Theorem absnidd

Description: A negative number is the negative of its own absolute value. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	resqrcld.1	|- ( ph -> A e. RR )
	absnidd.2	|- ( ph -> A <_ 0 )
Assertion	absnidd	|- ( ph -> ( abs ` A ) = -u A )

Proof

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