Theorem sqrtmulii

Description: Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999)

	Ref	Expression
Hypotheses	sqrtthi.1	⊢ 𝐴 ∈ ℝ
	sqr11.1	⊢ 𝐵 ∈ ℝ
	sqrmuli.1	⊢ 0 ≤ 𝐴
	sqrmuli.2	⊢ 0 ≤ 𝐵
Assertion	sqrtmulii	⊢ ( √ ‘ ( 𝐴 · 𝐵 ) ) = ( ( √ ‘ 𝐴 ) · ( √ ‘ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		sqrtthi.1	⊢ 𝐴 ∈ ℝ
2		sqr11.1	⊢ 𝐵 ∈ ℝ
3		sqrmuli.1	⊢ 0 ≤ 𝐴
4		sqrmuli.2	⊢ 0 ≤ 𝐵
5	1 2	sqrtmuli	⊢ ( ( 0 ≤ 𝐴 ∧ 0 ≤ 𝐵 ) → ( √ ‘ ( 𝐴 · 𝐵 ) ) = ( ( √ ‘ 𝐴 ) · ( √ ‘ 𝐵 ) ) )
6	3 4 5	mp2an	⊢ ( √ ‘ ( 𝐴 · 𝐵 ) ) = ( ( √ ‘ 𝐴 ) · ( √ ‘ 𝐵 ) )