Theorem rplogcld

Description: Closure of the logarithm function in the positive reals. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	relogefd.1	$$⊢\phi \to A\in ℝ$$
	rplogcld.2	$$⊢\phi \to 1<A$$
Assertion	rplogcld	$$⊢\phi \to \log A\in {ℝ}^{+}$$

Proof

Step	Hyp	Ref	Expression
1		relogefd.1	$$⊢\phi \to A\in ℝ$$
2		rplogcld.2	$$⊢\phi \to 1<A$$
3		rplogcl	$$⊢\left(A\in ℝ\wedge 1<A\right)\to \log A\in {ℝ}^{+}$$
4	1 2 3	syl2anc	$$⊢\phi \to \log A\in {ℝ}^{+}$$