Step | Hyp | Ref
| Expression |
1 | | ovex 7246 |
. . . 4
⊢ ({0, 1}
↑m (1...𝐾))
∈ V |
2 | | fveqeq2 6726 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → ((𝑄‘𝑥) = 1 ↔ (𝑄‘𝑦) = 1)) |
3 | 2 | elrab 3602 |
. . . . . 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 3995 |
. . . . . . . . . . . . . . . . . . 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 3898 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑦 ∈ (1...𝑁)) |
9 | | elfznn 13141 |
. . . . . . . . . . . . . . . . . 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 488 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑛 ∈ ℙ) |
12 | | prmuz2 16253 |
. . . . . . . . . . . . . . . . . 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 16469 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ ℕ → ((𝑄‘𝑦) ∈ ℕ ∧ ((𝑄‘𝑦)↑2) ∥ 𝑦 ∧ (𝑛 ∈ (ℤ≥‘2)
→ ¬ (𝑛↑2)
∥ (𝑦 / ((𝑄‘𝑦)↑2))))) |
16 | 15 | simp3d 1146 |
. . . . . . . . . . . . . . . . 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 7228 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑄‘𝑦)↑2) = (1↑2)) |
21 | | sq1 13764 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(1↑2) = 1 |
22 | 20, 21 | eqtrdi 2794 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑄‘𝑦)↑2) = 1) |
23 | 22 | oveq2d 7229 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑦 / ((𝑄‘𝑦)↑2)) = (𝑦 / 1)) |
24 | 10 | nncnd 11846 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑦 ∈ ℂ) |
25 | 24 | div1d 11600 |
. . . . . . . . . . . . . . . . . . 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 5065 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑛↑2) ∥ (𝑦 / ((𝑄‘𝑦)↑2)) ↔ (𝑛↑2) ∥ 𝑦)) |
28 | 10 | nnzd 12281 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑦 ∈ ℤ) |
29 | | 2nn0 12107 |
. . . . . . . . . . . . . . . . . . 19
⊢ 2 ∈
ℕ0 |
30 | 29 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 2 ∈
ℕ0) |
31 | | pcdvdsb 16422 |
. . . . . . . . . . . . . . . . . 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 285 |
. . . . . . . . . . . . . . . 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 327 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ¬ 2 ≤ (𝑛 pCnt 𝑦)) |
35 | 11, 10 | pccld 16403 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈
ℕ0) |
36 | 35 | nn0red 12151 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈ ℝ) |
37 | | 2re 11904 |
. . . . . . . . . . . . . . . 16
⊢ 2 ∈
ℝ |
38 | | ltnle 10912 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑛 pCnt 𝑦) ∈ ℝ ∧ 2 ∈ ℝ)
→ ((𝑛 pCnt 𝑦) < 2 ↔ ¬ 2 ≤
(𝑛 pCnt 𝑦))) |
39 | 36, 37, 38 | sylancl 589 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑛 pCnt 𝑦) < 2 ↔ ¬ 2 ≤ (𝑛 pCnt 𝑦))) |
40 | 34, 39 | mpbird 260 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) < 2) |
41 | | df-2 11893 |
. . . . . . . . . . . . . 14
⊢ 2 = (1 +
1) |
42 | 40, 41 | breqtrdi 5094 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) < (1 + 1)) |
43 | 35 | nn0zd 12280 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈ ℤ) |
44 | | 1z 12207 |
. . . . . . . . . . . . . 14
⊢ 1 ∈
ℤ |
45 | | zleltp1 12228 |
. . . . . . . . . . . . . 14
⊢ (((𝑛 pCnt 𝑦) ∈ ℤ ∧ 1 ∈ ℤ)
→ ((𝑛 pCnt 𝑦) ≤ 1 ↔ (𝑛 pCnt 𝑦) < (1 + 1))) |
46 | 43, 44, 45 | sylancl 589 |
. . . . . . . . . . . . 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 260 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ≤ 1) |
48 | | nn0uz 12476 |
. . . . . . . . . . . . . 14
⊢
ℕ0 = (ℤ≥‘0) |
49 | 35, 48 | eleqtrdi 2848 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈
(ℤ≥‘0)) |
50 | | elfz5 13104 |
. . . . . . . . . . . . 13
⊢ (((𝑛 pCnt 𝑦) ∈ (ℤ≥‘0)
∧ 1 ∈ ℤ) → ((𝑛 pCnt 𝑦) ∈ (0...1) ↔ (𝑛 pCnt 𝑦) ≤ 1)) |
51 | 49, 44, 50 | sylancl 589 |
. . . . . . . . . . . 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 260 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈ (0...1)) |
