This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.
Metamath Proof Explorer
Description: Swapping order of subtraction doesn't change the absolute value.
Example of Apostol p. 363. (Contributed by Mario Carneiro, 29-May-2016)
|
|
Ref |
Expression |
|
Hypotheses |
abscld.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
|
|
abssubd.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
|
Assertion |
abssubd |
⊢ ( 𝜑 → ( abs ‘ ( 𝐴 − 𝐵 ) ) = ( abs ‘ ( 𝐵 − 𝐴 ) ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
abscld.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
| 2 |
|
abssubd.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
| 3 |
|
abssub |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( abs ‘ ( 𝐴 − 𝐵 ) ) = ( abs ‘ ( 𝐵 − 𝐴 ) ) ) |
| 4 |
1 2 3
|
syl2anc |
⊢ ( 𝜑 → ( abs ‘ ( 𝐴 − 𝐵 ) ) = ( abs ‘ ( 𝐵 − 𝐴 ) ) ) |