Theorem rplogcld

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

	Ref	Expression
Hypotheses	relogefd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	rplogcld.2	⊢ ( 𝜑 → 1 < 𝐴 )
Assertion	rplogcld	⊢ ( 𝜑 → ( log ‘ 𝐴 ) ∈ ℝ+ )

Proof

Step	Hyp	Ref	Expression
1		relogefd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		rplogcld.2	⊢ ( 𝜑 → 1 < 𝐴 )
3		rplogcl	⊢ ( ( 𝐴 ∈ ℝ ∧ 1 < 𝐴 ) → ( log ‘ 𝐴 ) ∈ ℝ+ )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( log ‘ 𝐴 ) ∈ ℝ+ )