Theorem reeflog

Description: Relationship between the natural logarithm function and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007)

	Ref	Expression
Assertion	reeflog	⊢ ( 𝐴 ∈ ℝ+ → ( exp ‘ ( log ‘ 𝐴 ) ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		rpcnne0	⊢ ( 𝐴 ∈ ℝ+ → ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) )
2		eflog	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( exp ‘ ( log ‘ 𝐴 ) ) = 𝐴 )
3	1 2	syl	⊢ ( 𝐴 ∈ ℝ+ → ( exp ‘ ( log ‘ 𝐴 ) ) = 𝐴 )