Step | Hyp | Ref
| Expression |
1 | | simpl2 1065 |
. . . . . 6
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ (μ‘𝐴) = 0) → 𝐵 ∈ ℕ) |
2 | | mucl 24867 |
. . . . . 6
⊢ (𝐵 ∈ ℕ →
(μ‘𝐵) ∈
ℤ) |
3 | 1, 2 | syl 17 |
. . . . 5
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ (μ‘𝐴) = 0) → (μ‘𝐵) ∈ ℤ) |
4 | 3 | zcnd 11483 |
. . . 4
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ (μ‘𝐴) = 0) → (μ‘𝐵) ∈ ℂ) |
5 | 4 | mul02d 10234 |
. . 3
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ (μ‘𝐴) = 0) → (0 · (μ‘𝐵)) = 0) |
6 | | simpr 477 |
. . . 4
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ (μ‘𝐴) = 0) → (μ‘𝐴) = 0) |
7 | 6 | oveq1d 6665 |
. . 3
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ (μ‘𝐴) = 0) → ((μ‘𝐴) · (μ‘𝐵)) = (0 · (μ‘𝐵))) |
8 | | mumullem1 24905 |
. . . 4
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(μ‘𝐴) = 0) →
(μ‘(𝐴 ·
𝐵)) = 0) |
9 | 8 | 3adantl3 1219 |
. . 3
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ (μ‘𝐴) = 0) → (μ‘(𝐴 · 𝐵)) = 0) |
10 | 5, 7, 9 | 3eqtr4rd 2667 |
. 2
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ (μ‘𝐴) = 0) → (μ‘(𝐴 · 𝐵)) = ((μ‘𝐴) · (μ‘𝐵))) |
11 | | simpl1 1064 |
. . . . . 6
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ (μ‘𝐵) = 0) → 𝐴 ∈ ℕ) |
12 | | mucl 24867 |
. . . . . 6
⊢ (𝐴 ∈ ℕ →
(μ‘𝐴) ∈
ℤ) |
13 | 11, 12 | syl 17 |
. . . . 5
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ (μ‘𝐵) = 0) → (μ‘𝐴) ∈ ℤ) |
14 | 13 | zcnd 11483 |
. . . 4
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ (μ‘𝐵) = 0) → (μ‘𝐴) ∈ ℂ) |
15 | 14 | mul01d 10235 |
. . 3
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ (μ‘𝐵) = 0) → ((μ‘𝐴) · 0) = 0) |
16 | | simpr 477 |
. . . 4
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ (μ‘𝐵) = 0) → (μ‘𝐵) = 0) |
17 | 16 | oveq2d 6666 |
. . 3
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ (μ‘𝐵) = 0) → ((μ‘𝐴) · (μ‘𝐵)) = ((μ‘𝐴) · 0)) |
18 | | nncn 11028 |
. . . . . . . 8
⊢ (𝐴 ∈ ℕ → 𝐴 ∈
ℂ) |
19 | | nncn 11028 |
. . . . . . . 8
⊢ (𝐵 ∈ ℕ → 𝐵 ∈
ℂ) |
20 | | mulcom 10022 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴)) |
21 | 18, 19, 20 | syl2an 494 |
. . . . . . 7
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 · 𝐵) = (𝐵 · 𝐴)) |
22 | 21 | fveq2d 6195 |
. . . . . 6
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) →
(μ‘(𝐴 ·
𝐵)) = (μ‘(𝐵 · 𝐴))) |
23 | 22 | adantr 481 |
. . . . 5
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(μ‘𝐵) = 0) →
(μ‘(𝐴 ·
𝐵)) = (μ‘(𝐵 · 𝐴))) |
24 | | mumullem1 24905 |
. . . . . 6
⊢ (((𝐵 ∈ ℕ ∧ 𝐴 ∈ ℕ) ∧
(μ‘𝐵) = 0) →
(μ‘(𝐵 ·
𝐴)) = 0) |
25 | 24 | ancom1s 847 |
. . . . 5
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(μ‘𝐵) = 0) →
(μ‘(𝐵 ·
𝐴)) = 0) |
26 | 23, 25 | eqtrd 2656 |
. . . 4
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(μ‘𝐵) = 0) →
(μ‘(𝐴 ·
𝐵)) = 0) |
27 | 26 | 3adantl3 1219 |
. . 3
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ (μ‘𝐵) = 0) → (μ‘(𝐴 · 𝐵)) = 0) |
28 | 15, 17, 27 | 3eqtr4rd 2667 |
. 2
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ (μ‘𝐵) = 0) → (μ‘(𝐴 · 𝐵)) = ((μ‘𝐴) · (μ‘𝐵))) |
29 | | simpl1 1064 |
. . . . 5
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) → 𝐴 ∈ ℕ) |
30 | | simpl2 1065 |
. . . . 5
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) → 𝐵 ∈ ℕ) |
31 | 29, 30 | nnmulcld 11068 |
. . . 4
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) → (𝐴 · 𝐵) ∈ ℕ) |
