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	|- ( ph -> A e. CC )
	absne0d.2	|- ( ph -> A =/= 0 )
Assertion	absrpcld	|- ( ph -> ( abs ` A ) e. RR+ )

Proof

Step	Hyp	Ref	Expression
1		abscld.1	|- ( ph -> A e. CC )
2		absne0d.2	|- ( ph -> A =/= 0 )
3		absrpcl	|- ( ( A e. CC /\ A =/= 0 ) -> ( abs ` A ) e. RR+ )
4	1 2 3	syl2anc	|- ( ph -> ( abs ` A ) e. RR+ )