53 | | 0z 12187 |
. . . . . . . . . . . . 13
⊢ 0 ∈
ℤ |
54 | | fzpr 13167 |
. . . . . . . . . . . . 13
⊢ (0 ∈
ℤ → (0...(0 + 1)) = {0, (0 + 1)}) |
55 | 53, 54 | ax-mp 5 |
. . . . . . . . . . . 12
⊢ (0...(0 +
1)) = {0, (0 + 1)} |
56 | | 1e0p1 12335 |
. . . . . . . . . . . . 13
⊢ 1 = (0 +
1) |
57 | 56 | oveq2i 7224 |
. . . . . . . . . . . 12
⊢ (0...1) =
(0...(0 + 1)) |
58 | 56 | preq2i 4653 |
. . . . . . . . . . . 12
⊢ {0, 1} =
{0, (0 + 1)} |
59 | 55, 57, 58 | 3eqtr4i 2775 |
. . . . . . . . . . 11
⊢ (0...1) =
{0, 1} |
60 | 52, 59 | eleqtrdi 2848 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈ {0, 1}) |
61 | | c0ex 10827 |
. . . . . . . . . . . 12
⊢ 0 ∈
V |
62 | 61 | prid1 4678 |
. . . . . . . . . . 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 4474 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) → if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0) ∈ {0, 1}) |
65 | 64 | fmpttd 6932 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) → (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)):(1...𝐾)⟶{0, 1}) |
66 | | prex 5325 |
. . . . . . . . 9
⊢ {0, 1}
∈ V |
67 | | ovex 7246 |
. . . . . . . . 9
⊢
(1...𝐾) ∈
V |
68 | 66, 67 | elmap 8552 |
. . . . . . . 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 237 |
. . . . . . 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 416 |
. . . . . 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 | syl5bi 245 |
. . . . 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 6726 |
. . . . . . . 8
⊢ (𝑥 = 𝑧 → ((𝑄‘𝑥) = 1 ↔ (𝑄‘𝑧) = 1)) |
73 | 72 | elrab 3602 |
. . . . . . 7
⊢ (𝑧 ∈ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ↔ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1)) |
74 | 3, 73 | anbi12i 630 |
. . . . . 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 7246 |
. . . . . . . . . . . 12
⊢ (𝑛 pCnt 𝑦) ∈ V |
76 | 75, 61 | ifex 4489 |
. . . . . . . . . . 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 6521 |
. . . . . . . . . 10
⊢ (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) Fn (1...𝐾) |
79 | | ovex 7246 |
. . . . . . . . . . . 12
⊢ (𝑛 pCnt 𝑧) ∈ V |
80 | 79, 61 | ifex 4489 |
. . . . . . . . . . 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 6521 |
. . . . . . . . . 10
⊢ (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0)) Fn (1...𝐾) |
83 | | eqfnfv 6852 |
. . . . . . . . . 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 2820 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑝 → (𝑛 ∈ ℙ ↔ 𝑝 ∈ ℙ)) |
86 | | oveq1 7220 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑝 → (𝑛 pCnt 𝑦) = (𝑝 pCnt 𝑦)) |
87 | 85, 86 | ifbieq1d 4463 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑝 → if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0)) |
88 | | ovex 7246 |
. . . . . . . . . . . . 13
⊢ (𝑝 pCnt 𝑦) ∈ V |
89 | 88, 61 | ifex 4489 |
. . . . . . . . . . . 12
⊢ if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) ∈ V |
90 | 87, 77, 89 | fvmpt 6818 |
. . . . . . . . . . 11
⊢ (𝑝 ∈ (1...𝐾) → ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0))‘𝑝) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0)) |
91 | | oveq1 7220 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑝 → (𝑛 pCnt 𝑧) = (𝑝 pCnt 𝑧)) |
92 | 85, 91 | ifbieq1d 4463 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑝 → if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0)) |
93 | | ovex 7246 |
. . . . . . . . . . . . 13
⊢ (𝑝 pCnt 𝑧) ∈ V |
94 | 93, 61 | ifex 4489 |
. . . . . . . . . . . 12
⊢ if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ∈ V |
95 | 92, 81, 94 | fvmpt 6818 |
. . . . . . . . . . 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 3087 |
. . . . . . . . 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 278 |
