Theorem sqrtdivd

Description: Square root distributes over division. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	resqrcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	resqrcld.2	⊢ ( 𝜑 → 0 ≤ 𝐴 )
	sqrdivd.3	⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
Assertion	sqrtdivd	⊢ ( 𝜑 → ( √ ‘ ( 𝐴 / 𝐵 ) ) = ( ( √ ‘ 𝐴 ) / ( √ ‘ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		resqrcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		resqrcld.2	⊢ ( 𝜑 → 0 ≤ 𝐴 )
3		sqrdivd.3	⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
4		sqrtdiv	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ 𝐵 ∈ ℝ+ ) → ( √ ‘ ( 𝐴 / 𝐵 ) ) = ( ( √ ‘ 𝐴 ) / ( √ ‘ 𝐵 ) ) )
5	1 2 3 4	syl21anc	⊢ ( 𝜑 → ( √ ‘ ( 𝐴 / 𝐵 ) ) = ( ( √ ‘ 𝐴 ) / ( √ ‘ 𝐵 ) ) )