Theorem absexpd

Description: Absolute value of positive integer exponentiation. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	abscld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	absexpd.2	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
Assertion	absexpd	⊢ ( 𝜑 → ( abs ‘ ( 𝐴 ↑ 𝑁 ) ) = ( ( abs ‘ 𝐴 ) ↑ 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		abscld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		absexpd.2	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
3		absexp	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) → ( abs ‘ ( 𝐴 ↑ 𝑁 ) ) = ( ( abs ‘ 𝐴 ) ↑ 𝑁 ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( abs ‘ ( 𝐴 ↑ 𝑁 ) ) = ( ( abs ‘ 𝐴 ) ↑ 𝑁 ) )