Theorem rpmulcld

Description: Closure law for multiplication of positive reals. Part of Axiom 7 of Apostol p. 20. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	rpred.1	|- ( ph -> A e. RR+ )
	rpaddcld.1	|- ( ph -> B e. RR+ )
Assertion	rpmulcld	|- ( ph -> ( A x. B ) e. RR+ )

Proof

Step	Hyp	Ref	Expression
1		rpred.1	|- ( ph -> A e. RR+ )
2		rpaddcld.1	|- ( ph -> B e. RR+ )
3		rpmulcl	|- ( ( A e. RR+ /\ B e. RR+ ) -> ( A x. B ) e. RR+ )
4	1 2 3	syl2anc	|- ( ph -> ( A x. B ) e. RR+ )