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Metamath Proof Explorer

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+ )