Step | Hyp | Ref
| Expression |
1 | | ovex 6678 |
. . . 4
⊢ ({0, 1}
↑𝑚 (1...𝐾)) ∈ V |
2 | | fveq2 6191 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → (𝑄‘𝑥) = (𝑄‘𝑦)) |
3 | 2 | eqeq1d 2624 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → ((𝑄‘𝑥) = 1 ↔ (𝑄‘𝑦) = 1)) |
4 | 3 | elrab 3363 |
. . . . . 6
⊢ (𝑦 ∈ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ↔ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) |
5 | | prmrec.4 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 𝑀 = {𝑛 ∈ (1...𝑁) ∣ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑛} |
6 | | ssrab2 3687 |
. . . . . . . . . . . . . . . . . . . 20
⊢ {𝑛 ∈ (1...𝑁) ∣ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑛} ⊆ (1...𝑁) |
7 | 5, 6 | eqsstri 3635 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝑀 ⊆ (1...𝑁) |
8 | | simprl 794 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) → 𝑦 ∈ 𝑀) |
9 | 8 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑦 ∈ 𝑀) |
10 | 7, 9 | sseldi 3601 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑦 ∈ (1...𝑁)) |
11 | | elfznn 12370 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ (1...𝑁) → 𝑦 ∈ ℕ) |
12 | 10, 11 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑦 ∈ ℕ) |
13 | | simpr 477 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑛 ∈ ℙ) |
14 | | prmuz2 15408 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ ℙ → 𝑛 ∈
(ℤ≥‘2)) |
15 | 13, 14 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑛 ∈
(ℤ≥‘2)) |
16 | | prmreclem2.5 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝑄 = (𝑛 ∈ ℕ ↦ sup({𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑛}, ℝ, <
)) |
17 | 16 | prmreclem1 15620 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ ℕ → ((𝑄‘𝑦) ∈ ℕ ∧ ((𝑄‘𝑦)↑2) ∥ 𝑦 ∧ (𝑛 ∈ (ℤ≥‘2)
→ ¬ (𝑛↑2)
∥ (𝑦 / ((𝑄‘𝑦)↑2))))) |
18 | 17 | simp3d 1075 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ ℕ → (𝑛 ∈
(ℤ≥‘2) → ¬ (𝑛↑2) ∥ (𝑦 / ((𝑄‘𝑦)↑2)))) |
19 | 12, 15, 18 | sylc 65 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ¬ (𝑛↑2) ∥ (𝑦 / ((𝑄‘𝑦)↑2))) |
20 | | simprr 796 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) → (𝑄‘𝑦) = 1) |
21 | 20 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑄‘𝑦) = 1) |
22 | 21 | oveq1d 6665 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑄‘𝑦)↑2) = (1↑2)) |
23 | | sq1 12958 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(1↑2) = 1 |
24 | 22, 23 | syl6eq 2672 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑄‘𝑦)↑2) = 1) |
25 | 24 | oveq2d 6666 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑦 / ((𝑄‘𝑦)↑2)) = (𝑦 / 1)) |
26 | 12 | nncnd 11036 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑦 ∈ ℂ) |
27 | 26 | div1d 10793 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑦 / 1) = 𝑦) |
28 | 25, 27 | eqtrd 2656 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑦 / ((𝑄‘𝑦)↑2)) = 𝑦) |
29 | 28 | breq2d 4665 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑛↑2) ∥ (𝑦 / ((𝑄‘𝑦)↑2)) ↔ (𝑛↑2) ∥ 𝑦)) |
30 | 12 | nnzd 11481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑦 ∈ ℤ) |
31 | | 2nn0 11309 |
. . . . . . . . . . . . . . . . . . 19
⊢ 2 ∈
ℕ0 |
