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Metamath Proof Explorer

Theorem absrpcld

Description: The absolute value of a nonzero number is a positive real. (Contributed by Mario Carneiro, 29-May-2016)

Step	Hyp	Ref	Expression
1		abscld.1	$⊢ φ \to A \in ℂ$
2		absne0d.2	$⊢ φ \to A \neq 0$
3		absrpcl	$⊢ (A \in ℂ \land A \neq 0) \to \|A\| \in ℝ^{+}$
4	1 2 3	syl2anc	$⊢ φ \to \|A\| \in ℝ^{+}$