Proof of Theorem 3lcm2e6woprm
Step | Hyp | Ref
| Expression |
1 | | 3cn 12052 |
. . . 4
⊢ 3 ∈
ℂ |
2 | | 2cn 12046 |
. . . 4
⊢ 2 ∈
ℂ |
3 | 1, 2 | mulcli 10980 |
. . 3
⊢ (3
· 2) ∈ ℂ |
4 | | 3z 12351 |
. . . 4
⊢ 3 ∈
ℤ |
5 | | 2z 12350 |
. . . 4
⊢ 2 ∈
ℤ |
6 | | lcmcl 16304 |
. . . . 5
⊢ ((3
∈ ℤ ∧ 2 ∈ ℤ) → (3 lcm 2) ∈
ℕ0) |
7 | 6 | nn0cnd 12293 |
. . . 4
⊢ ((3
∈ ℤ ∧ 2 ∈ ℤ) → (3 lcm 2) ∈
ℂ) |
8 | 4, 5, 7 | mp2an 689 |
. . 3
⊢ (3 lcm 2)
∈ ℂ |
9 | 4, 5 | pm3.2i 471 |
. . . . 5
⊢ (3 ∈
ℤ ∧ 2 ∈ ℤ) |
10 | | 2ne0 12075 |
. . . . . . 7
⊢ 2 ≠
0 |
11 | 10 | neii 2945 |
. . . . . 6
⊢ ¬ 2
= 0 |
12 | 11 | intnan 487 |
. . . . 5
⊢ ¬ (3
= 0 ∧ 2 = 0) |
13 | | gcdn0cl 16207 |
. . . . . 6
⊢ (((3
∈ ℤ ∧ 2 ∈ ℤ) ∧ ¬ (3 = 0 ∧ 2 = 0)) →
(3 gcd 2) ∈ ℕ) |
14 | 13 | nncnd 11987 |
. . . . 5
⊢ (((3
∈ ℤ ∧ 2 ∈ ℤ) ∧ ¬ (3 = 0 ∧ 2 = 0)) →
(3 gcd 2) ∈ ℂ) |
15 | 9, 12, 14 | mp2an 689 |
. . . 4
⊢ (3 gcd 2)
∈ ℂ |
16 | 9, 12, 13 | mp2an 689 |
. . . . 5
⊢ (3 gcd 2)
∈ ℕ |
17 | 16 | nnne0i 12011 |
. . . 4
⊢ (3 gcd 2)
≠ 0 |
18 | 15, 17 | pm3.2i 471 |
. . 3
⊢ ((3 gcd
2) ∈ ℂ ∧ (3 gcd 2) ≠ 0) |
19 | | 3nn 12050 |
. . . . . . 7
⊢ 3 ∈
ℕ |
20 | | 2nn 12044 |
. . . . . . 7
⊢ 2 ∈
ℕ |
21 | 19, 20 | pm3.2i 471 |
. . . . . 6
⊢ (3 ∈
ℕ ∧ 2 ∈ ℕ) |
22 | | lcmgcdnn 16314 |
. . . . . . 7
⊢ ((3
∈ ℕ ∧ 2 ∈ ℕ) → ((3 lcm 2) · (3 gcd 2)) =
(3 · 2)) |
23 | 22 | eqcomd 2744 |
. . . . . 6
⊢ ((3
∈ ℕ ∧ 2 ∈ ℕ) → (3 · 2) = ((3 lcm 2)
· (3 gcd 2))) |
24 | 21, 23 | mp1i 13 |
. . . . 5
⊢ (((3
· 2) ∈ ℂ ∧ (3 lcm 2) ∈ ℂ ∧ ((3 gcd 2)
∈ ℂ ∧ (3 gcd 2) ≠ 0)) → (3 · 2) = ((3 lcm 2)
· (3 gcd 2))) |
25 | | divmul3 11636 |
. . . . 5
⊢ (((3
· 2) ∈ ℂ ∧ (3 lcm 2) ∈ ℂ ∧ ((3 gcd 2)
∈ ℂ ∧ (3 gcd 2) ≠ 0)) → (((3 · 2) / (3 gcd 2)) =
(3 lcm 2) ↔ (3 · 2) = ((3 lcm 2) · (3 gcd
2)))) |
26 | 24, 25 | mpbird 256 |
. . . 4
⊢ (((3
· 2) ∈ ℂ ∧ (3 lcm 2) ∈ ℂ ∧ ((3 gcd 2)
∈ ℂ ∧ (3 gcd 2) ≠ 0)) → ((3 · 2) / (3 gcd 2)) =
(3 lcm 2)) |
27 | 26 | eqcomd 2744 |
. . 3
⊢ (((3