32 | 31 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 2 ∈
ℕ0) |
33 | | pcdvdsb 15573 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑛 ∈ ℙ ∧ 𝑦 ∈ ℤ ∧ 2 ∈
ℕ0) → (2 ≤ (𝑛 pCnt 𝑦) ↔ (𝑛↑2) ∥ 𝑦)) |
34 | 13, 30, 32, 33 | syl3anc 1326 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (2 ≤ (𝑛 pCnt 𝑦) ↔ (𝑛↑2) ∥ 𝑦)) |
35 | 29, 34 | bitr4d 271 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑛↑2) ∥ (𝑦 / ((𝑄‘𝑦)↑2)) ↔ 2 ≤ (𝑛 pCnt 𝑦))) |
36 | 19, 35 | mtbid 314 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ¬ 2 ≤ (𝑛 pCnt 𝑦)) |
37 | 13, 12 | pccld 15555 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈
ℕ0) |
38 | 37 | nn0red 11352 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈ ℝ) |
39 | | 2re 11090 |
. . . . . . . . . . . . . . . 16
⊢ 2 ∈
ℝ |
40 | | ltnle 10117 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑛 pCnt 𝑦) ∈ ℝ ∧ 2 ∈ ℝ)
→ ((𝑛 pCnt 𝑦) < 2 ↔ ¬ 2 ≤
(𝑛 pCnt 𝑦))) |
41 | 38, 39, 40 | sylancl 694 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑛 pCnt 𝑦) < 2 ↔ ¬ 2 ≤ (𝑛 pCnt 𝑦))) |
42 | 36, 41 | mpbird 247 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) < 2) |
43 | | df-2 11079 |
. . . . . . . . . . . . . 14
⊢ 2 = (1 +
1) |
44 | 42, 43 | syl6breq 4694 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) < (1 + 1)) |
45 | 37 | nn0zd 11480 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈ ℤ) |
46 | | 1z 11407 |
. . . . . . . . . . . . . 14
⊢ 1 ∈
ℤ |
47 | | zleltp1 11428 |
. . . . . . . . . . . . . 14
⊢ (((𝑛 pCnt 𝑦) ∈ ℤ ∧ 1 ∈ ℤ)
→ ((𝑛 pCnt 𝑦) ≤ 1 ↔ (𝑛 pCnt 𝑦) < (1 + 1))) |
48 | 45, 46, 47 | sylancl 694 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑛 pCnt 𝑦) ≤ 1 ↔ (𝑛 pCnt 𝑦) < (1 + 1))) |
49 | 44, 48 | mpbird 247 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ≤ 1) |
50 | | nn0uz 11722 |
. . . . . . . . . . . . . 14
⊢
ℕ0 = (ℤ≥‘0) |
51 | 37, 50 | syl6eleq 2711 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈
(ℤ≥‘0)) |
52 | | elfz5 12334 |
. . . . . . . . . . . . 13
⊢ (((𝑛 pCnt 𝑦) ∈ (ℤ≥‘0)
∧ 1 ∈ ℤ) → ((𝑛 pCnt 𝑦) ∈ (0...1) ↔ (𝑛 pCnt 𝑦) ≤ 1)) |
53 | 51, 46, 52 | sylancl 694 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑛 pCnt 𝑦) ∈ (0...1) ↔ (𝑛 pCnt 𝑦) ≤ 1)) |
54 | 49, 53 | mpbird 247 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈ (0...1)) |
55 | | 0z 11388 |
. . . . . . . . . . . . 13
⊢ 0 ∈
ℤ |
56 | | fzpr 12396 |
. . . . . . . . . . . . 13
⊢ (0 ∈
ℤ → (0...(0 + 1)) = {0, (0 + 1)}) |
57 | 55, 56 | ax-mp 5 |
. . . . . . . . . . . 12
⊢ (0...(0 +
1)) = {0, (0 + 1)} |
58 | | 1e0p1 11552 |
. . . . . . . . . . . . 13
⊢ 1 = (0 +
1) |
59 | 58 | oveq2i 6661 |
. . . . . . . . . . . 12
⊢ (0...1) =
(0...(0 + 1)) |
60 | 58 | preq2i 4272 |
. . . . . . . . . . . 12
⊢ {0, 1} =
{0, (0 + 1)} |
61 | 57, 59, 60 | 3eqtr4i 2654 |
. . . . . . . . . . 11
⊢ (0...1) =
{0, 1} |
62 | 54, 61 | syl6eleq 2711 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈ {0, 1}) |
63 | | c0ex 10034 |
. . . . . . . . . . . 12
⊢ 0 ∈
V |
64 | 63 | prid1 4297 |
. . . . . . . . . . 11
⊢ 0 ∈
{0, 1} |
65 | 64 | a1i 11 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ ¬ 𝑛 ∈ ℙ) → 0 ∈ {0,
1}) |
66 | 62, 65 | ifclda 4120 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) → if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0) ∈ {0, 1}) |
