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Metamath Proof Explorer

Theorem expdivd

Description: Nonnegative integer exponentiation of a quotient. (Contributed by Mario Carneiro, 28-May-2016)

Ref Expression
Hypotheses expcld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
mulexpd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
sqdivd.3 ⊢ ( 𝜑 → 𝐵 ≠ 0 )
expdivd.3 ⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
Assertion expdivd ( 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 sqdivd.3 ⊢ ( 𝜑 → 𝐵 ≠ 0 )
4 expdivd.3 ⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
5 expdiv ⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ 𝑁 ∈ ℕ0 ) → ( ( 𝐴 / 𝐵 ) ↑ 𝑁 ) = ( ( 𝐴 ↑ 𝑁 ) / ( 𝐵 ↑ 𝑁 ) ) )
6 1 2 3 4 5 syl121anc ⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) ↑ 𝑁 ) = ( ( 𝐴 ↑ 𝑁 ) / ( 𝐵 ↑ 𝑁 ) ) )