Step | Hyp | Ref
| Expression |
1 | | nnmulcl 8899 |
. . . . 5
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 · 𝑁) ∈ ℕ) |
2 | 1 | nnred 8891 |
. . . 4
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 · 𝑁) ∈ ℝ) |
3 | | nnz 9231 |
. . . . . . 7
⊢ (𝑀 ∈ ℕ → 𝑀 ∈
ℤ) |
4 | 3 | adantr 274 |
. . . . . 6
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑀 ∈
ℤ) |
5 | 4 | zred 9334 |
. . . . 5
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑀 ∈
ℝ) |
6 | | nnz 9231 |
. . . . . . 7
⊢ (𝑁 ∈ ℕ → 𝑁 ∈
ℤ) |
7 | 6 | adantl 275 |
. . . . . 6
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈
ℤ) |
8 | 7 | zred 9334 |
. . . . 5
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈
ℝ) |
9 | | 0red 7921 |
. . . . . . 7
⊢ (𝑀 ∈ ℕ → 0 ∈
ℝ) |
10 | | nnre 8885 |
. . . . . . 7
⊢ (𝑀 ∈ ℕ → 𝑀 ∈
ℝ) |
11 | | nngt0 8903 |
. . . . . . 7
⊢ (𝑀 ∈ ℕ → 0 <
𝑀) |
12 | 9, 10, 11 | ltled 8038 |
. . . . . 6
⊢ (𝑀 ∈ ℕ → 0 ≤
𝑀) |
13 | 12 | adantr 274 |
. . . . 5
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 0 ≤
𝑀) |
14 | | 0red 7921 |
. . . . . . 7
⊢ (𝑁 ∈ ℕ → 0 ∈
ℝ) |
15 | | nnre 8885 |
. . . . . . 7
⊢ (𝑁 ∈ ℕ → 𝑁 ∈
ℝ) |
16 | | nngt0 8903 |
. . . . . . 7
⊢ (𝑁 ∈ ℕ → 0 <
𝑁) |
17 | 14, 15, 16 | ltled 8038 |
. . . . . 6
⊢ (𝑁 ∈ ℕ → 0 ≤
𝑁) |
18 | 17 | adantl 275 |
. . . . 5
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 0 ≤
𝑁) |
19 | 5, 8, 13, 18 | mulge0d 8540 |
. . . 4
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 0 ≤
(𝑀 · 𝑁)) |
20 | 2, 19 | absidd 11131 |
. . 3
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) →
(abs‘(𝑀 ·
𝑁)) = (𝑀 · 𝑁)) |
21 | 3, 6 | anim12i 336 |
. . . . . 6
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 ∈ ℤ ∧ 𝑁 ∈
ℤ)) |
22 | | nnne0 8906 |
. . . . . . . . 9
⊢ (𝑀 ∈ ℕ → 𝑀 ≠ 0) |
23 | 22 | neneqd 2361 |
. . . . . . . 8
⊢ (𝑀 ∈ ℕ → ¬
𝑀 = 0) |
24 | | nnne0 8906 |
. . . . . . . . 9
⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0) |
25 | 24 | neneqd 2361 |
. . . . . . . 8
⊢ (𝑁 ∈ ℕ → ¬
𝑁 = 0) |
26 | 23, 25 | anim12i 336 |
. . . . . . 7
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (¬
𝑀 = 0 ∧ ¬ 𝑁 = 0)) |
27 | | ioran 747 |
. . . . . . 7
⊢ (¬
(𝑀 = 0 ∨ 𝑁 = 0) ↔ (¬ 𝑀 = 0 ∧ ¬ 𝑁 = 0)) |
28 | 26, 27 | sylibr 133 |
. . . . . 6
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ¬
(𝑀 = 0 ∨ 𝑁 = 0)) |
29 | | lcmn0val 12020 |
. . . . . 6
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm 𝑁) = inf({𝑥 ∈ ℕ ∣ (𝑀 ∥ 𝑥 ∧ 𝑁 ∥ 𝑥)}, ℝ, < )) |
30 | 21, 28, 29 | syl2anc 409 |
. . . . 5
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 lcm 𝑁) = inf({𝑥 ∈ ℕ ∣ (𝑀 ∥ 𝑥 ∧ 𝑁 ∥ 𝑥)}, ℝ, < )) |
31 | | lttri3 7999 |
. . . . . . 7
⊢ ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓))) |
32 | 31 | adantl 275 |
. . . . . 6
⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓))) |
33 | | gcddvds 11918 |
. . . . . . . . . . 11
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁)) |
34 | 33 | simpld 111 |
. . . . . . . . . 10
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ∥ 𝑀) |
35 | | gcdcl 11921 |
. . . . . . . . . . . 12
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ∈
ℕ0) |
36 | 35 | nn0zd 9332 |
. . . . . . . . . . 11
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ∈ ℤ) |
37 | | dvdsmultr1 11793 |
. . . . . . . . . . . 12
⊢ (((𝑀 gcd 𝑁) ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd 𝑁) ∥ 𝑀 → (𝑀 gcd 𝑁) ∥ (𝑀 · 𝑁))) |
38 | 37 | 3expb 1199 |
. . . . . . . . . . 11
⊢ (((𝑀 gcd 𝑁) ∈ ℤ ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑀 gcd 𝑁) ∥ 𝑀 → (𝑀 gcd 𝑁) ∥ (𝑀 · 𝑁))) |
39 | 36, 38 | mpancom 420 |
. . . . . . . . . 10
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd 𝑁) ∥ 𝑀 → (𝑀 gcd 𝑁) ∥ (𝑀 · 𝑁))) |
40 | 34, 39 | mpd 13 |