67 | | eqid 2622 |
. . . . . . . . 9
⊢ (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) = (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) |
68 | 66, 67 | fmptd 6385 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) → (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)):(1...𝐾)⟶{0, 1}) |
69 | | prex 4909 |
. . . . . . . . 9
⊢ {0, 1}
∈ V |
70 | | ovex 6678 |
. . . . . . . . 9
⊢
(1...𝐾) ∈
V |
71 | 69, 70 | elmap 7886 |
. . . . . . . 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}) |
72 | 68, 71 | sylibr 224 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) → (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) ∈ ({0, 1}
↑𝑚 (1...𝐾))) |
73 | 72 | ex 450 |
. . . . . 6
⊢ (𝜑 → ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) → (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) ∈ ({0, 1}
↑𝑚 (1...𝐾)))) |
74 | 4, 73 | syl5bi 232 |
. . . . 5
⊢ (𝜑 → (𝑦 ∈ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} → (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) ∈ ({0, 1}
↑𝑚 (1...𝐾)))) |
75 | | fveq2 6191 |
. . . . . . . . 9
⊢ (𝑥 = 𝑧 → (𝑄‘𝑥) = (𝑄‘𝑧)) |
76 | 75 | eqeq1d 2624 |
. . . . . . . 8
⊢ (𝑥 = 𝑧 → ((𝑄‘𝑥) = 1 ↔ (𝑄‘𝑧) = 1)) |
77 | 76 | elrab 3363 |
. . . . . . 7
⊢ (𝑧 ∈ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ↔ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1)) |
78 | 4, 77 | anbi12i 733 |
. . . . . 6
⊢ ((𝑦 ∈ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ∧ 𝑧 ∈ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1}) ↔ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) |
79 | | ovex 6678 |
. . . . . . . . . . . 12
⊢ (𝑛 pCnt 𝑦) ∈ V |
80 | 79, 63 | ifex 4156 |
. . . . . . . . . . 11
⊢ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0) ∈ V |
81 | 80, 67 | fnmpti 6022 |
. . . . . . . . . 10
⊢ (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) Fn (1...𝐾) |
82 | | ovex 6678 |
. . . . . . . . . . . 12
⊢ (𝑛 pCnt 𝑧) ∈ V |
83 | 82, 63 | ifex 4156 |
. . . . . . . . . . 11
⊢ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0) ∈ V |
84 | | eqid 2622 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0)) = (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0)) |
85 | 83, 84 | fnmpti 6022 |
. . . . . . . . . 10
⊢ (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0)) Fn (1...𝐾) |
86 | | eqfnfv 6311 |
. . . . . . . . . 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))) |
87 | 81, 85, 86 | mp2an 708 |
. . . . . . . . 9
⊢ ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) = (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0)) ↔ ∀𝑝 ∈ (1...𝐾)((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0))‘𝑝) = ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0))‘𝑝)) |
88 | | eleq1 2689 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑝 → (𝑛 ∈ ℙ ↔ 𝑝 ∈ ℙ)) |
89 | | oveq1 6657 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑝 → (𝑛 pCnt 𝑦) = (𝑝 pCnt 𝑦)) |
90 | 88, 89 | ifbieq1d 4109 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑝 → if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0)) |
91 | | ovex 6678 |
. . . . . . . . . . . . 13
