Metamath Proof Explorer
Description: Absolute value distributes over division. (Contributed by NM, 26-Mar-2005)
|
|
Ref |
Expression |
|
Hypotheses |
absvalsqi.1 |
⊢ 𝐴 ∈ ℂ |
|
|
abssub.2 |
⊢ 𝐵 ∈ ℂ |
|
Assertion |
absdivzi |
⊢ ( 𝐵 ≠ 0 → ( abs ‘ ( 𝐴 / 𝐵 ) ) = ( ( abs ‘ 𝐴 ) / ( abs ‘ 𝐵 ) ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
absvalsqi.1 |
⊢ 𝐴 ∈ ℂ |
2 |
|
abssub.2 |
⊢ 𝐵 ∈ ℂ |
3 |
|
absdiv |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( abs ‘ ( 𝐴 / 𝐵 ) ) = ( ( abs ‘ 𝐴 ) / ( abs ‘ 𝐵 ) ) ) |
4 |
1 2 3
|
mp3an12 |
⊢ ( 𝐵 ≠ 0 → ( abs ‘ ( 𝐴 / 𝐵 ) ) = ( ( abs ‘ 𝐴 ) / ( abs ‘ 𝐵 ) ) ) |