. . . . . . . . 9
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ∥ (𝑀 · 𝑁)) |
41 | 21, 40 | syl 14 |
. . . . . . . 8
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 gcd 𝑁) ∥ (𝑀 · 𝑁)) |
42 | | gcdnncl 11922 |
. . . . . . . . 9
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 gcd 𝑁) ∈ ℕ) |
43 | | nndivdvds 11758 |
. . . . . . . . 9
⊢ (((𝑀 · 𝑁) ∈ ℕ ∧ (𝑀 gcd 𝑁) ∈ ℕ) → ((𝑀 gcd 𝑁) ∥ (𝑀 · 𝑁) ↔ ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∈ ℕ)) |
44 | 1, 42, 43 | syl2anc 409 |
. . . . . . . 8
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 gcd 𝑁) ∥ (𝑀 · 𝑁) ↔ ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∈ ℕ)) |
45 | 41, 44 | mpbid 146 |
. . . . . . 7
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∈ ℕ) |
46 | 45 | nnred 8891 |
. . . . . 6
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∈ ℝ) |
47 | 33 | simprd 113 |
. . . . . . . . . . . 12
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ∥ 𝑁) |
48 | 21, 47 | syl 14 |
. . . . . . . . . . 11
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 gcd 𝑁) ∥ 𝑁) |
49 | 21, 36 | syl 14 |
. . . . . . . . . . . 12
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 gcd 𝑁) ∈ ℤ) |
50 | 42 | nnne0d 8923 |
. . . . . . . . . . . 12
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 gcd 𝑁) ≠ 0) |
51 | | dvdsval2 11752 |
. . . . . . . . . . . 12
⊢ (((𝑀 gcd 𝑁) ∈ ℤ ∧ (𝑀 gcd 𝑁) ≠ 0 ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd 𝑁) ∥ 𝑁 ↔ (𝑁 / (𝑀 gcd 𝑁)) ∈ ℤ)) |
52 | 49, 50, 7, 51 | syl3anc 1233 |
. . . . . . . . . . 11
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 gcd 𝑁) ∥ 𝑁 ↔ (𝑁 / (𝑀 gcd 𝑁)) ∈ ℤ)) |
53 | 48, 52 | mpbid 146 |
. . . . . . . . . 10
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑁 / (𝑀 gcd 𝑁)) ∈ ℤ) |
54 | | dvdsmul1 11775 |
. . . . . . . . . 10
⊢ ((𝑀 ∈ ℤ ∧ (𝑁 / (𝑀 gcd 𝑁)) ∈ ℤ) → 𝑀 ∥ (𝑀 · (𝑁 / (𝑀 gcd 𝑁)))) |
55 | 4, 53, 54 | syl2anc 409 |
. . . . . . . . 9
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑀 ∥ (𝑀 · (𝑁 / (𝑀 gcd 𝑁)))) |
56 | | nncn 8886 |
. . . . . . . . . . 11
⊢ (𝑀 ∈ ℕ → 𝑀 ∈
ℂ) |
57 | 56 | adantr 274 |
. . . . . . . . . 10
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑀 ∈
ℂ) |
58 | | nncn 8886 |
. . . . . . . . . . 11
⊢ (𝑁 ∈ ℕ → 𝑁 ∈
ℂ) |
59 | 58 | adantl 275 |
. . . . . . . . . 10
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈
ℂ) |
60 | 42 | nncnd 8892 |
. . . . . . . . . 10
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 gcd 𝑁) ∈ ℂ) |
61 | 42 | nnap0d 8924 |
. . . . . . . . . 10
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 gcd 𝑁) # 0) |
62 | 57, 59, 60, 61 | divassapd 8743 |
. . . . . . . . 9
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) = (𝑀 · (𝑁 / (𝑀 gcd 𝑁)))) |
63 | 55, 62 | breqtrrd 4017 |
. . . . . . . 8
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑀 ∥ ((𝑀 · 𝑁) / (𝑀 gcd 𝑁))) |
64 | 21, 34 | syl 14 |
. . . . . . . . . . 11
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 gcd 𝑁) ∥ 𝑀) |
65 | | dvdsval2 11752 |
. . . . . . . . . . . 12
⊢ (((𝑀 gcd 𝑁) ∈ ℤ ∧ (𝑀 gcd 𝑁) ≠ 0 ∧ 𝑀 ∈ ℤ) → ((𝑀 gcd 𝑁) ∥ 𝑀 ↔ (𝑀 / (𝑀 gcd 𝑁)) ∈ ℤ)) |
66 | 49, 50, 4, 65 | syl3anc 1233 |
. . . . . . . . . . 11
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 gcd 𝑁) ∥ 𝑀 ↔ (𝑀 / (𝑀 gcd 𝑁)) ∈ ℤ)) |
67 | 64, 66 | mpbid 146 |
. . . . . . . . . 10
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 / (𝑀 gcd 𝑁)) ∈ ℤ) |
68 | | dvdsmul1 11775 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℤ ∧ (𝑀 / (𝑀 gcd 𝑁)) ∈ ℤ) → 𝑁 ∥ (𝑁 · (𝑀 / (𝑀 gcd 𝑁)))) |
69 | 7, 67, 68 | syl2anc 409 |
. . . . . . . . 9
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑁 ∥ (𝑁 · (𝑀 / (𝑀 gcd 𝑁)))) |
70 | 57, 59 | mulcomd 7941 |
