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	⊢ ( 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 ) )