Theorem reexplog

Description: Exponentiation of a positive real number to an integer power. (Contributed by Steve Rodriguez, 25-Nov-2007)

	Ref	Expression
Assertion	reexplog	⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ ) → ( 𝐴 ↑ 𝑁 ) = ( exp ‘ ( 𝑁 · ( log ‘ 𝐴 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		relogcl	⊢ ( 𝐴 ∈ ℝ+ → ( log ‘ 𝐴 ) ∈ ℝ )
2	1	recnd	⊢ ( 𝐴 ∈ ℝ+ → ( log ‘ 𝐴 ) ∈ ℂ )
3		efexp	⊢ ( ( ( log ‘ 𝐴 ) ∈ ℂ ∧ 𝑁 ∈ ℤ ) → ( exp ‘ ( 𝑁 · ( log ‘ 𝐴 ) ) ) = ( ( exp ‘ ( log ‘ 𝐴 ) ) ↑ 𝑁 ) )
4	2 3	sylan	⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ ) → ( exp ‘ ( 𝑁 · ( log ‘ 𝐴 ) ) ) = ( ( exp ‘ ( log ‘ 𝐴 ) ) ↑ 𝑁 ) )
5		reeflog	⊢ ( 𝐴 ∈ ℝ+ → ( exp ‘ ( log ‘ 𝐴 ) ) = 𝐴 )
6	5	oveq1d	⊢ ( 𝐴 ∈ ℝ+ → ( ( exp ‘ ( log ‘ 𝐴 ) ) ↑ 𝑁 ) = ( 𝐴 ↑ 𝑁 ) )
7	6	adantr	⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ ) → ( ( exp ‘ ( log ‘ 𝐴 ) ) ↑ 𝑁 ) = ( 𝐴 ↑ 𝑁 ) )
8	4 7	eqtr2d	⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ ) → ( 𝐴 ↑ 𝑁 ) = ( exp ‘ ( 𝑁 · ( log ‘ 𝐴 ) ) ) )