. . . . . . . . . . 11
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 · 𝑁) = (𝑁 · 𝑀)) |
71 | 70 | oveq1d 5868 |
. . . . . . . . . 10
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) = ((𝑁 · 𝑀) / (𝑀 gcd 𝑁))) |
72 | 59, 57, 60, 61 | divassapd 8743 |
. . . . . . . . . 10
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑁 · 𝑀) / (𝑀 gcd 𝑁)) = (𝑁 · (𝑀 / (𝑀 gcd 𝑁)))) |
73 | 71, 72 | eqtrd 2203 |
. . . . . . . . 9
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) = (𝑁 · (𝑀 / (𝑀 gcd 𝑁)))) |
74 | 69, 73 | breqtrrd 4017 |
. . . . . . . 8
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑁 ∥ ((𝑀 · 𝑁) / (𝑀 gcd 𝑁))) |
75 | 63, 74 | jca 304 |
. . . . . . 7
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 ∥ ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∧ 𝑁 ∥ ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)))) |
76 | | breq2 3993 |
. . . . . . . . 9
⊢ (𝑥 = ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) → (𝑀 ∥ 𝑥 ↔ 𝑀 ∥ ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)))) |
77 | | breq2 3993 |
. . . . . . . . 9
⊢ (𝑥 = ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) → (𝑁 ∥ 𝑥 ↔ 𝑁 ∥ ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)))) |
78 | 76, 77 | anbi12d 470 |
. . . . . . . 8
⊢ (𝑥 = ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) → ((𝑀 ∥ 𝑥 ∧ 𝑁 ∥ 𝑥) ↔ (𝑀 ∥ ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∧ 𝑁 ∥ ((𝑀 · 𝑁) / (𝑀 gcd 𝑁))))) |
79 | 78 | elrab 2886 |
. . . . . . 7
⊢ (((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∈ {𝑥 ∈ ℕ ∣ (𝑀 ∥ 𝑥 ∧ 𝑁 ∥ 𝑥)} ↔ (((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∈ ℕ ∧ (𝑀 ∥ ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∧ 𝑁 ∥ ((𝑀 · 𝑁) / (𝑀 gcd 𝑁))))) |
80 | 45, 75, 79 | sylanbrc 415 |
. . . . . 6
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∈ {𝑥 ∈ ℕ ∣ (𝑀 ∥ 𝑥 ∧ 𝑁 ∥ 𝑥)}) |
81 | 46 | adantr 274 |
. . . . . . 7
⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ {𝑥 ∈ ℕ ∣ (𝑀 ∥ 𝑥 ∧ 𝑁 ∥ 𝑥)}) → ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∈ ℝ) |
82 | | elrabi 2883 |
. . . . . . . . 9
⊢ (𝑛 ∈ {𝑥 ∈ ℕ ∣ (𝑀 ∥ 𝑥 ∧ 𝑁 ∥ 𝑥)} → 𝑛 ∈ ℕ) |
83 | 82 | nnred 8891 |
. . . . . . . 8
⊢ (𝑛 ∈ {𝑥 ∈ ℕ ∣ (𝑀 ∥ 𝑥 ∧ 𝑁 ∥ 𝑥)} → 𝑛 ∈ ℝ) |
84 | 83 | adantl 275 |
. . . . . . 7
⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ {𝑥 ∈ ℕ ∣ (𝑀 ∥ 𝑥 ∧ 𝑁 ∥ 𝑥)}) → 𝑛 ∈ ℝ) |
85 | | breq2 3993 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑛 → (𝑀 ∥ 𝑥 ↔ 𝑀 ∥ 𝑛)) |
86 | | breq2 3993 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑛 → (𝑁 ∥ 𝑥 ↔ 𝑁 ∥ 𝑛)) |
87 | 85, 86 | anbi12d 470 |
. . . . . . . . 9
⊢ (𝑥 = 𝑛 → ((𝑀 ∥ 𝑥 ∧ 𝑁 ∥ 𝑥) ↔ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛))) |
88 | 87 | elrab 2886 |
. . . . . . . 8
⊢ (𝑛 ∈ {𝑥 ∈ ℕ ∣ (𝑀 ∥ 𝑥 ∧ 𝑁 ∥ 𝑥)} ↔ (𝑛 ∈ ℕ ∧ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛))) |
89 | | bezout 11966 |
. . . . . . . . . . . . 13
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) →
∃𝑥 ∈ ℤ
∃𝑦 ∈ ℤ
(𝑀 gcd 𝑁) = ((𝑀 · 𝑥) + (𝑁 · 𝑦))) |
90 | 21, 89 | syl 14 |
. . . . . . . . . . . 12
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) →
∃𝑥 ∈ ℤ
∃𝑦 ∈ ℤ
(𝑀 gcd 𝑁) = ((𝑀 · 𝑥) + (𝑁 · 𝑦))) |
91 | 90 | adantr 274 |
. . . . . . . . . . 11
⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛))) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ (𝑀 gcd 𝑁) = ((𝑀 · 𝑥) + (𝑁 · 𝑦))) |
92 | | nncn 8886 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℂ) |
93 | 92 | ad2antlr 486 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑛 ∈
ℂ) |
94 | 1 | nncnd 8892 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 · 𝑁) ∈ ℂ) |
95 | 94 | ad2antrr 485 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝑀 · 𝑁) ∈ ℂ) |
