Theorem mulexpd

Description: Nonnegative integer exponentiation of a product. Proposition 10-4.2(c) of Gleason p. 135, restricted to nonnegative integer exponents. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	expcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	mulexpd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	mulexpd.3	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
Assertion	mulexpd	⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) ↑ 𝑁 ) = ( ( 𝐴 ↑ 𝑁 ) · ( 𝐵 ↑ 𝑁 ) ) )

Proof

Step	Hyp	Ref	Expression
1		expcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		mulexpd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		mulexpd.3	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
4		mulexp	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) → ( ( 𝐴 · 𝐵 ) ↑ 𝑁 ) = ( ( 𝐴 ↑ 𝑁 ) · ( 𝐵 ↑ 𝑁 ) ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) ↑ 𝑁 ) = ( ( 𝐴 ↑ 𝑁 ) · ( 𝐵 ↑ 𝑁 ) ) )