Theorem absne0d

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 )
	absne0d.2	⊢ ( 𝜑 → 𝐴 ≠ 0 )
Assertion	absne0d	⊢ ( 𝜑 → ( abs ‘ 𝐴 ) ≠ 0 )

Proof

Step	Hyp	Ref	Expression
1		abscld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		absne0d.2	⊢ ( 𝜑 → 𝐴 ≠ 0 )
3	1	abs00ad	⊢ ( 𝜑 → ( ( abs ‘ 𝐴 ) = 0 ↔ 𝐴 = 0 ) )
4	3	necon3bid	⊢ ( 𝜑 → ( ( abs ‘ 𝐴 ) ≠ 0 ↔ 𝐴 ≠ 0 ) )
5	2 4	mpbird	⊢ ( 𝜑 → ( abs ‘ 𝐴 ) ≠ 0 )