96 | 60 | ad2antrr 485 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝑀 gcd 𝑁) ∈ ℂ) |
97 | 57 | ad2antrr 485 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑀 ∈
ℂ) |
98 | 58 | ad3antlr 490 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑁 ∈
ℂ) |
99 | | simplll 528 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑀 ∈
ℕ) |
100 | 99 | nnap0d 8924 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑀 # 0) |
101 | | simpllr 529 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑁 ∈
ℕ) |
102 | 101 | nnap0d 8924 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑁 # 0) |
103 | 97, 98, 100, 102 | mulap0d 8576 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝑀 · 𝑁) # 0) |
104 | 61 | ad2antrr 485 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝑀 gcd 𝑁) # 0) |
105 | 93, 95, 96, 103, 104 | divdivap2d 8740 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝑛 / ((𝑀 · 𝑁) / (𝑀 gcd 𝑁))) = ((𝑛 · (𝑀 gcd 𝑁)) / (𝑀 · 𝑁))) |
106 | 105 | adantr 274 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝑀 ∈
ℕ ∧ 𝑁 ∈
ℕ) ∧ 𝑛 ∈
ℕ) ∧ (𝑥 ∈
ℤ ∧ 𝑦 ∈
ℤ)) ∧ (𝑀 gcd
𝑁) = ((𝑀 · 𝑥) + (𝑁 · 𝑦))) → (𝑛 / ((𝑀 · 𝑁) / (𝑀 gcd 𝑁))) = ((𝑛 · (𝑀 gcd 𝑁)) / (𝑀 · 𝑁))) |
107 | | oveq2 5861 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑀 gcd 𝑁) = ((𝑀 · 𝑥) + (𝑁 · 𝑦)) → (𝑛 · (𝑀 gcd 𝑁)) = (𝑛 · ((𝑀 · 𝑥) + (𝑁 · 𝑦)))) |
108 | 107 | oveq1d 5868 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑀 gcd 𝑁) = ((𝑀 · 𝑥) + (𝑁 · 𝑦)) → ((𝑛 · (𝑀 gcd 𝑁)) / (𝑀 · 𝑁)) = ((𝑛 · ((𝑀 · 𝑥) + (𝑁 · 𝑦))) / (𝑀 · 𝑁))) |
109 | | zcn 9217 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑥 ∈ ℤ → 𝑥 ∈
ℂ) |
110 | 109 | ad2antrl 487 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑥 ∈
ℂ) |
111 | 97, 110 | mulcld 7940 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝑀 · 𝑥) ∈ ℂ) |
112 | | zcn 9217 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑦 ∈ ℤ → 𝑦 ∈
ℂ) |
113 | 112 | ad2antll 488 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑦 ∈
ℂ) |
114 | 98, 113 | mulcld 7940 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝑁 · 𝑦) ∈ ℂ) |
115 | 93, 111, 114 | adddid 7944 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝑛 · ((𝑀 · 𝑥) + (𝑁 · 𝑦))) = ((𝑛 · (𝑀 · 𝑥)) + (𝑛 · (𝑁 · 𝑦)))) |
116 | 115 | oveq1d 5868 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝑛 · ((𝑀 · 𝑥) + (𝑁 · 𝑦))) / (𝑀 · 𝑁)) = (((𝑛 · (𝑀 · 𝑥)) + (𝑛 · (𝑁 · 𝑦))) / (𝑀 · 𝑁))) |
117 | 93, 111 | mulcld 7940 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝑛 · (𝑀 · 𝑥)) ∈ ℂ) |
118 | 93, 114 | mulcld 7940 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝑛 · (𝑁 · 𝑦)) ∈ ℂ) |
119 | 117, 118,
95, 103 | divdirapd 8746 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) →
(((𝑛 · (𝑀 · 𝑥)) + (𝑛 · (𝑁 · 𝑦))) / (𝑀 · 𝑁)) = (((𝑛 · (𝑀 · 𝑥)) / (𝑀 · 𝑁)) + ((𝑛 · (𝑁 · 𝑦)) / (𝑀 · 𝑁)))) |
120 | 116, 119 | eqtrd 2203 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝑛 · ((𝑀 · 𝑥) + (𝑁 · 𝑦))) / (𝑀 · 𝑁)) = (((𝑛 · (𝑀 · 𝑥)) / (𝑀 · 𝑁)) + ((𝑛 · (𝑁 · 𝑦)) / (𝑀 · 𝑁)))) |
121 | 108, 120 | sylan9eqr 2225 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝑀 ∈
ℕ ∧ 𝑁 ∈
ℕ) ∧ 𝑛 ∈
ℕ) ∧ (𝑥 ∈
ℤ ∧ 𝑦 ∈
ℤ)) ∧ (𝑀 gcd
𝑁) = ((𝑀 · 𝑥) + (𝑁 · 𝑦))) → ((𝑛 · (𝑀 gcd 𝑁)) / (𝑀 · 𝑁)) = (((𝑛 · (𝑀 · 𝑥)) / (𝑀 · 𝑁)) + ((𝑛 · (𝑁 · 𝑦)) / (𝑀 · 𝑁)))) |
122 | 93, 97, 110 | mul12d 8071 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝑛 · (𝑀 · 𝑥)) = (𝑀 · (𝑛 · 𝑥))) |
123 | 122 | oveq1d 5868 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝑛 · (𝑀 · 𝑥)) / (𝑀 · 𝑁)) = ((𝑀 · (𝑛 · 𝑥)) / (𝑀 · 𝑁))) |
124 | 93, 110 | mulcld 7940 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝑛 · 𝑥) ∈ ℂ) |
