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	$$⊢\phi \to A\in {ℝ}^{+}$$
	rpaddcld.1	$$⊢\phi \to B\in {ℝ}^{+}$$
Assertion	rpmulcld	$$⊢\phi \to AB\in {ℝ}^{+}$$

Proof

Step	Hyp	Ref	Expression
1		rpred.1	$$⊢\phi \to A\in {ℝ}^{+}$$
2		rpaddcld.1	$$⊢\phi \to B\in {ℝ}^{+}$$
3		rpmulcl	$$⊢\left(A\in {ℝ}^{+}\wedge B\in {ℝ}^{+}\right)\to AB\in {ℝ}^{+}$$
4	1 2 3	syl2anc	$$⊢\phi \to AB\in {ℝ}^{+}$$