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Metamath Proof Explorer
Description: A complex number is zero iff its absolute value is zero. Deduction form
of abs00 . (Contributed by David Moews, 28-Feb-2017)
|
|
Ref |
Expression |
|
Hypothesis |
abs00ad.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
|
Assertion |
abs00ad |
⊢ ( 𝜑 → ( ( abs ‘ 𝐴 ) = 0 ↔ 𝐴 = 0 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
abs00ad.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
| 2 |
|
abs00 |
⊢ ( 𝐴 ∈ ℂ → ( ( abs ‘ 𝐴 ) = 0 ↔ 𝐴 = 0 ) ) |
| 3 |
1 2
|
syl |
⊢ ( 𝜑 → ( ( abs ‘ 𝐴 ) = 0 ↔ 𝐴 = 0 ) ) |