125 | 124, 98, 97, 102, 100 | divcanap5d 8734 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝑀 · (𝑛 · 𝑥)) / (𝑀 · 𝑁)) = ((𝑛 · 𝑥) / 𝑁)) |
126 | 123, 125 | eqtrd 2203 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝑛 · (𝑀 · 𝑥)) / (𝑀 · 𝑁)) = ((𝑛 · 𝑥) / 𝑁)) |
127 | 93, 98, 113 | mul12d 8071 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝑛 · (𝑁 · 𝑦)) = (𝑁 · (𝑛 · 𝑦))) |
128 | 127 | oveq1d 5868 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝑛 · (𝑁 · 𝑦)) / (𝑀 · 𝑁)) = ((𝑁 · (𝑛 · 𝑦)) / (𝑀 · 𝑁))) |
129 | 70 | ad2antrr 485 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝑀 · 𝑁) = (𝑁 · 𝑀)) |
130 | 129 | oveq2d 5869 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝑁 · (𝑛 · 𝑦)) / (𝑀 · 𝑁)) = ((𝑁 · (𝑛 · 𝑦)) / (𝑁 · 𝑀))) |
131 | 93, 113 | mulcld 7940 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝑛 · 𝑦) ∈ ℂ) |
132 | 131, 97, 98, 100, 102 | divcanap5d 8734 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝑁 · (𝑛 · 𝑦)) / (𝑁 · 𝑀)) = ((𝑛 · 𝑦) / 𝑀)) |
133 | 128, 130,
132 | 3eqtrd 2207 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝑛 · (𝑁 · 𝑦)) / (𝑀 · 𝑁)) = ((𝑛 · 𝑦) / 𝑀)) |
134 | 126, 133 | oveq12d 5871 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) →
(((𝑛 · (𝑀 · 𝑥)) / (𝑀 · 𝑁)) + ((𝑛 · (𝑁 · 𝑦)) / (𝑀 · 𝑁))) = (((𝑛 · 𝑥) / 𝑁) + ((𝑛 · 𝑦) / 𝑀))) |
135 | 134 | adantr 274 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝑀 ∈
ℕ ∧ 𝑁 ∈
ℕ) ∧ 𝑛 ∈
ℕ) ∧ (𝑥 ∈
ℤ ∧ 𝑦 ∈
ℤ)) ∧ (𝑀 gcd
𝑁) = ((𝑀 · 𝑥) + (𝑁 · 𝑦))) → (((𝑛 · (𝑀 · 𝑥)) / (𝑀 · 𝑁)) + ((𝑛 · (𝑁 · 𝑦)) / (𝑀 · 𝑁))) = (((𝑛 · 𝑥) / 𝑁) + ((𝑛 · 𝑦) / 𝑀))) |
136 | 106, 121,
135 | 3eqtrd 2207 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝑀 ∈
ℕ ∧ 𝑁 ∈
ℕ) ∧ 𝑛 ∈
ℕ) ∧ (𝑥 ∈
ℤ ∧ 𝑦 ∈
ℤ)) ∧ (𝑀 gcd
𝑁) = ((𝑀 · 𝑥) + (𝑁 · 𝑦))) → (𝑛 / ((𝑀 · 𝑁) / (𝑀 gcd 𝑁))) = (((𝑛 · 𝑥) / 𝑁) + ((𝑛 · 𝑦) / 𝑀))) |
137 | 136 | ex 114 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝑀 gcd 𝑁) = ((𝑀 · 𝑥) + (𝑁 · 𝑦)) → (𝑛 / ((𝑀 · 𝑁) / (𝑀 gcd 𝑁))) = (((𝑛 · 𝑥) / 𝑁) + ((𝑛 · 𝑦) / 𝑀)))) |
138 | 137 | adantlrr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛))) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝑀 gcd 𝑁) = ((𝑀 · 𝑥) + (𝑁 · 𝑦)) → (𝑛 / ((𝑀 · 𝑁) / (𝑀 gcd 𝑁))) = (((𝑛 · 𝑥) / 𝑁) + ((𝑛 · 𝑦) / 𝑀)))) |
139 | 138 | imp 123 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝑀 ∈
ℕ ∧ 𝑁 ∈
ℕ) ∧ (𝑛 ∈
ℕ ∧ (𝑀 ∥
𝑛 ∧ 𝑁 ∥ 𝑛))) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑀 gcd 𝑁) = ((𝑀 · 𝑥) + (𝑁 · 𝑦))) → (𝑛 / ((𝑀 · 𝑁) / (𝑀 gcd 𝑁))) = (((𝑛 · 𝑥) / 𝑁) + ((𝑛 · 𝑦) / 𝑀))) |
140 | 6 | ad3antlr 490 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑁 ∈
ℤ) |
141 | | nnz 9231 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℤ) |
142 | 141 | ad2antlr 486 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑛 ∈
ℤ) |
143 | | simprl 526 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑥 ∈
ℤ) |
144 | | dvdsmultr1 11793 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑥 ∈ ℤ) → (𝑁 ∥ 𝑛 → 𝑁 ∥ (𝑛 · 𝑥))) |
145 | 140, 142,
143, 144 | syl3anc 1233 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝑁 ∥ 𝑛 → 𝑁 ∥ (𝑛 · 𝑥))) |
146 | 24 | ad3antlr 490 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑁 ≠ 0) |
147 | 142, 143 | zmulcld 9340 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝑛 · 𝑥) ∈ ℤ) |
148 | | dvdsval2 11752 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ∧ (𝑛 · 𝑥) ∈ ℤ) → (𝑁 ∥ (𝑛 · 𝑥) ↔ ((𝑛 · 𝑥) / 𝑁) ∈ ℤ)) |
149 | 140, 146,
147, 148 | syl3anc 1233 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝑁 ∥ (𝑛 · 𝑥) ↔ ((𝑛 · 𝑥) / 𝑁) ∈ ℤ)) |
150 | 145, 149 | sylibd 148 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝑁 ∥ 𝑛 → ((𝑛 · 𝑥) / 𝑁) ∈ ℤ)) |
151 | 150 | adantld 276 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛) → ((𝑛 · 𝑥) / 𝑁) ∈ ℤ)) |
