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Metamath Proof Explorer

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 ) )