Theorem relogcld

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

	Ref	Expression
Hypothesis	relogcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
Assertion	relogcld	⊢ ( 𝜑 → ( log ‘ 𝐴 ) ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		relogcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
2		relogcl	⊢ ( 𝐴 ∈ ℝ+ → ( log ‘ 𝐴 ) ∈ ℝ )
3	1 2	syl	⊢ ( 𝜑 → ( log ‘ 𝐴 ) ∈ ℝ )