⊢ (𝑝 pCnt 𝑦) ∈ V |
92 | 91, 63 | ifex 4156 |
. . . . . . . . . . . 12
⊢ if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) ∈ V |
93 | 90, 67, 92 | fvmpt 6282 |
. . . . . . . . . . 11
⊢ (𝑝 ∈ (1...𝐾) → ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0))‘𝑝) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0)) |
94 | | oveq1 6657 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑝 → (𝑛 pCnt 𝑧) = (𝑝 pCnt 𝑧)) |
95 | 88, 94 | ifbieq1d 4109 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑝 → if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0)) |
96 | | ovex 6678 |
. . . . . . . . . . . . 13
⊢ (𝑝 pCnt 𝑧) ∈ V |
97 | 96, 63 | ifex 4156 |
. . . . . . . . . . . 12
⊢ if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ∈ V |
98 | 95, 84, 97 | fvmpt 6282 |
. . . . . . . . . . 11
⊢ (𝑝 ∈ (1...𝐾) → ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0))‘𝑝) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0)) |
99 | 93, 98 | eqeq12d 2637 |
. . . . . . . . . 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))) |
100 | 99 | ralbiia 2979 |
. . . . . . . . 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)) |
101 | 87, 100 | bitri 264 |
. . . . . . . 8
⊢ ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) = (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0)) ↔ ∀𝑝 ∈ (1...𝐾)if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0)) |
102 | | simprll 802 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → 𝑦 ∈ 𝑀) |
103 | | breq2 4657 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 = 𝑦 → (𝑝 ∥ 𝑛 ↔ 𝑝 ∥ 𝑦)) |
104 | 103 | notbid 308 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑦 → (¬ 𝑝 ∥ 𝑛 ↔ ¬ 𝑝 ∥ 𝑦)) |
105 | 104 | ralbidv 2986 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑦 → (∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑛 ↔ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑦)) |
106 | 105, 5 | elrab2 3366 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ 𝑀 ↔ (𝑦 ∈ (1...𝑁) ∧ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑦)) |
107 | 106 | simprbi 480 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ 𝑀 → ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑦) |
108 | 102, 107 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑦) |
109 | | simprrl 804 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → 𝑧 ∈ 𝑀) |
110 | | breq2 4657 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 = 𝑧 → (𝑝 ∥ 𝑛 ↔ 𝑝 ∥ 𝑧)) |
111 | 110 | notbid 308 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑧 → (¬ 𝑝 ∥ 𝑛 ↔ ¬ 𝑝 ∥ 𝑧)) |
112 | 111 | ralbidv 2986 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑧 → (∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑛 ↔ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑧)) |
113 | 112, 5 | elrab2 3366 |
. . . . . . . . . . . . . 14
⊢ (𝑧 ∈ 𝑀 ↔ (𝑧 ∈ (1...𝑁) ∧ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑧)) |
114 | 113 | simprbi 480 |
. . . . . . . . . . . . 13
⊢ (𝑧 ∈ 𝑀 → ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑧) |
115 | 109, 114 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑧) |
116 | | r19.26 3064 |
. . . . . . . . . . . . 13
⊢
(∀𝑝 ∈