152 | 151 | 3impia 1195 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)) → ((𝑛 · 𝑥) / 𝑁) ∈ ℤ) |
153 | 3 | ad3antrrr 489 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑀 ∈
ℤ) |
154 | | simprr 527 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑦 ∈
ℤ) |
155 | | dvdsmultr1 11793 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑀 ∥ 𝑛 → 𝑀 ∥ (𝑛 · 𝑦))) |
156 | 153, 142,
154, 155 | syl3anc 1233 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝑀 ∥ 𝑛 → 𝑀 ∥ (𝑛 · 𝑦))) |
157 | 22 | ad3antrrr 489 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑀 ≠ 0) |
158 | 142, 154 | zmulcld 9340 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝑛 · 𝑦) ∈ ℤ) |
159 | | dvdsval2 11752 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0 ∧ (𝑛 · 𝑦) ∈ ℤ) → (𝑀 ∥ (𝑛 · 𝑦) ↔ ((𝑛 · 𝑦) / 𝑀) ∈ ℤ)) |
160 | 153, 157,
158, 159 | syl3anc 1233 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝑀 ∥ (𝑛 · 𝑦) ↔ ((𝑛 · 𝑦) / 𝑀) ∈ ℤ)) |
161 | 156, 160 | sylibd 148 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝑀 ∥ 𝑛 → ((𝑛 · 𝑦) / 𝑀) ∈ ℤ)) |
162 | 161 | adantrd 277 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛) → ((𝑛 · 𝑦) / 𝑀) ∈ ℤ)) |
163 | 162 | 3impia 1195 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)) → ((𝑛 · 𝑦) / 𝑀) ∈ ℤ) |
164 | 152, 163 | zaddcld 9338 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)) → (((𝑛 · 𝑥) / 𝑁) + ((𝑛 · 𝑦) / 𝑀)) ∈ ℤ) |
165 | 164 | 3expia 1200 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛) → (((𝑛 · 𝑥) / 𝑁) + ((𝑛 · 𝑦) / 𝑀)) ∈ ℤ)) |
166 | 165 | an32s 563 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ 𝑛 ∈ ℕ) → ((𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛) → (((𝑛 · 𝑥) / 𝑁) + ((𝑛 · 𝑦) / 𝑀)) ∈ ℤ)) |
167 | 166 | impr 377 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑛 ∈ ℕ ∧ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛))) → (((𝑛 · 𝑥) / 𝑁) + ((𝑛 · 𝑦) / 𝑀)) ∈ ℤ) |
168 | 167 | an32s 563 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛))) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (((𝑛 · 𝑥) / 𝑁) + ((𝑛 · 𝑦) / 𝑀)) ∈ ℤ) |
169 | 168 | adantr 274 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝑀 ∈
ℕ ∧ 𝑁 ∈
ℕ) ∧ (𝑛 ∈
ℕ ∧ (𝑀 ∥
𝑛 ∧ 𝑁 ∥ 𝑛))) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑀 gcd 𝑁) = ((𝑀 · 𝑥) + (𝑁 · 𝑦))) → (((𝑛 · 𝑥) / 𝑁) + ((𝑛 · 𝑦) / 𝑀)) ∈ ℤ) |
170 | 139, 169 | eqeltrd 2247 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝑀 ∈
ℕ ∧ 𝑁 ∈
ℕ) ∧ (𝑛 ∈
ℕ ∧ (𝑀 ∥
𝑛 ∧ 𝑁 ∥ 𝑛))) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑀 gcd 𝑁) = ((𝑀 · 𝑥) + (𝑁 · 𝑦))) → (𝑛 / ((𝑀 · 𝑁) / (𝑀 gcd 𝑁))) ∈ ℤ) |
171 | 45 | nnzd 9333 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∈ ℤ) |
172 | 171 | ad2antrr 485 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛))) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∈ ℤ) |
173 | 45 | nnne0d 8923 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ≠ 0) |
174 | 173 | ad2antrr 485 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛))) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ≠ 0) |
175 | 142 | adantlrr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛))) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑛 ∈ ℤ) |
176 | | dvdsval2 11752 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∈ ℤ ∧ ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ≠ 0 ∧ 𝑛 ∈ ℤ) → (((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∥ 𝑛 ↔ (𝑛 / ((𝑀 · 𝑁) / (𝑀 gcd 𝑁))) ∈ ℤ)) |
177 | 172, 174,
175, 176 | syl3anc 1233 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛))) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∥ 𝑛 ↔ (𝑛 / ((𝑀 · 𝑁) / (𝑀 gcd 𝑁))) ∈ ℤ)) |
