Theorem expmuld

Description: Product of exponents law for positive integer exponentiation. Proposition 10-4.2(b) of Gleason p. 135, restricted to nonnegative integer exponents. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	expcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	expcld.2	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
	expaddd.2	⊢ ( 𝜑 → 𝑀 ∈ ℕ0 )
Assertion	expmuld	⊢ ( 𝜑 → ( 𝐴 ↑ ( 𝑀 · 𝑁 ) ) = ( ( 𝐴 ↑ 𝑀 ) ↑ 𝑁 ) )

Proof

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