Theorem rpdivcld

Description: Closure law for division of positive reals. (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	rpdivcld	$$⊢\phi \to \frac{A}{B}\in {ℝ}^{+}$$

Proof

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