178 | 177 | adantr 274 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝑀 ∈
ℕ ∧ 𝑁 ∈
ℕ) ∧ (𝑛 ∈
ℕ ∧ (𝑀 ∥
𝑛 ∧ 𝑁 ∥ 𝑛))) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑀 gcd 𝑁) = ((𝑀 · 𝑥) + (𝑁 · 𝑦))) → (((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∥ 𝑛 ↔ (𝑛 / ((𝑀 · 𝑁) / (𝑀 gcd 𝑁))) ∈ ℤ)) |
179 | 170, 178 | mpbird 166 |
. . . . . . . . . . . . . . 15
⊢
(((((𝑀 ∈
ℕ ∧ 𝑁 ∈
ℕ) ∧ (𝑛 ∈
ℕ ∧ (𝑀 ∥
𝑛 ∧ 𝑁 ∥ 𝑛))) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑀 gcd 𝑁) = ((𝑀 · 𝑥) + (𝑁 · 𝑦))) → ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∥ 𝑛) |
180 | 179 | ex 114 |
. . . . . . . . . . . . . 14
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛))) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝑀 gcd 𝑁) = ((𝑀 · 𝑥) + (𝑁 · 𝑦)) → ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∥ 𝑛)) |
181 | 180 | anassrs 398 |
. . . . . . . . . . . . 13
⊢
(((((𝑀 ∈
ℕ ∧ 𝑁 ∈
ℕ) ∧ (𝑛 ∈
ℕ ∧ (𝑀 ∥
𝑛 ∧ 𝑁 ∥ 𝑛))) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℤ) → ((𝑀 gcd 𝑁) = ((𝑀 · 𝑥) + (𝑁 · 𝑦)) → ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∥ 𝑛)) |
182 | 181 | reximdva 2572 |
. . . . . . . . . . . 12
⊢ ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛))) ∧ 𝑥 ∈ ℤ) → (∃𝑦 ∈ ℤ (𝑀 gcd 𝑁) = ((𝑀 · 𝑥) + (𝑁 · 𝑦)) → ∃𝑦 ∈ ℤ ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∥ 𝑛)) |
183 | 182 | reximdva 2572 |
. . . . . . . . . . 11
⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛))) → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ (𝑀 gcd 𝑁) = ((𝑀 · 𝑥) + (𝑁 · 𝑦)) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∥ 𝑛)) |
184 | 91, 183 | mpd 13 |
. . . . . . . . . 10
⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛))) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∥ 𝑛) |
185 | | 1z 9238 |
. . . . . . . . . . . 12
⊢ 1 ∈
ℤ |
186 | | elex2 2746 |
. . . . . . . . . . . 12
⊢ (1 ∈
ℤ → ∃𝑤
𝑤 ∈
ℤ) |
187 | | r19.9rmv 3506 |
. . . . . . . . . . . 12
⊢
(∃𝑤 𝑤 ∈ ℤ → (((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∥ 𝑛 ↔ ∃𝑦 ∈ ℤ ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∥ 𝑛)) |
188 | 185, 186,
187 | mp2b 8 |
. . . . . . . . . . 11
⊢ (((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∥ 𝑛 ↔ ∃𝑦 ∈ ℤ ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∥ 𝑛) |
189 | | r19.9rmv 3506 |
. . . . . . . . . . . 12
⊢
(∃𝑤 𝑤 ∈ ℤ →
(∃𝑦 ∈ ℤ
((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∥ 𝑛 ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∥ 𝑛)) |
190 | 185, 186,
189 | mp2b 8 |
. . . . . . . . . . 11
⊢
(∃𝑦 ∈
ℤ ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∥ 𝑛 ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∥ 𝑛) |
191 | 188, 190 | bitri 183 |
. . . . . . . . . 10
⊢ (((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∥ 𝑛 ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∥ 𝑛) |
192 | 184, 191 | sylibr 133 |
. . . . . . . . 9
⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛))) → ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∥ 𝑛) |
193 | 171 | adantr 274 |
. . . . . . . . . 10
⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛))) → ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∈ ℤ) |
194 | | simprl 526 |
. . . . . . . . . 10
⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛))) → 𝑛 ∈ ℕ) |
195 | | dvdsle 11804 |
. . . . . . . . . 10
⊢ ((((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∈ ℤ ∧ 𝑛 ∈ ℕ) → (((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∥ 𝑛 → ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ≤ 𝑛)) |
196 | 193, 194,
195 | syl2anc 409 |
. . . . . . . . 9
⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛))) → (((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∥ 𝑛 → ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ≤ 𝑛)) |