(ℙ ∖ (1...𝐾))(¬ 𝑝 ∥ 𝑦 ∧ ¬ 𝑝 ∥ 𝑧) ↔ (∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑦 ∧ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑧)) |
117 | | eldifi 3732 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑝 ∈ (ℙ ∖
(1...𝐾)) → 𝑝 ∈
ℙ) |
118 | | fz1ssnn 12372 |
. . . . . . . . . . . . . . . . . . 19
⊢
(1...𝑁) ⊆
ℕ |
119 | 7, 118 | sstri 3612 |
. . . . . . . . . . . . . . . . . 18
⊢ 𝑀 ⊆
ℕ |
120 | 119, 102 | sseldi 3601 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → 𝑦 ∈ ℕ) |
121 | | pceq0 15575 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑝 ∈ ℙ ∧ 𝑦 ∈ ℕ) → ((𝑝 pCnt 𝑦) = 0 ↔ ¬ 𝑝 ∥ 𝑦)) |
122 | 117, 120,
121 | syl2anr 495 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) ∧ 𝑝 ∈ (ℙ ∖ (1...𝐾))) → ((𝑝 pCnt 𝑦) = 0 ↔ ¬ 𝑝 ∥ 𝑦)) |
123 | 119, 109 | sseldi 3601 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → 𝑧 ∈ ℕ) |
124 | | pceq0 15575 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑝 ∈ ℙ ∧ 𝑧 ∈ ℕ) → ((𝑝 pCnt 𝑧) = 0 ↔ ¬ 𝑝 ∥ 𝑧)) |
125 | 117, 123,
124 | syl2anr 495 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) ∧ 𝑝 ∈ (ℙ ∖ (1...𝐾))) → ((𝑝 pCnt 𝑧) = 0 ↔ ¬ 𝑝 ∥ 𝑧)) |
126 | 122, 125 | anbi12d 747 |
. . . . . . . . . . . . . . 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))) |
127 | | eqtr3 2643 |
. . . . . . . . . . . . . . 15
⊢ (((𝑝 pCnt 𝑦) = 0 ∧ (𝑝 pCnt 𝑧) = 0) → (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧)) |
128 | 126, 127 | syl6bir 244 |
. . . . . . . . . . . . . 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))) |
129 | 128 | ralimdva 2962 |
. . . . . . . . . . . . 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))) |
130 | 116, 129 | syl5bir 233 |
. . . . . . . . . . . 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))) |
131 | 108, 115,
130 | mp2and 715 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → ∀𝑝 ∈ (ℙ ∖ (1...𝐾))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧)) |
132 | 131 | biantrud 528 |
. . . . . . . . . 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)))) |
133 | | incom 3805 |
. . . . . . . . . . . . . . 15
⊢ (ℙ
∩ (1...𝐾)) =
((1...𝐾) ∩
ℙ) |
134 | 133 | uneq1i 3763 |
. . . . . . . . . . . . . 14
⊢ ((ℙ
∩ (1...𝐾)) ∪
((1...𝐾) ∖ ℙ))
= (((1...𝐾) ∩ ℙ)
∪ ((1...𝐾) ∖
ℙ)) |
135 | | inundif 4046 |
. . . . . . . . . . . . . 14
⊢
(((1...𝐾) ∩
ℙ) ∪ ((1...𝐾)
∖ ℙ)) = (1...𝐾) |
136 | 134, 135 | eqtri 2644 |
. . . . . . . . . . . . 13
⊢ ((ℙ
∩ (1...𝐾)) ∪
((1...𝐾) ∖ ℙ))
= (1...𝐾) |
137 | 136 | raleqi 3142 |
. . . . . . . . . . . 12
⊢
(∀𝑝 ∈
((ℙ ∩ (1...𝐾))
∪ ((1...𝐾) ∖
ℙ))if(𝑝 ∈
ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ↔ ∀𝑝 ∈ (1...𝐾)if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0)) |
138 | | ralunb 3794 |
. . . . . . . . . . . 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))) |
139 | 137, 138 | bitr3i 266 |
. . . . . . . . . . 11
⊢
(∀𝑝 ∈
(1...𝐾)if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ↔ (∀𝑝 ∈ (ℙ ∩ (1...𝐾))if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ∧ ∀𝑝 ∈ ((1...𝐾) ∖ ℙ)if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0))) |
140 | | eldifn 3733 |
. . . . . . . . . . . . . . 15
⊢ (𝑝 ∈ ((1...𝐾) ∖ ℙ) → ¬ 𝑝 ∈
ℙ) |
141 | | iffalse 4095 |
