Metamath Proof Explorer
Description: The absolute value of a real number is either that number or its
negative. (Contributed by Mario Carneiro, 29-May-2016)
|
|
Ref |
Expression |
|
Hypothesis |
resqrcld.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
|
Assertion |
absord |
⊢ ( 𝜑 → ( ( abs ‘ 𝐴 ) = 𝐴 ∨ ( abs ‘ 𝐴 ) = - 𝐴 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
resqrcld.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
absor |
⊢ ( 𝐴 ∈ ℝ → ( ( abs ‘ 𝐴 ) = 𝐴 ∨ ( abs ‘ 𝐴 ) = - 𝐴 ) ) |
3 |
1 2
|
syl |
⊢ ( 𝜑 → ( ( abs ‘ 𝐴 ) = 𝐴 ∨ ( abs ‘ 𝐴 ) = - 𝐴 ) ) |