32 | | mumullem2 24906 |
. . . 4
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) → (μ‘(𝐴 · 𝐵)) ≠ 0) |
33 | | muval2 24860 |
. . . 4
⊢ (((𝐴 · 𝐵) ∈ ℕ ∧ (μ‘(𝐴 · 𝐵)) ≠ 0) → (μ‘(𝐴 · 𝐵)) = (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ (𝐴 · 𝐵)}))) |
34 | 31, 32, 33 | syl2anc 693 |
. . 3
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) → (μ‘(𝐴 · 𝐵)) = (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ (𝐴 · 𝐵)}))) |
35 | | neg1cn 11124 |
. . . . . 6
⊢ -1 ∈
ℂ |
36 | 35 | a1i 11 |
. . . . 5
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) → -1 ∈
ℂ) |
37 | | fzfi 12771 |
. . . . . . 7
⊢
(1...𝐵) ∈
Fin |
38 | | prmnn 15388 |
. . . . . . . . . 10
⊢ (𝑝 ∈ ℙ → 𝑝 ∈
ℕ) |
39 | 38 | ssriv 3607 |
. . . . . . . . 9
⊢ ℙ
⊆ ℕ |
40 | | rabss2 3685 |
. . . . . . . . 9
⊢ (ℙ
⊆ ℕ → {𝑝
∈ ℙ ∣ 𝑝
∥ 𝐵} ⊆ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵}) |
41 | 39, 40 | ax-mp 5 |
. . . . . . . 8
⊢ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐵} ⊆ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵} |
42 | | dvdsssfz1 15040 |
. . . . . . . . 9
⊢ (𝐵 ∈ ℕ → {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵} ⊆ (1...𝐵)) |
43 | 30, 42 | syl 17 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) → {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐵} ⊆ (1...𝐵)) |
44 | 41, 43 | syl5ss 3614 |
. . . . . . 7
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐵} ⊆ (1...𝐵)) |
45 | | ssfi 8180 |
. . . . . . 7
⊢
(((1...𝐵) ∈ Fin
∧ {𝑝 ∈ ℙ
∣ 𝑝 ∥ 𝐵} ⊆ (1...𝐵)) → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐵} ∈ Fin) |
46 | 37, 44, 45 | sylancr 695 |
. . . . . 6
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐵} ∈ Fin) |
47 | | hashcl 13147 |
. . . . . 6
⊢ ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐵} ∈ Fin → (#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐵}) ∈
ℕ0) |
48 | 46, 47 | syl 17 |
. . . . 5
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) → (#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐵}) ∈
ℕ0) |
49 | | fzfi 12771 |
. . . . . . 7
⊢
(1...𝐴) ∈
Fin |
50 | | rabss2 3685 |
. . . . . . . . 9
⊢ (ℙ
⊆ ℕ → {𝑝
∈ ℙ ∣ 𝑝
∥ 𝐴} ⊆ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴}) |
51 | 39, 50 | ax-mp 5 |
. . . . . . . 8
⊢ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ⊆ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} |
52 | | dvdsssfz1 15040 |
. . . . . . . . 9
⊢ (𝐴 ∈ ℕ → {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} ⊆ (1...𝐴)) |
53 | 29, 52 | syl 17 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) → {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝐴} ⊆ (1...𝐴)) |
54 | 51, 53 | syl5ss 3614 |
. . . . . . 7
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ⊆ (1...𝐴)) |
55 | | ssfi 8180 |
. . . . . . 7
⊢
(((1...𝐴) ∈ Fin
∧ {𝑝 ∈ ℙ
∣ 𝑝 ∥ 𝐴} ⊆ (1...𝐴)) → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ Fin) |
56 | 49, 54, 55 | sylancr 695 |
. . . . . 6
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ Fin) |
57 | | hashcl 13147 |
. . . . . 6
⊢ ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ Fin → (#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) ∈
ℕ0) |
58 | 56, 57 | syl 17 |
. . . . 5
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) → (#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) ∈
ℕ0) |
59 | 36, 48, 58 | expaddd 13010 |
. . . 4
