ∈ (1...𝐾)if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ↔ 𝑦 = 𝑧)) |
| 162 | 98, 161 | bitrid 283 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) = (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0)) ↔ 𝑦 = 𝑧)) |
| 163 | 162 | ex 412 |
. . . . . 6
⊢ (𝜑 → (((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1)) → ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) = (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0)) ↔ 𝑦 = 𝑧))) |
| 164 | 74, 163 | biimtrid 242 |
. . . . 5
⊢ (𝜑 → ((𝑦 ∈ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ∧ 𝑧 ∈ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1}) → ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) = (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0)) ↔ 𝑦 = 𝑧))) |
| 165 | 71, 164 | dom2d 9033 |
. . . 4
⊢ (𝜑 → (({0, 1}
↑m (1...𝐾))
∈ V → {𝑥 ∈
𝑀 ∣ (𝑄‘𝑥) = 1} ≼ ({0, 1} ↑m
(1...𝐾)))) |
| 166 | 1, 165 | mpi 20 |
. . 3
⊢ (𝜑 → {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ≼ ({0, 1} ↑m
(1...𝐾))) |
| 167 | | fzfi 14013 |
. . . . . . 7
⊢
(1...𝑁) ∈
Fin |
| 168 | | ssrab2 4080 |
. . . . . . 7
⊢ {𝑛 ∈ (1...𝑁) ∣ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑛} ⊆ (1...𝑁) |
| 169 | | ssfi 9213 |
. . . . . . 7
⊢
(((1...𝑁) ∈ Fin
∧ {𝑛 ∈ (1...𝑁) ∣ ∀𝑝 ∈ (ℙ ∖
(1...𝐾)) ¬ 𝑝 ∥ 𝑛} ⊆ (1...𝑁)) → {𝑛 ∈ (1...𝑁) ∣ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑛} ∈ Fin) |
| 170 | 167, 168,
169 | mp2an 692 |
. . . . . 6
⊢ {𝑛 ∈ (1...𝑁) ∣ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑛} ∈ Fin |
| 171 | 4, 170 | eqeltri 2837 |
. . . . 5
⊢ 𝑀 ∈ Fin |
| 172 | | ssrab2 4080 |
. . . . 5
⊢ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ⊆ 𝑀 |
| 173 | | ssfi 9213 |
. . . . 5
⊢ ((𝑀 ∈ Fin ∧ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ⊆ 𝑀) → {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ∈ Fin) |
| 174 | 171, 172,
173 | mp2an 692 |
. . . 4
⊢ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ∈ Fin |
| 175 | | prfi 9363 |
. . . . 5
⊢ {0, 1}
∈ Fin |
| 176 | | fzfid 14014 |
. . . . 5
⊢ (𝜑 → (1...𝐾) ∈ Fin) |
| 177 | | mapfi 9388 |
. . . . 5
⊢ (({0, 1}
∈ Fin ∧ (1...𝐾)
∈ Fin) → ({0, 1} ↑m (1...𝐾)) ∈ Fin) |
| 178 | 175, 176,
177 | sylancr 587 |
. . . 4
⊢ (𝜑 → ({0, 1} ↑m
(1...𝐾)) ∈
Fin) |
| 179 | | hashdom 14418 |
. . . 4
⊢ (({𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ∈ Fin ∧ ({0, 1}
↑m (1...𝐾))
∈ Fin) → ((♯‘{𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1}) ≤ (♯‘({0, 1}
↑m (1...𝐾))) ↔ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ≼ ({0, 1} ↑m
(1...𝐾)))) |
| 180 | 174, 178,
179 | sylancr 587 |
. . 3
⊢ (𝜑 → ((♯‘{𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1}) ≤ (♯‘({0, 1}
↑m (1...𝐾))) ↔ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ≼ ({0, 1} ↑m
(1...𝐾)))) |
| 181 | 166, 180 | mpbird 257 |
. 2
⊢ (𝜑 → (♯‘{𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1}) ≤ (♯‘({0, 1}
↑m (1...𝐾)))) |
| 182 | | hashmap 14474 |
. . . 4
⊢ (({0, 1}
∈ Fin ∧ (1...𝐾)
∈ Fin) → (♯‘({0, 1} ↑m (1...𝐾))) = ((♯‘{0,
1})↑(♯‘(1...𝐾)))) |
| 183 | 175, 176,
182 | sylancr 587 |
. . 3
⊢ (𝜑 → (♯‘({0, 1}
↑m (1...𝐾))) = ((♯‘{0,
1})↑(♯‘(1...𝐾)))) |
| 184 | | prhash2ex 14438 |
. . . . 5
⊢
(♯‘{0, 1}) = 2 |
| 185 | 184 | a1i 11 |
. . . 4
⊢ (𝜑 → (♯‘{0, 1}) =
2) |
| 186 | | prmrec.2 |
. . . . . 6
⊢ (𝜑 → 𝐾 ∈ ℕ) |
| 187 | 186 | nnnn0d 12587 |
. . . . 5
⊢ (𝜑 → 𝐾 ∈
ℕ0) |
| 188 | | hashfz1 14385 |
. . . . 5
⊢ (𝐾 ∈ ℕ0
→ (♯‘(1...𝐾)) = 𝐾) |
| 189 | 187, 188 | syl 17 |
. . . 4
⊢ (𝜑 → (♯‘(1...𝐾)) = 𝐾) |
| 190 | 185, 189 | oveq12d 7449 |
. . 3
⊢ (𝜑 → ((♯‘{0,
1})↑(♯‘(1...𝐾))) = (2↑𝐾)) |
| 191 | 183, 190 | eqtrd 2777 |
. 2
⊢ (𝜑 → (♯‘({0, 1}
↑m (1...𝐾))) = (2↑𝐾)) |
| 192 | 181, 191 | breqtrd 5169 |
1
⊢ (𝜑 → (♯‘{𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1}) ≤ (2↑𝐾)) |