This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) ) )