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Metamath Proof Explorer

Theorem denexp

Description: Elevating to a nonnegative power commutes with canonical denominator. Similar to densq , extended to nonnegative exponents. (Contributed by Steven Nguyen, 5-Apr-2023)

		Ref	Expression
	Assertion	denexp	⊢ ( ( 𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ₀ ) → ( denom ‘ ( 𝐴 ↑ 𝑁 ) ) = ( ( denom ‘ 𝐴 ) ↑ 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		numdenexp	⊢ ( ( 𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ₀ ) → ( ( numer ‘ ( 𝐴 ↑ 𝑁 ) ) = ( ( numer ‘ 𝐴 ) ↑ 𝑁 ) ∧ ( denom ‘ ( 𝐴 ↑ 𝑁 ) ) = ( ( denom ‘ 𝐴 ) ↑ 𝑁 ) ) )
2	1	simprd	⊢ ( ( 𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ₀ ) → ( denom ‘ ( 𝐴 ↑ 𝑁 ) ) = ( ( denom ‘ 𝐴 ) ↑ 𝑁 ) )