Theorem expaddd

Description: Sum of exponents law for nonnegative integer exponentiation. Proposition 10-4.2(a) of Gleason p. 135. (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	expaddd	⊢ ( 𝜑 → ( 𝐴 ↑ ( 𝑀 + 𝑁 ) ) = ( ( 𝐴 ↑ 𝑀 ) · ( 𝐴 ↑ 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		expadd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → ( 𝐴 ↑ ( 𝑀 + 𝑁 ) ) = ( ( 𝐴 ↑ 𝑀 ) · ( 𝐴 ↑ 𝑁 ) ) )
5	1 3 2 4	syl3anc	⊢ ( 𝜑 → ( 𝐴 ↑ ( 𝑀 + 𝑁 ) ) = ( ( 𝐴 ↑ 𝑀 ) · ( 𝐴 ↑ 𝑁 ) ) )