Theorem rpmulcl

Description: Closure law for multiplication of positive reals. Part of Axiom 7 of Apostol p. 20. (Contributed by NM, 27-Oct-2007)

	Ref	Expression
Assertion	rpmulcl	$$⊢\left(A\in {ℝ}^{+}\wedge B\in {ℝ}^{+}\right)\to AB\in {ℝ}^{+}$$

Proof

Step	Hyp	Ref	Expression
1		rpre	$$⊢A\in {ℝ}^{+}\to A\in ℝ$$
2		rpre	$$⊢B\in {ℝ}^{+}\to B\in ℝ$$
3		remulcl	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to AB\in ℝ$$
4	1 2 3	syl2an	$$⊢\left(A\in {ℝ}^{+}\wedge B\in {ℝ}^{+}\right)\to AB\in ℝ$$
5		elrp	$$⊢A\in {ℝ}^{+}\leftrightarrow \left(A\in ℝ\wedge 0<A\right)$$
6		elrp	$$⊢B\in {ℝ}^{+}\leftrightarrow \left(B\in ℝ\wedge 0<B\right)$$
7		mulgt0	$$⊢\left(\left(A\in ℝ\wedge 0<A\right)\wedge \left(B\in ℝ\wedge 0<B\right)\right)\to 0<AB$$
8	5 6 7	syl2anb	$$⊢\left(A\in {ℝ}^{+}\wedge B\in {ℝ}^{+}\right)\to 0<AB$$
9		elrp	$$⊢AB\in {ℝ}^{+}\leftrightarrow \left(AB\in ℝ\wedge 0<AB\right)$$
10	4 8 9	sylanbrc	$$⊢\left(A\in {ℝ}^{+}\wedge B\in {ℝ}^{+}\right)\to AB\in {ℝ}^{+}$$