Theorem rpaddcl

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

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

Proof

Step	Hyp	Ref	Expression
1		rpre	$$⊢A\in {ℝ}^{+}\to A\in ℝ$$
2		rpre	$$⊢B\in {ℝ}^{+}\to B\in ℝ$$
3		readdcl	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to A+B\in ℝ$$
4	1 2 3	syl2an	$$⊢\left(A\in {ℝ}^{+}\wedge B\in {ℝ}^{+}\right)\to A+B\in ℝ$$
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		addgt0	$$⊢\left(\left(A\in ℝ\wedge B\in ℝ\right)\wedge \left(0<A\wedge 0<B\right)\right)\to 0<A+B$$
8	7	an4s	$$⊢\left(\left(A\in ℝ\wedge 0<A\right)\wedge \left(B\in ℝ\wedge 0<B\right)\right)\to 0<A+B$$
9	5 6 8	syl2anb	$$⊢\left(A\in {ℝ}^{+}\wedge B\in {ℝ}^{+}\right)\to 0<A+B$$
10		elrp	$$⊢A+B\in {ℝ}^{+}\leftrightarrow \left(A+B\in ℝ\wedge 0<A+B\right)$$
11	4 9 10	sylanbrc	$$⊢\left(A\in {ℝ}^{+}\wedge B\in {ℝ}^{+}\right)\to A+B\in {ℝ}^{+}$$