Theorem abslei

Description: Absolute value and 'less than or equal to' relation. (Contributed by NM, 6-Apr-2005)

	Ref	Expression
Hypotheses	sqrtthi.1	|- A e. RR
	sqr11.1	|- B e. RR
Assertion	abslei	|- ( ( abs ` A ) <_ B <-> ( -u B <_ A /\ A <_ B ) )

Proof

Step	Hyp	Ref	Expression
1		sqrtthi.1	|- A e. RR
2		sqr11.1	|- B e. RR
3		absle	|- ( ( A e. RR /\ B e. RR ) -> ( ( abs ` A ) <_ B <-> ( -u B <_ A /\ A <_ B ) ) )
4	1 2 3	mp2an	|- ( ( abs ` A ) <_ B <-> ( -u B <_ A /\ A <_ B ) )