Proof of Theorem 3lcm2e6woprm
Step | Hyp | Ref
| Expression |
1 | | 3cn 8908 |
. . . 4
⊢ 3 ∈
ℂ |
2 | | 2cn 8904 |
. . . 4
⊢ 2 ∈
ℂ |
3 | 1, 2 | mulcli 7883 |
. . 3
⊢ (3
· 2) ∈ ℂ |
4 | | 3z 9196 |
. . . 4
⊢ 3 ∈
ℤ |
5 | | 2z 9195 |
. . . 4
⊢ 2 ∈
ℤ |
6 | | lcmcl 11949 |
. . . . 5
⊢ ((3
∈ ℤ ∧ 2 ∈ ℤ) → (3 lcm 2) ∈
ℕ0) |
7 | 6 | nn0cnd 9145 |
. . . 4
⊢ ((3
∈ ℤ ∧ 2 ∈ ℤ) → (3 lcm 2) ∈
ℂ) |
8 | 4, 5, 7 | mp2an 423 |
. . 3
⊢ (3 lcm 2)
∈ ℂ |
9 | 4, 5 | pm3.2i 270 |
. . . . 5
⊢ (3 ∈
ℤ ∧ 2 ∈ ℤ) |
10 | | 2ne0 8925 |
. . . . . . 7
⊢ 2 ≠
0 |
11 | 10 | neii 2329 |
. . . . . 6
⊢ ¬ 2
= 0 |
12 | 11 | intnan 915 |
. . . . 5
⊢ ¬ (3
= 0 ∧ 2 = 0) |
13 | | gcdn0cl 11846 |
. . . . . 6
⊢ (((3
∈ ℤ ∧ 2 ∈ ℤ) ∧ ¬ (3 = 0 ∧ 2 = 0)) →
(3 gcd 2) ∈ ℕ) |
14 | 13 | nncnd 8847 |
. . . . 5
⊢ (((3
∈ ℤ ∧ 2 ∈ ℤ) ∧ ¬ (3 = 0 ∧ 2 = 0)) →
(3 gcd 2) ∈ ℂ) |
15 | 9, 12, 14 | mp2an 423 |
. . . 4
⊢ (3 gcd 2)
∈ ℂ |
16 | 9, 12, 13 | mp2an 423 |
. . . . . 6
⊢ (3 gcd 2)
∈ ℕ |
17 | 16 | nnne0i 8865 |
. . . . 5
⊢ (3 gcd 2)
≠ 0 |
18 | 16 | nnzi 9188 |
. . . . . 6
⊢ (3 gcd 2)
∈ ℤ |
19 | | 0z 9178 |
. . . . . 6
⊢ 0 ∈
ℤ |
20 | | zapne 9238 |
. . . . . 6
⊢ (((3 gcd
2) ∈ ℤ ∧ 0 ∈ ℤ) → ((3 gcd 2) # 0 ↔ (3 gcd
2) ≠ 0)) |
21 | 18, 19, 20 | mp2an 423 |
. . . . 5
⊢ ((3 gcd
2) # 0 ↔ (3 gcd 2) ≠ 0) |
22 | 17, 21 | mpbir 145 |
. . . 4
⊢ (3 gcd 2)
# 0 |
23 | 15, 22 | pm3.2i 270 |
. . 3
⊢ ((3 gcd
2) ∈ ℂ ∧ (3 gcd 2) # 0) |
24 | | 3nn 8995 |
. . . . . . 7
⊢ 3 ∈
ℕ |
25 | | 2nn 8994 |
. . . . . . 7
⊢ 2 ∈
ℕ |
26 | 24, 25 | pm3.2i 270 |
. . . . . 6
⊢ (3 ∈
ℕ ∧ 2 ∈ ℕ) |
27 | | lcmgcdnn 11959 |
. . . . . . 7
⊢ ((3
∈ ℕ ∧ 2 ∈ ℕ) → ((3 lcm 2) · (3 gcd 2)) =
(3 · 2)) |
28 | 27 | eqcomd 2163 |
. . . . . 6
⊢ ((3
∈ ℕ ∧ 2 ∈ ℕ) → (3 · 2) = ((3 lcm 2)
· (3 gcd 2))) |
29 | 26, 28 | mp1i 10 |
. . . . 5
⊢ (((3
· 2) ∈ ℂ ∧ (3 lcm 2) ∈ ℂ ∧ ((3 gcd 2)
∈ ℂ ∧ (3 gcd 2) # 0)) → (3 · 2) = ((3 lcm 2)
· (3 gcd 2))) |
30 | | divmulap3 8550 |
. . . . 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)))) |
31 | 29, 30 | mpbird 166 |
. . . 4
⊢ (((3
· 2) ∈ ℂ ∧ (3 lcm 2) ∈ ℂ ∧ ((3 gcd 2)
