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Metamath Proof Explorer

Theorem reeflogd

Description: Relationship between the natural logarithm function and the exponential function. (Contributed by Mario Carneiro, 29-May-2016)

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

Proof

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