. . . . . . . . . . . . . . . 16
⊢ (¬
𝑝 ∈ ℙ →
if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = 0) |
142 | | iffalse 4095 |
. . . . . . . . . . . . . . . 16
⊢ (¬
𝑝 ∈ ℙ →
if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) = 0) |
143 | 141, 142 | eqtr4d 2659 |
. . . . . . . . . . . . . . 15
⊢ (¬
𝑝 ∈ ℙ →
if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0)) |
144 | 140, 143 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑝 ∈ ((1...𝐾) ∖ ℙ) → if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0)) |
145 | 144 | rgen 2922 |
. . . . . . . . . . . . 13
⊢
∀𝑝 ∈
((1...𝐾) ∖
ℙ)if(𝑝 ∈
ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) |
146 | 145 | biantru 526 |
. . . . . . . . . . . 12
⊢
(∀𝑝 ∈
(ℙ ∩ (1...𝐾))if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ↔ (∀𝑝 ∈ (ℙ ∩ (1...𝐾))if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ∧ ∀𝑝 ∈ ((1...𝐾) ∖ ℙ)if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0))) |
147 | | elinel1 3799 |
. . . . . . . . . . . . . 14
⊢ (𝑝 ∈ (ℙ ∩
(1...𝐾)) → 𝑝 ∈
ℙ) |
148 | | iftrue 4092 |
. . . . . . . . . . . . . . 15
⊢ (𝑝 ∈ ℙ → if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = (𝑝 pCnt 𝑦)) |
149 | | iftrue 4092 |
. . . . . . . . . . . . . . 15
⊢ (𝑝 ∈ ℙ → if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) = (𝑝 pCnt 𝑧)) |
150 | 148, 149 | eqeq12d 2637 |
. . . . . . . . . . . . . 14
⊢ (𝑝 ∈ ℙ → (if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ↔ (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧))) |
151 | 147, 150 | 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))) |
152 | 151 | ralbiia 2979 |
. . . . . . . . . . . 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)) |
153 | 146, 152 | bitr3i 266 |
. . . . . . . . . . 11
⊢
((∀𝑝 ∈
(ℙ ∩ (1...𝐾))if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ∧ ∀𝑝 ∈ ((1...𝐾) ∖ ℙ)if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0)) ↔ ∀𝑝 ∈ (ℙ ∩ (1...𝐾))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧)) |
154 | 139, 153 | bitri 264 |
. . . . . . . . . 10
⊢
(∀𝑝 ∈
(1...𝐾)if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ↔ ∀𝑝 ∈ (ℙ ∩ (1...𝐾))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧)) |
155 | | inundif 4046 |
. . . . . . . . . . . 12
⊢ ((ℙ
∩ (1...𝐾)) ∪
(ℙ ∖ (1...𝐾)))
= ℙ |
156 | 155 | raleqi 3142 |
. . . . . . . . . . 11
⊢
(∀𝑝 ∈
((ℙ ∩ (1...𝐾))
∪ (ℙ ∖ (1...𝐾)))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧) ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧)) |
157 | | ralunb 3794 |
. . . . . . . . . . 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))) |
158 | 156, 157 | bitr3i 266 |
. . . . . . . . . 10
⊢
(∀𝑝 ∈
ℙ (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧) ↔ (∀𝑝 ∈ (ℙ ∩ (1...𝐾))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧) ∧ ∀𝑝 ∈ (ℙ ∖ (1...𝐾))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧))) |
159 | 132, 154,
158 | 3bitr4g 303 |
. . . . . . . . 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))) |
160 | 120 | nnnn0d 11351 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → 𝑦 ∈ ℕ0) |
161 | 123 | nnnn0d 11351 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → 𝑧 ∈ ℕ0) |
162 | | pc11 15584 |
. . . . . . . . . 10
