Theorem abssuble0d

Description: Absolute value of a nonpositive difference. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	absltd.1	|- ( ph -> A e. RR )
	absltd.2	|- ( ph -> B e. RR )
	abssubge0d.2	|- ( ph -> A <_ B )
Assertion	abssuble0d	|- ( ph -> ( abs ` ( A - B ) ) = ( B - A ) )

Proof

Step	Hyp	Ref	Expression
1		absltd.1	|- ( ph -> A e. RR )
2		absltd.2	|- ( ph -> B e. RR )
3		abssubge0d.2	|- ( ph -> A <_ B )
4		abssuble0	|- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( abs ` ( A - B ) ) = ( B - A ) )
5	1 2 3 4	syl3anc	|- ( ph -> ( abs ` ( A - B ) ) = ( B - A ) )