. . . . . . . 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 5057 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 = 𝑦 → (𝑝 ∥ 𝑛 ↔ 𝑝 ∥ 𝑦)) |
101 | 100 | notbid 321 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑦 → (¬ 𝑝 ∥ 𝑛 ↔ ¬ 𝑝 ∥ 𝑦)) |
102 | 101 | ralbidv 3118 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑦 → (∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑛 ↔ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑦)) |
103 | 102, 4 | elrab2 3605 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ 𝑀 ↔ (𝑦 ∈ (1...𝑁) ∧ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑦)) |
104 | 103 | simprbi 500 |
. . . . . . . . . . . . 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 5057 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 = 𝑧 → (𝑝 ∥ 𝑛 ↔ 𝑝 ∥ 𝑧)) |
108 | 107 | notbid 321 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑧 → (¬ 𝑝 ∥ 𝑛 ↔ ¬ 𝑝 ∥ 𝑧)) |
109 | 108 | ralbidv 3118 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑧 → (∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑛 ↔ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑧)) |
110 | 109, 4 | elrab2 3605 |
. . . . . . . . . . . . . 14
⊢ (𝑧 ∈ 𝑀 ↔ (𝑧 ∈ (1...𝑁) ∧ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑧)) |
111 | 110 | simprbi 500 |
. . . . . . . . . . . . 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 3092 |
. . . . . . . . . . . . 13
⊢
(∀𝑝 ∈
(ℙ ∖ (1...𝐾))(¬ 𝑝 ∥ 𝑦 ∧ ¬ 𝑝 ∥ 𝑧) ↔ (∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑦 ∧ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑧)) |
114 | | eldifi 4041 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑝 ∈ (ℙ ∖
(1...𝐾)) → 𝑝 ∈
ℙ) |
115 | | fz1ssnn 13143 |
. . . . . . . . . . . . . . . . . . 19
⊢
(1...𝑁) ⊆
ℕ |
116 | 5, 115 | sstri 3910 |
. . . . . . . . . . . . . . . . . 18
⊢ 𝑀 ⊆
ℕ |
117 | 116, 99 | sselid 3898 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → 𝑦 ∈ ℕ) |
118 | | pceq0 16424 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑝 ∈ ℙ ∧ 𝑦 ∈ ℕ) → ((𝑝 pCnt 𝑦) = 0 ↔ ¬ 𝑝 ∥ 𝑦)) |
119 | 114, 117,
118 | syl2anr 600 |
. . . . . . . . . . . . . . . 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 3898 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → 𝑧 ∈ ℕ) |
121 | | pceq0 16424 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑝 ∈ ℙ ∧ 𝑧 ∈ ℕ) → ((𝑝 pCnt 𝑧) = 0 ↔ ¬ 𝑝 ∥ 𝑧)) |
122 | 114, 120,
121 | syl2anr 600 |
. . . . . . . . . . . . . . . 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 634 |
. . . . . . . . . . . . . . 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 | syl6bir 257 |
. . . . . . . . . . . . . 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 3100 |
. . . . . . . . . . . . 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 | syl5bir 246 |
. . . . . . . . . . . 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 535 |
. . . . . . . . . 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 4115 |
. . . . . . . . . . . . . . 15
⊢ (ℙ
∩ (1...𝐾)) =
((1...𝐾) ∩
ℙ) |
131 | 130 | uneq1i 4073 |
. . . . . . . . . . . . . 14
⊢ ((ℙ
∩ (1...𝐾)) ∪
((1...𝐾) ∖ ℙ))
= (((1...𝐾) ∩ ℙ)
∪ ((1...𝐾) ∖
ℙ)) |
132 | | inundif 4393 |
. . . . . . . . . . . . . 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 3323 |
. . . . . . . . . . . 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 4105 |
. . . . . . . . . . . 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 280 |
. . . . . . . . . . 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 4042 |
. . . . . . . . . . . . . . 15
⊢ (𝑝 ∈ ((1...𝐾) ∖ ℙ) → ¬ 𝑝 ∈
ℙ) |
138 | | iffalse 4448 |
. . . . . . . . . . . . . . . 16
⊢ (¬
𝑝 ∈ ℙ →
if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = 0) |
139 | | iffalse 4448 |
. . . . . . . . . . . . . . . 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 3071 |
. . . . . . . . . . . . 13