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) → (-1↑((#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) + (#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐵}))) = ((-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴})) · (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐵})))) |
60 | | simpr 477 |
. . . . . . . . . 10
⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℙ) |
61 | | simpl1 1064 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → 𝐴 ∈ ℕ) |
62 | 61 | nnzd 11481 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → 𝐴 ∈ ℤ) |
63 | 62 | adantlr 751 |
. . . . . . . . . 10
⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) ∧ 𝑝 ∈ ℙ) → 𝐴 ∈ ℤ) |
64 | | simpl2 1065 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → 𝐵 ∈ ℕ) |
65 | 64 | nnzd 11481 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → 𝐵 ∈ ℤ) |
66 | 65 | adantlr 751 |
. . . . . . . . . 10
⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) ∧ 𝑝 ∈ ℙ) → 𝐵 ∈ ℤ) |
67 | | euclemma 15425 |
. . . . . . . . . 10
⊢ ((𝑝 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝑝 ∥ (𝐴 · 𝐵) ↔ (𝑝 ∥ 𝐴 ∨ 𝑝 ∥ 𝐵))) |
68 | 60, 63, 66, 67 | syl3anc 1326 |
. . . . . . . . 9
⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ (𝐴 · 𝐵) ↔ (𝑝 ∥ 𝐴 ∨ 𝑝 ∥ 𝐵))) |
69 | 68 | rabbidva 3188 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) → {𝑝 ∈ ℙ ∣ 𝑝 ∥ (𝐴 · 𝐵)} = {𝑝 ∈ ℙ ∣ (𝑝 ∥ 𝐴 ∨ 𝑝 ∥ 𝐵)}) |
70 | | unrab 3898 |
. . . . . . . 8
⊢ ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐵}) = {𝑝 ∈ ℙ ∣ (𝑝 ∥ 𝐴 ∨ 𝑝 ∥ 𝐵)} |
71 | 69, 70 | syl6eqr 2674 |
. . . . . . 7
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) → {𝑝 ∈ ℙ ∣ 𝑝 ∥ (𝐴 · 𝐵)} = ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐵})) |
72 | 71 | fveq2d 6195 |
. . . . . 6
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) → (#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ (𝐴 · 𝐵)}) = (#‘({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐵}))) |
73 | | inrab 3899 |
. . . . . . . 8
⊢ ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∩ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐵}) = {𝑝 ∈ ℙ ∣ (𝑝 ∥ 𝐴 ∧ 𝑝 ∥ 𝐵)} |
74 | | nprmdvds1 15418 |
. . . . . . . . . . . 12
⊢ (𝑝 ∈ ℙ → ¬
𝑝 ∥
1) |
75 | 74 | adantl 482 |
. . . . . . . . . . 11
⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) ∧ 𝑝 ∈ ℙ) → ¬ 𝑝 ∥ 1) |
76 | | prmz 15389 |
. . . . . . . . . . . . . 14
⊢ (𝑝 ∈ ℙ → 𝑝 ∈
ℤ) |
77 | 76 | adantl 482 |
. . . . . . . . . . . . 13
⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℤ) |
78 | | dvdsgcd 15261 |
. . . . . . . . . . . . 13
⊢ ((𝑝 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑝 ∥ 𝐴 ∧ 𝑝 ∥ 𝐵) → 𝑝 ∥ (𝐴 gcd 𝐵))) |
79 | 77, 63, 66, 78 | syl3anc 1326 |
. . . . . . . . . . . 12
⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) ∧ 𝑝 ∈ ℙ) → ((𝑝 ∥ 𝐴 ∧ 𝑝 ∥ 𝐵) → 𝑝 ∥ (𝐴 gcd 𝐵))) |
80 | | simpll3 1102 |
. . . . . . . . . . . . 13
⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) ∧ 𝑝 ∈ ℙ) → (𝐴 gcd 𝐵) = 1) |
81 | 80 | breq2d 4665 |
. . . . . . . . . . . 12
⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ (𝐴 gcd 𝐵) ↔ 𝑝 ∥ 1)) |
82 | 79, 81 | sylibd 229 |
. . . . . . . . . . 11
⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) ∧ 𝑝 ∈ ℙ) → ((𝑝 ∥ 𝐴 ∧ 𝑝 ∥ 𝐵) → 𝑝 ∥ 1)) |
83 | 75, 82 | mtod 189 |
. . . . . . . . . 10
⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) ∧ 𝑝 ∈ ℙ) → ¬ (𝑝 ∥ 𝐴 ∧ 𝑝 ∥ 𝐵)) |
