Proof of Theorem lcmgcd
| Step | Hyp | Ref
| Expression |
| 1 | | gcdcl 16543 |
. . . . . . . 8
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ∈
ℕ0) |
| 2 | 1 | nn0cnd 12589 |
. . . . . . 7
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ∈ ℂ) |
| 3 | 2 | mul02d 11459 |
. . . . . 6
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0
· (𝑀 gcd 𝑁)) = 0) |
| 4 | | 0z 12624 |
. . . . . . . . . 10
⊢ 0 ∈
ℤ |
| 5 | | lcmcom 16630 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℤ ∧ 0 ∈
ℤ) → (𝑁 lcm 0) =
(0 lcm 𝑁)) |
| 6 | 4, 5 | mpan2 691 |
. . . . . . . . 9
⊢ (𝑁 ∈ ℤ → (𝑁 lcm 0) = (0 lcm 𝑁)) |
| 7 | | lcm0val 16631 |
. . . . . . . . 9
⊢ (𝑁 ∈ ℤ → (𝑁 lcm 0) = 0) |
| 8 | 6, 7 | eqtr3d 2779 |
. . . . . . . 8
⊢ (𝑁 ∈ ℤ → (0 lcm
𝑁) = 0) |
| 9 | 8 | adantl 481 |
. . . . . . 7
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 lcm
𝑁) = 0) |
| 10 | 9 | oveq1d 7446 |
. . . . . 6
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((0 lcm
𝑁) · (𝑀 gcd 𝑁)) = (0 · (𝑀 gcd 𝑁))) |
| 11 | | zcn 12618 |
. . . . . . . . 9
⊢ (𝑁 ∈ ℤ → 𝑁 ∈
ℂ) |
| 12 | 11 | adantl 481 |
. . . . . . . 8
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈
ℂ) |
| 13 | 12 | mul02d 11459 |
. . . . . . 7
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0
· 𝑁) =
0) |
| 14 | 13 | abs00bd 15330 |
. . . . . 6
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) →
(abs‘(0 · 𝑁))
= 0) |
| 15 | 3, 10, 14 | 3eqtr4d 2787 |
. . . . 5
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((0 lcm
𝑁) · (𝑀 gcd 𝑁)) = (abs‘(0 · 𝑁))) |
| 16 | 15 | adantr 480 |
. . . 4
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → ((0 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(0 · 𝑁))) |
| 17 | | simpr 484 |
. . . . . 6
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → 𝑀 = 0) |
| 18 | 17 | oveq1d 7446 |
. . . . 5
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝑀 lcm 𝑁) = (0 lcm 𝑁)) |
| 19 | 18 | oveq1d 7446 |
. . . 4
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = ((0 lcm 𝑁) · (𝑀 gcd 𝑁))) |
| 20 | 17 | oveq1d 7446 |
. . . . 5
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝑀 · 𝑁) = (0 · 𝑁)) |
| 21 | 20 | fveq2d 6910 |
. . . 4
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (abs‘(𝑀 · 𝑁)) = (abs‘(0 · 𝑁))) |
| 22 | 16, 19, 21 | 3eqtr4d 2787 |
. . 3
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁))) |
| 23 | | lcm0val 16631 |
. . . . . . . 8
⊢ (𝑀 ∈ ℤ → (𝑀 lcm 0) = 0) |
| 24 | 23 | adantr 480 |
. . . . . . 7
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 0) = 0) |
| 25 | 24 | oveq1d 7446 |
. . . . . 6
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 0) · (𝑀 gcd 𝑁)) = (0 · (𝑀 gcd 𝑁))) |
| 26 | | zcn 12618 |
. . . . . . . . 9
⊢ (𝑀 ∈ ℤ → 𝑀 ∈
ℂ) |
| 27 | 26 | adantr 480 |
. . . . . . . 8
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∈
ℂ) |
| 28 | 27 | mul01d 11460 |
. . . . . . 7
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 · 0) =
0) |
| 29 | 28 | abs00bd 15330 |
. . . . . 6
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) →
(abs‘(𝑀 · 0))
= 0) |
| 30 | 3, 25, 29 | 3eqtr4d 2787 |
. . . . 5
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 0) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 0))) |
| 31 | 30 | adantr 480 |
. . . 4
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → ((𝑀 lcm 0) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 0))) |
| 32 | | simpr 484 |
. . . . . 6
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → 𝑁 = 0) |
| 33 | 32 | oveq2d 7447 |
. . . . 5
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → (𝑀 lcm 𝑁) = (𝑀 lcm 0)) |
| 34 | 33 | oveq1d 7446 |
. . . 4
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = ((𝑀 lcm 0) · (𝑀 gcd 𝑁))) |
| 35 | 32 | oveq2d 7447 |
. . . . 5
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → (𝑀 · 𝑁) = (𝑀 · 0)) |
| 36 | 35 | fveq2d 6910 |
. . . 4
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → (abs‘(𝑀 · 𝑁)) = (abs‘(𝑀 · 0))) |
| 37 | 31, 34, 36 | 3eqtr4d 2787 |
. . 3