⊢
∀𝑝 ∈
((1...𝐾) ∖
ℙ)if(𝑝 ∈
ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) |
143 | 142 | biantru 533 |
. . . . . . . . . . . 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 4109 |
. . . . . . . . . . . . . 14
⊢ (𝑝 ∈ (ℙ ∩
(1...𝐾)) → 𝑝 ∈
ℙ) |
145 | | iftrue 4445 |
. . . . . . . . . . . . . . 15
⊢ (𝑝 ∈ ℙ → if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = (𝑝 pCnt 𝑦)) |
146 | | iftrue 4445 |
. . . . . . . . . . . . . . 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 3087 |
. . . . . . . . . . . 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 280 |
. . . . . . . . . . 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 278 |
. . . . . . . . . 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 4393 |
. . . . . . . . . . . 12
⊢ ((ℙ
∩ (1...𝐾)) ∪
(ℙ ∖ (1...𝐾)))
= ℙ |
153 | 152 | raleqi 3323 |
. . . . . . . . . . 11
⊢
(∀𝑝 ∈
((ℙ ∩ (1...𝐾))
∪ (ℙ ∖ (1...𝐾)))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧) ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧)) |
154 | | ralunb 4105 |
. . . . . . . . . . 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 280 |
. . . . . . . . . 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 317 |
. . . . . . . . 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 12150 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → 𝑦 ∈ ℕ0) |
158 | 120 | nnnn0d 12150 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → 𝑧 ∈ ℕ0) |
159 | | pc11 16433 |
. . . . . . . . . 10
⊢ ((𝑦 ∈ ℕ0
∧ 𝑧 ∈
ℕ0) → (𝑦 = 𝑧 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧))) |
160 | 157, 158,
159 | syl2anc 587 |
. . . . . . . . 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 285 |
. . . . . . . 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 | syl5bb 286 |
. . . . . . 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 416 |
. . . . . 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 | syl5bi 245 |
. . . . 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 8669 |
. . . 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 13545 |
. . . . . . 7
⊢
(1...𝑁) ∈
Fin |
168 | | ssrab2 3993 |
. . . . . . 7
⊢ {𝑛 ∈ (1...𝑁) ∣ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑛} ⊆ (1...𝑁) |
169 | | ssfi 8851 |
. . . . . . 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 2834 |
. . . . 5
⊢ 𝑀 ∈ Fin |
172 | | ssrab2 3993 |
. . . . 5
⊢ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ⊆ 𝑀 |
173 | | ssfi 8851 |
. . . . 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 8946 |
. . . . 5
⊢ {0, 1}
∈ Fin |
176 | | fzfid 13546 |
. . . . 5
⊢ (𝜑 → (1...𝐾) ∈ Fin) |
177 | | mapfi 8972 |
. . . . 5
⊢ (({0, 1}
∈ Fin ∧ (1...𝐾)
∈ Fin) → ({0, 1} ↑m (1...𝐾)) ∈ Fin) |
178 | 175, 176,
177 | sylancr 590 |
. . . 4
⊢ (𝜑 → ({0, 1} ↑m
(1...𝐾)) ∈
Fin) |
179 | | hashdom 13946 |
. . . 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 590 |
. . 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 260 |
. 2
⊢ (𝜑 → (♯‘{𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1}) ≤ (♯‘({0, 1}
↑m (1...𝐾)))) |
182 | | hashmap 14002 |
. . . 4
⊢ (({0, 1}
∈ Fin ∧ (1...𝐾)
∈ Fin) → (♯‘({0, 1} ↑m (1...𝐾))) = ((♯‘{0,
1})↑(♯‘(1...𝐾)))) |
183 | 175, 176,
182 | sylancr 590 |
. . 3
⊢ (𝜑 → (♯‘({0, 1}
↑m (1...𝐾))) = ((♯‘{0,
1})↑(♯‘(1...𝐾)))) |
184 | | prhash2ex 13966 |
. . . . 5
⊢
(♯‘{0, 1}) = 2 |
185 | 184 | a1i 11 |
. . . 4
⊢ (𝜑 → (♯‘{0, 1}) =
2) |
186 | | prmrec.2 |
. . . . . 6
⊢ (𝜑 → 𝐾 ∈ ℕ) |
187 | 186 | nnnn0d 12150 |
. . . . 5
⊢ (𝜑 → 𝐾 ∈
ℕ0) |
188 | | hashfz1 13912 |
. . . . 5
⊢ (𝐾 ∈ ℕ0
→ (♯‘(1...𝐾)) = 𝐾) |
189 | 187, 188 | syl 17 |
. . . 4
⊢ (𝜑 → (♯‘(1...𝐾)) = 𝐾) |
190 | 185, 189 | oveq12d 7231 |
. . 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 5079 |
1
⊢ (𝜑 → (♯‘{𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1}) ≤ (2↑𝐾)) |