84 | 83 | ralrimiva 2966 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) → ∀𝑝 ∈ ℙ ¬ (𝑝 ∥ 𝐴 ∧ 𝑝 ∥ 𝐵)) |
85 | | rabeq0 3957 |
. . . . . . . . 9
⊢ ({𝑝 ∈ ℙ ∣ (𝑝 ∥ 𝐴 ∧ 𝑝 ∥ 𝐵)} = ∅ ↔ ∀𝑝 ∈ ℙ ¬ (𝑝 ∥ 𝐴 ∧ 𝑝 ∥ 𝐵)) |
86 | 84, 85 | sylibr 224 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) → {𝑝 ∈ ℙ ∣ (𝑝 ∥ 𝐴 ∧ 𝑝 ∥ 𝐵)} = ∅) |
87 | 73, 86 | syl5eq 2668 |
. . . . . . 7
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) → ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∩ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐵}) = ∅) |
88 | | hashun 13171 |
. . . . . . 7
⊢ (({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ Fin ∧ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐵} ∈ Fin ∧ ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∩ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐵}) = ∅) → (#‘({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐵})) = ((#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) + (#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐵}))) |
89 | 56, 46, 87, 88 | syl3anc 1326 |
. . . . . 6
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) → (#‘({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐵})) = ((#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) + (#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐵}))) |
90 | 72, 89 | eqtrd 2656 |
. . . . 5
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) → (#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ (𝐴 · 𝐵)}) = ((#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) + (#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐵}))) |
91 | 90 | oveq2d 6666 |
. . . 4
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) → (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ (𝐴 · 𝐵)})) = (-1↑((#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) + (#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐵})))) |
92 | | simprl 794 |
. . . . . 6
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) → (μ‘𝐴) ≠ 0) |
93 | | muval2 24860 |
. . . . . 6
⊢ ((𝐴 ∈ ℕ ∧
(μ‘𝐴) ≠ 0)
→ (μ‘𝐴) =
(-1↑(#‘{𝑝 ∈
ℙ ∣ 𝑝 ∥
𝐴}))) |
94 | 29, 92, 93 | syl2anc 693 |
. . . . 5
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) → (μ‘𝐴) = (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))) |
95 | | simprr 796 |
. . . . . 6
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) → (μ‘𝐵) ≠ 0) |
96 | | muval2 24860 |
. . . . . 6
⊢ ((𝐵 ∈ ℕ ∧
(μ‘𝐵) ≠ 0)
→ (μ‘𝐵) =
(-1↑(#‘{𝑝 ∈
ℙ ∣ 𝑝 ∥
𝐵}))) |
97 | 30, 95, 96 | syl2anc 693 |
. . . . 5
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) → (μ‘𝐵) = (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐵}))) |
98 | 94, 97 | oveq12d 6668 |
. . . 4
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) → ((μ‘𝐴) · (μ‘𝐵)) = ((-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴})) · (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐵})))) |
99 | 59, 91, 98 | 3eqtr4rd 2667 |
. . 3
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) → ((μ‘𝐴) · (μ‘𝐵)) = (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ (𝐴 · 𝐵)}))) |
100 | 34, 99 | eqtr4d 2659 |
. 2
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) → (μ‘(𝐴 · 𝐵)) = ((μ‘𝐴) · (μ‘𝐵))) |
101 | 10, 28, 100 | pm2.61da2ne 2882 |
1
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (μ‘(𝐴 · 𝐵)) = ((μ‘𝐴) · (μ‘𝐵))) |