Theorem abs00d

Description: The absolute value of a number is zero iff the number is zero. Proposition 10-3.7(c) of Gleason p. 133. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	abscld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	abs00d.2	⊢ ( 𝜑 → ( abs ‘ 𝐴 ) = 0 )
Assertion	abs00d	⊢ ( 𝜑 → 𝐴 = 0 )

Proof

Step	Hyp	Ref	Expression
1		abscld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		abs00d.2	⊢ ( 𝜑 → ( abs ‘ 𝐴 ) = 0 )
3	1	abs00ad	⊢ ( 𝜑 → ( ( abs ‘ 𝐴 ) = 0 ↔ 𝐴 = 0 ) )
4	2 3	mpbid	⊢ ( 𝜑 → 𝐴 = 0 )