Theorem sqrtnegd

Description: The square root of a negative number. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	resqrcld.1	|- ( ph -> A e. RR )
	resqrcld.2	|- ( ph -> 0 <_ A )
Assertion	sqrtnegd	|- ( ph -> ( sqrt ` -u A ) = ( _i x. ( sqrt ` A ) ) )

Proof

Step	Hyp	Ref	Expression
1		resqrcld.1	|- ( ph -> A e. RR )
2		resqrcld.2	|- ( ph -> 0 <_ A )
3		sqrtneg	|- ( ( A e. RR /\ 0 <_ A ) -> ( sqrt ` -u A ) = ( _i x. ( sqrt ` A ) ) )
4	1 2 3	syl2anc	|- ( ph -> ( sqrt ` -u A ) = ( _i x. ( sqrt ` A ) ) )