197 | 192, 196 | mpd 13 |
. . . . . . . 8
⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛))) → ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ≤ 𝑛) |
198 | 88, 197 | sylan2b 285 |
. . . . . . 7
⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ {𝑥 ∈ ℕ ∣ (𝑀 ∥ 𝑥 ∧ 𝑁 ∥ 𝑥)}) → ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ≤ 𝑛) |
199 | 81, 84, 198 | lensymd 8041 |
. . . . . 6
⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑛 ∈ {𝑥 ∈ ℕ ∣ (𝑀 ∥ 𝑥 ∧ 𝑁 ∥ 𝑥)}) → ¬ 𝑛 < ((𝑀 · 𝑁) / (𝑀 gcd 𝑁))) |
200 | 32, 46, 80, 199 | infminti 7004 |
. . . . 5
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) →
inf({𝑥 ∈ ℕ
∣ (𝑀 ∥ 𝑥 ∧ 𝑁 ∥ 𝑥)}, ℝ, < ) = ((𝑀 · 𝑁) / (𝑀 gcd 𝑁))) |
201 | 30, 200 | eqtr2d 2204 |
. . . 4
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) = (𝑀 lcm 𝑁)) |
202 | 201, 45 | eqeltrrd 2248 |
. . . . . 6
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 lcm 𝑁) ∈ ℕ) |
203 | 202 | nncnd 8892 |
. . . . 5
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 lcm 𝑁) ∈ ℂ) |
204 | 94, 203, 60, 61 | divmulap3d 8742 |
. . . 4
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) = (𝑀 lcm 𝑁) ↔ (𝑀 · 𝑁) = ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)))) |
205 | 201, 204 | mpbid 146 |
. . 3
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 · 𝑁) = ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁))) |
206 | 20, 205 | eqtr2d 2204 |
. 2
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁))) |
207 | | simprl 526 |
. . . 4
⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝐾 ∈ ℕ ∧ (𝑀 ∥ 𝐾 ∧ 𝑁 ∥ 𝐾))) → 𝐾 ∈ ℕ) |
208 | | eleq1 2233 |
. . . . . . . 8
⊢ (𝑛 = 𝐾 → (𝑛 ∈ ℕ ↔ 𝐾 ∈ ℕ)) |
209 | | breq2 3993 |
. . . . . . . . 9
⊢ (𝑛 = 𝐾 → (𝑀 ∥ 𝑛 ↔ 𝑀 ∥ 𝐾)) |
210 | | breq2 3993 |
. . . . . . . . 9
⊢ (𝑛 = 𝐾 → (𝑁 ∥ 𝑛 ↔ 𝑁 ∥ 𝐾)) |
211 | 209, 210 | anbi12d 470 |
. . . . . . . 8
⊢ (𝑛 = 𝐾 → ((𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛) ↔ (𝑀 ∥ 𝐾 ∧ 𝑁 ∥ 𝐾))) |
212 | 208, 211 | anbi12d 470 |
. . . . . . 7
⊢ (𝑛 = 𝐾 → ((𝑛 ∈ ℕ ∧ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)) ↔ (𝐾 ∈ ℕ ∧ (𝑀 ∥ 𝐾 ∧ 𝑁 ∥ 𝐾)))) |
213 | 212 | anbi2d 461 |
. . . . . 6
⊢ (𝑛 = 𝐾 → (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛))) ↔ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝐾 ∈ ℕ ∧ (𝑀 ∥ 𝐾 ∧ 𝑁 ∥ 𝐾))))) |
214 | | breq2 3993 |
. . . . . 6
⊢ (𝑛 = 𝐾 → ((𝑀 lcm 𝑁) ∥ 𝑛 ↔ (𝑀 lcm 𝑁) ∥ 𝐾)) |
215 | 213, 214 | imbi12d 233 |
. . . . 5
⊢ (𝑛 = 𝐾 → ((((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛))) → (𝑀 lcm 𝑁) ∥ 𝑛) ↔ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝐾 ∈ ℕ ∧ (𝑀 ∥ 𝐾 ∧ 𝑁 ∥ 𝐾))) → (𝑀 lcm 𝑁) ∥ 𝐾))) |
216 | 201 | breq1d 3999 |
. . . . . . 7
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∥ 𝑛 ↔ (𝑀 lcm 𝑁) ∥ 𝑛)) |
217 | 216 | adantr 274 |
. . . . . 6
⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛))) → (((𝑀 · 𝑁) / (𝑀 gcd 𝑁)) ∥ 𝑛 ↔ (𝑀 lcm 𝑁) ∥ 𝑛)) |
218 | 192, 217 | mpbid 146 |
. . . . 5
⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑛 ∈ ℕ ∧ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛))) → (𝑀 lcm 𝑁) ∥ 𝑛) |
219 | 215, 218 | vtoclg 2790 |
. . . 4
⊢ (𝐾 ∈ ℕ → (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝐾 ∈ ℕ ∧ (𝑀 ∥ 𝐾 ∧ 𝑁 ∥ 𝐾))) → (𝑀 lcm 𝑁) ∥ 𝐾)) |
220 | 207, 219 | mpcom 36 |
. . 3
⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝐾 ∈ ℕ ∧ (𝑀 ∥ 𝐾 ∧ 𝑁 ∥ 𝐾))) → (𝑀 lcm 𝑁) ∥ 𝐾) |
221 | 220 | ex 114 |
. 2
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐾 ∈ ℕ ∧ (𝑀 ∥ 𝐾 ∧ 𝑁 ∥ 𝐾)) → (𝑀 lcm 𝑁) ∥ 𝐾)) |
222 | 206, 221 | jca 304 |
1
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁)) ∧ ((𝐾 ∈ ℕ ∧ (𝑀 ∥ 𝐾 ∧ 𝑁 ∥ 𝐾)) → (𝑀 lcm 𝑁) ∥ 𝐾))) |