Theorem absgt0i

Description: The absolute value of a nonzero number is positive. Remark in Apostol p. 363. (Contributed by NM, 1-Oct-1999)

	Ref	Expression
Hypothesis	absvalsqi.1	⊢ 𝐴 ∈ ℂ
Assertion	absgt0i	⊢ ( 𝐴 ≠ 0 ↔ 0 < ( abs ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		absvalsqi.1	⊢ 𝐴 ∈ ℂ
2		absgt0	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ≠ 0 ↔ 0 < ( abs ‘ 𝐴 ) ) )
3	1 2	ax-mp	⊢ ( 𝐴 ≠ 0 ↔ 0 < ( abs ‘ 𝐴 ) )