Theorem absrpcld

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

	Ref	Expression
Hypotheses	abscld.1	$$⊢\phi \to A\in ℂ$$
	absne0d.2	$$⊢\phi \to A\ne 0$$
Assertion	absrpcld	$$⊢\phi \to \left|A\right|\in {ℝ}^{+}$$

Proof

Step	Hyp	Ref	Expression
1		abscld.1	$$⊢\phi \to A\in ℂ$$
2		absne0d.2	$$⊢\phi \to A\ne 0$$
3		absrpcl	$$⊢\left(A\in ℂ\wedge A\ne 0\right)\to \left|A\right|\in {ℝ}^{+}$$
4	1 2 3	syl2anc	$$⊢\phi \to \left|A\right|\in {ℝ}^{+}$$