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Metamath Proof Explorer

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	$⊢ A \in ℂ$
	Assertion	absgt0i	$⊢ A \neq 0 \leftrightarrow 0 < \|A\|$

Step	Hyp	Ref	Expression
1		absvalsqi.1	$⊢ A \in ℂ$
2		absgt0	$⊢ A \in ℂ \to (A \neq 0 \leftrightarrow 0 < \|A\|)$
3	1 2	ax-mp	$⊢ A \neq 0 \leftrightarrow 0 < \|A\|$