Theorem abs00ad

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	$$⊢\phi \to A\in ℂ$$
Assertion	abs00ad	$$⊢\phi \to \left(\left|A\right|=0\leftrightarrow A=0\right)$$

Proof

Step	Hyp	Ref	Expression
1		abs00ad.1	$$⊢\phi \to A\in ℂ$$
2		abs00	$$⊢A\in ℂ\to \left(\left|A\right|=0\leftrightarrow A=0\right)$$
3	1 2	syl	$$⊢\phi \to \left(\left|A\right|=0\leftrightarrow A=0\right)$$