Theorem absled

Description: Absolute value and 'less than or equal to' relation. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	absltd.1	|- ( ph -> A e. RR )
	absltd.2	|- ( ph -> B e. RR )
Assertion	absled	|- ( ph -> ( ( abs ` A ) <_ B <-> ( -u B <_ A /\ A <_ B ) ) )

Proof

Step	Hyp	Ref	Expression
1		absltd.1	|- ( ph -> A e. RR )
2		absltd.2	|- ( ph -> B e. RR )
3		absle	|- ( ( A e. RR /\ B e. RR ) -> ( ( abs ` A ) <_ B <-> ( -u B <_ A /\ A <_ B ) ) )
4	1 2 3	syl2anc	|- ( ph -> ( ( abs ` A ) <_ B <-> ( -u B <_ A /\ A <_ B ) ) )