Theorem sqrtdivd

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

	Ref	Expression
Hypotheses	resqrcld.1	|- ( ph -> A e. RR )
	resqrcld.2	|- ( ph -> 0 <_ A )
	sqrdivd.3	|- ( ph -> B e. RR+ )
Assertion	sqrtdivd	|- ( ph -> ( sqrt ` ( A / B ) ) = ( ( sqrt ` A ) / ( sqrt ` B ) ) )

Proof

Step	Hyp	Ref	Expression
1		resqrcld.1	|- ( ph -> A e. RR )
2		resqrcld.2	|- ( ph -> 0 <_ A )
3		sqrdivd.3	|- ( ph -> B e. RR+ )
4		sqrtdiv	|- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( sqrt ` ( A / B ) ) = ( ( sqrt ` A ) / ( sqrt ` B ) ) )
5	1 2 3 4	syl21anc	|- ( ph -> ( sqrt ` ( A / B ) ) = ( ( sqrt ` A ) / ( sqrt ` B ) ) )