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁))) |
| 38 | 22, 37 | jaodan 960 |
. 2
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 = 0 ∨ 𝑁 = 0)) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁))) |
| 39 | | neanior 3035 |
. . . . 5
⊢ ((𝑀 ≠ 0 ∧ 𝑁 ≠ 0) ↔ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) |
| 40 | | nnabscl 15364 |
. . . . . . 7
⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (abs‘𝑀) ∈
ℕ) |
| 41 | | nnabscl 15364 |
. . . . . . 7
⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (abs‘𝑁) ∈
ℕ) |
| 42 | 40, 41 | anim12i 613 |
. . . . . 6
⊢ (((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) →
((abs‘𝑀) ∈
ℕ ∧ (abs‘𝑁)
∈ ℕ)) |
| 43 | 42 | an4s 660 |
. . . . 5
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → ((abs‘𝑀) ∈ ℕ ∧
(abs‘𝑁) ∈
ℕ)) |
| 44 | 39, 43 | sylan2br 595 |
. . . 4
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑀 = 0 ∨ 𝑁 = 0)) → ((abs‘𝑀) ∈ ℕ ∧
(abs‘𝑁) ∈
ℕ)) |
| 45 | | lcmgcdlem 16643 |
. . . . 5
⊢
(((abs‘𝑀)
∈ ℕ ∧ (abs‘𝑁) ∈ ℕ) → ((((abs‘𝑀) lcm (abs‘𝑁)) · ((abs‘𝑀) gcd (abs‘𝑁))) =
(abs‘((abs‘𝑀)
· (abs‘𝑁)))
∧ ((0 ∈ ℕ ∧ ((abs‘𝑀) ∥ 0 ∧ (abs‘𝑁) ∥ 0)) →
((abs‘𝑀) lcm
(abs‘𝑁)) ∥
0))) |
| 46 | 45 | simpld 494 |
. . . 4
⊢
(((abs‘𝑀)
∈ ℕ ∧ (abs‘𝑁) ∈ ℕ) → (((abs‘𝑀) lcm (abs‘𝑁)) · ((abs‘𝑀) gcd (abs‘𝑁))) =
(abs‘((abs‘𝑀)
· (abs‘𝑁)))) |
| 47 | 44, 46 | syl 17 |
. . 3
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑀 = 0 ∨ 𝑁 = 0)) → (((abs‘𝑀) lcm (abs‘𝑁)) · ((abs‘𝑀) gcd (abs‘𝑁))) =
(abs‘((abs‘𝑀)
· (abs‘𝑁)))) |
| 48 | | lcmabs 16642 |
. . . . 5
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) →
((abs‘𝑀) lcm
(abs‘𝑁)) = (𝑀 lcm 𝑁)) |
| 49 | | gcdabs 16568 |
. . . . 5
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) →
((abs‘𝑀) gcd
(abs‘𝑁)) = (𝑀 gcd 𝑁)) |
| 50 | 48, 49 | oveq12d 7449 |
. . . 4
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) →
(((abs‘𝑀) lcm
(abs‘𝑁)) ·
((abs‘𝑀) gcd
(abs‘𝑁))) = ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁))) |
| 51 | 50 | adantr 480 |
. . 3
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑀 = 0 ∨ 𝑁 = 0)) → (((abs‘𝑀) lcm (abs‘𝑁)) · ((abs‘𝑀) gcd (abs‘𝑁))) = ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁))) |
| 52 | | absidm 15362 |
. . . . . . 7
⊢ (𝑀 ∈ ℂ →
(abs‘(abs‘𝑀)) =
(abs‘𝑀)) |
| 53 | | absidm 15362 |
. . . . . . 7
⊢ (𝑁 ∈ ℂ →
(abs‘(abs‘𝑁)) =
(abs‘𝑁)) |
| 54 | 52, 53 | oveqan12d 7450 |
. . . . . 6
⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) →
((abs‘(abs‘𝑀))
· (abs‘(abs‘𝑁))) = ((abs‘𝑀) · (abs‘𝑁))) |
| 55 | 26, 11, 54 | syl2an 596 |
. . . . 5
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) →
((abs‘(abs‘𝑀))
· (abs‘(abs‘𝑁))) = ((abs‘𝑀) · (abs‘𝑁))) |
| 56 | | nn0abscl 15351 |
. . . . . . . 8
⊢ (𝑀 ∈ ℤ →
(abs‘𝑀) ∈
ℕ0) |
| 57 | 56 | nn0cnd 12589 |
. . . . . . 7
⊢ (𝑀 ∈ ℤ →
(abs‘𝑀) ∈
ℂ) |
| 58 | 57 | adantr 480 |
. . . . . 6
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) →
(abs‘𝑀) ∈
ℂ) |
| 59 | | nn0abscl 15351 |
. . . . . . . 8
⊢ (𝑁 ∈ ℤ →
(abs‘𝑁) ∈
ℕ0) |
| 60 | 59 | nn0cnd 12589 |
. . . . . . 7
⊢ (𝑁 ∈ ℤ →
(abs‘𝑁) ∈
ℂ) |
| 61 | 60 | adantl 481 |
. . . . . 6
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) →
(abs‘𝑁) ∈
ℂ) |
| 62 | 58, 61 | absmuld 15493 |
. . . . 5
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) →
(abs‘((abs‘𝑀)
· (abs‘𝑁))) =
((abs‘(abs‘𝑀))
· (abs‘(abs‘𝑁)))) |
| 63 | 27, 12 | absmuld 15493 |
. . . . 5
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) →
(abs‘(𝑀 ·
𝑁)) = ((abs‘𝑀) · (abs‘𝑁))) |
| 64 | 55, 62, 63 | 3eqtr4d 2787 |
. . . 4
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) →
(abs‘((abs‘𝑀)
· (abs‘𝑁))) =
(abs‘(𝑀 ·
𝑁))) |
| 65 | 64 | adantr 480 |
. . 3
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑀 = 0 ∨ 𝑁 = 0)) →
(abs‘((abs‘𝑀)
· (abs‘𝑁))) =
(abs‘(𝑀 ·
𝑁))) |
| 66 | 47, 51, 65 | 3eqtr3d 2785 |
. 2
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑀 = 0 ∨ 𝑁 = 0)) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁))) |
| 67 | 38, 66 | pm2.61dan 813 |
1
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁))) |