∈ ℂ ∧ (3 gcd 2) # 0)) → ((3 · 2) / (3 gcd 2)) = (3
lcm 2)) |
32 | 31 | eqcomd 2163 |
. . 3
⊢ (((3
· 2) ∈ ℂ ∧ (3 lcm 2) ∈ ℂ ∧ ((3 gcd 2)
∈ ℂ ∧ (3 gcd 2) # 0)) → (3 lcm 2) = ((3 · 2) / (3
gcd 2))) |
33 | 3, 8, 23, 32 | mp3an 1319 |
. 2
⊢ (3 lcm 2)
= ((3 · 2) / (3 gcd 2)) |
34 | | gcdcom 11857 |
. . . . 5
⊢ ((3
∈ ℤ ∧ 2 ∈ ℤ) → (3 gcd 2) = (2 gcd
3)) |
35 | 4, 5, 34 | mp2an 423 |
. . . 4
⊢ (3 gcd 2)
= (2 gcd 3) |
36 | | 1z 9193 |
. . . . . . . . 9
⊢ 1 ∈
ℤ |
37 | | gcdid 11870 |
. . . . . . . . 9
⊢ (1 ∈
ℤ → (1 gcd 1) = (abs‘1)) |
38 | 36, 37 | ax-mp 5 |
. . . . . . . 8
⊢ (1 gcd 1)
= (abs‘1) |
39 | | abs1 10972 |
. . . . . . . 8
⊢
(abs‘1) = 1 |
40 | 38, 39 | eqtr2i 2179 |
. . . . . . 7
⊢ 1 = (1
gcd 1) |
41 | | gcdadd 11869 |
. . . . . . . 8
⊢ ((1
∈ ℤ ∧ 1 ∈ ℤ) → (1 gcd 1) = (1 gcd (1 +
1))) |
42 | 36, 36, 41 | mp2an 423 |
. . . . . . 7
⊢ (1 gcd 1)
= (1 gcd (1 + 1)) |
43 | | 1p1e2 8950 |
. . . . . . . 8
⊢ (1 + 1) =
2 |
44 | 43 | oveq2i 5835 |
. . . . . . 7
⊢ (1 gcd (1
+ 1)) = (1 gcd 2) |
45 | 40, 42, 44 | 3eqtri 2182 |
. . . . . 6
⊢ 1 = (1
gcd 2) |
46 | | gcdcom 11857 |
. . . . . . 7
⊢ ((1
∈ ℤ ∧ 2 ∈ ℤ) → (1 gcd 2) = (2 gcd
1)) |
47 | 36, 5, 46 | mp2an 423 |
. . . . . 6
⊢ (1 gcd 2)
= (2 gcd 1) |
48 | | gcdadd 11869 |
. . . . . . 7
⊢ ((2
∈ ℤ ∧ 1 ∈ ℤ) → (2 gcd 1) = (2 gcd (1 +
2))) |
49 | 5, 36, 48 | mp2an 423 |
. . . . . 6
⊢ (2 gcd 1)
= (2 gcd (1 + 2)) |
50 | 45, 47, 49 | 3eqtri 2182 |
. . . . 5
⊢ 1 = (2
gcd (1 + 2)) |
51 | | 1p2e3 8967 |
. . . . . 6
⊢ (1 + 2) =
3 |
52 | 51 | oveq2i 5835 |
. . . . 5
⊢ (2 gcd (1
+ 2)) = (2 gcd 3) |
53 | 50, 52 | eqtr2i 2179 |
. . . 4
⊢ (2 gcd 3)
= 1 |
54 | 35, 53 | eqtri 2178 |
. . 3
⊢ (3 gcd 2)
= 1 |
55 | 54 | oveq2i 5835 |
. 2
⊢ ((3
· 2) / (3 gcd 2)) = ((3 · 2) / 1) |
56 | | 3t2e6 8989 |
. . . 4
⊢ (3
· 2) = 6 |
57 | 56 | oveq1i 5834 |
. . 3
⊢ ((3
· 2) / 1) = (6 / 1) |
58 | | 6cn 8915 |
. . . 4
⊢ 6 ∈
ℂ |
59 | 58 | div1i 8613 |
. . 3
⊢ (6 / 1) =
6 |
60 | 57, 59 | eqtri 2178 |
. 2
⊢ ((3
· 2) / 1) = 6 |
61 | 33, 55, 60 | 3eqtri 2182 |
1
⊢ (3 lcm 2)
= 6 |