· 2) ∈ ℂ ∧ (3 lcm 2) ∈ ℂ ∧ ((3 gcd 2)
∈ ℂ ∧ (3 gcd 2) ≠ 0)) → (3 lcm 2) = ((3 · 2) / (3
gcd 2))) |
28 | 3, 8, 18, 27 | mp3an 1460 |
. 2
⊢ (3 lcm 2)
= ((3 · 2) / (3 gcd 2)) |
29 | | gcdcom 16218 |
. . . . 5
⊢ ((3
∈ ℤ ∧ 2 ∈ ℤ) → (3 gcd 2) = (2 gcd
3)) |
30 | 4, 5, 29 | mp2an 689 |
. . . 4
⊢ (3 gcd 2)
= (2 gcd 3) |
31 | | 1z 12348 |
. . . . . . . . 9
⊢ 1 ∈
ℤ |
32 | | gcdid 16232 |
. . . . . . . . 9
⊢ (1 ∈
ℤ → (1 gcd 1) = (abs‘1)) |
33 | 31, 32 | ax-mp 5 |
. . . . . . . 8
⊢ (1 gcd 1)
= (abs‘1) |
34 | | abs1 15007 |
. . . . . . . 8
⊢
(abs‘1) = 1 |
35 | 33, 34 | eqtr2i 2767 |
. . . . . . 7
⊢ 1 = (1
gcd 1) |
36 | | gcdadd 16231 |
. . . . . . . 8
⊢ ((1
∈ ℤ ∧ 1 ∈ ℤ) → (1 gcd 1) = (1 gcd (1 +
1))) |
37 | 31, 31, 36 | mp2an 689 |
. . . . . . 7
⊢ (1 gcd 1)
= (1 gcd (1 + 1)) |
38 | | 1p1e2 12096 |
. . . . . . . 8
⊢ (1 + 1) =
2 |
39 | 38 | oveq2i 7288 |
. . . . . . 7
⊢ (1 gcd (1
+ 1)) = (1 gcd 2) |
40 | 35, 37, 39 | 3eqtri 2770 |
. . . . . 6
⊢ 1 = (1
gcd 2) |
41 | | gcdcom 16218 |
. . . . . . 7
⊢ ((1
∈ ℤ ∧ 2 ∈ ℤ) → (1 gcd 2) = (2 gcd
1)) |
42 | 31, 5, 41 | mp2an 689 |
. . . . . 6
⊢ (1 gcd 2)
= (2 gcd 1) |
43 | | gcdadd 16231 |
. . . . . . 7
⊢ ((2
∈ ℤ ∧ 1 ∈ ℤ) → (2 gcd 1) = (2 gcd (1 +
2))) |
44 | 5, 31, 43 | mp2an 689 |
. . . . . 6
⊢ (2 gcd 1)
= (2 gcd (1 + 2)) |
45 | 40, 42, 44 | 3eqtri 2770 |
. . . . 5
⊢ 1 = (2
gcd (1 + 2)) |
46 | | 1p2e3 12114 |
. . . . . 6
⊢ (1 + 2) =
3 |
47 | 46 | oveq2i 7288 |
. . . . 5
⊢ (2 gcd (1
+ 2)) = (2 gcd 3) |
48 | 45, 47 | eqtr2i 2767 |
. . . 4
⊢ (2 gcd 3)
= 1 |
49 | 30, 48 | eqtri 2766 |
. . 3
⊢ (3 gcd 2)
= 1 |
50 | 49 | oveq2i 7288 |
. 2
⊢ ((3
· 2) / (3 gcd 2)) = ((3 · 2) / 1) |
51 | | 3t2e6 12137 |
. . . 4
⊢ (3
· 2) = 6 |
52 | 51 | oveq1i 7287 |
. . 3
⊢ ((3
· 2) / 1) = (6 / 1) |
53 | | 6cn 12062 |
. . . 4
⊢ 6 ∈
ℂ |
54 | 53 | div1i 11701 |
. . 3
⊢ (6 / 1) =
6 |
55 | 52, 54 | eqtri 2766 |
. 2
⊢ ((3
· 2) / 1) = 6 |
56 | 28, 50, 55 | 3eqtri 2770 |
1
⊢ (3 lcm 2)
= 6 |