⊢ ((𝑦 ∈ ℕ0
∧ 𝑧 ∈
ℕ0) → (𝑦 = 𝑧 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧))) |
163 | 160, 161,
162 | syl2anc 693 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → (𝑦 = 𝑧 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧))) |
164 | 159, 163 | bitr4d 271 |
. . . . . . . 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)) |
165 | 101, 164 | syl5bb 272 |
. . . . . . 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)) |
166 | 165 | ex 450 |
. . . . . 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))) |
167 | 78, 166 | syl5bi 232 |
. . . . 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))) |
168 | 74, 167 | dom2d 7996 |
. . . 4
⊢ (𝜑 → (({0, 1}
↑𝑚 (1...𝐾)) ∈ V → {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ≼ ({0, 1}
↑𝑚 (1...𝐾)))) |
169 | 1, 168 | mpi 20 |
. . 3
⊢ (𝜑 → {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ≼ ({0, 1}
↑𝑚 (1...𝐾))) |
170 | | fzfi 12771 |
. . . . . . 7
⊢
(1...𝑁) ∈
Fin |
171 | | ssfi 8180 |
. . . . . . 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) |
172 | 170, 6, 171 | mp2an 708 |
. . . . . 6
⊢ {𝑛 ∈ (1...𝑁) ∣ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑛} ∈ Fin |
173 | 5, 172 | eqeltri 2697 |
. . . . 5
⊢ 𝑀 ∈ Fin |
174 | | ssrab2 3687 |
. . . . 5
⊢ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ⊆ 𝑀 |
175 | | ssfi 8180 |
. . . . 5
⊢ ((𝑀 ∈ Fin ∧ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ⊆ 𝑀) → {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ∈ Fin) |
176 | 173, 174,
175 | mp2an 708 |
. . . 4
⊢ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ∈ Fin |
177 | | prfi 8235 |
. . . . 5
⊢ {0, 1}
∈ Fin |
178 | | fzfid 12772 |
. . . . 5
⊢ (𝜑 → (1...𝐾) ∈ Fin) |
179 | | mapfi 8262 |
. . . . 5
⊢ (({0, 1}
∈ Fin ∧ (1...𝐾)
∈ Fin) → ({0, 1} ↑𝑚 (1...𝐾)) ∈ Fin) |
180 | 177, 178,
179 | sylancr 695 |
. . . 4
⊢ (𝜑 → ({0, 1}
↑𝑚 (1...𝐾)) ∈ Fin) |
181 | | hashdom 13168 |
. . . 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)))) |
182 | 176, 180,
181 | sylancr 695 |
. . 3
⊢ (𝜑 → ((#‘{𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1}) ≤ (#‘({0, 1}
↑𝑚 (1...𝐾))) ↔ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ≼ ({0, 1}
↑𝑚 (1...𝐾)))) |
183 | 169, 182 | mpbird 247 |
. 2
⊢ (𝜑 → (#‘{𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1}) ≤ (#‘({0, 1}
↑𝑚 (1...𝐾)))) |
184 | | hashmap 13222 |
. . . 4
⊢ (({0, 1}
∈ Fin ∧ (1...𝐾)
∈ Fin) → (#‘({0, 1} ↑𝑚 (1...𝐾))) = ((#‘{0,
1})↑(#‘(1...𝐾)))) |
185 | 177, 178,
184 | sylancr 695 |
. . 3
⊢ (𝜑 → (#‘({0, 1}
↑𝑚 (1...𝐾))) = ((#‘{0,
1})↑(#‘(1...𝐾)))) |
186 | | prhash2ex 13187 |
. . . . 5
⊢
(#‘{0, 1}) = 2 |
187 | 186 | a1i 11 |
. . . 4
⊢ (𝜑 → (#‘{0, 1}) =
2) |
188 | | prmrec.2 |
. . . . . 6
⊢ (𝜑 → 𝐾 ∈ ℕ) |
189 | 188 | nnnn0d 11351 |
. . . . 5
⊢ (𝜑 → 𝐾 ∈
ℕ0) |
190 | | hashfz1 13134 |
. . . . 5
⊢ (𝐾 ∈ ℕ0
→ (#‘(1...𝐾)) =
𝐾) |
191 | 189, 190 | syl 17 |
. . . 4
⊢ (𝜑 → (#‘(1...𝐾)) = 𝐾) |
192 | 187, 191 | oveq12d 6668 |
. . 3
⊢ (𝜑 → ((#‘{0,
1})↑(#‘(1...𝐾)))
= (2↑𝐾)) |
193 | 185, 192 | eqtrd 2656 |
. 2
⊢ (𝜑 → (#‘({0, 1}
↑𝑚 (1...𝐾))) = (2↑𝐾)) |
194 | 183, 193 | breqtrd 4679 |
1
⊢ (𝜑 → (#‘{𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1}) ≤ (2↑𝐾)) |