Theorem leneg2d

Description: Negative of one side of 'less than or equal to'. (Contributed by Glauco Siliprandi, 2-Jan-2022)

	Ref	Expression
Hypotheses	leneg2d.1	|- ( ph -> A e. RR )
	leneg2d.2	|- ( ph -> B e. RR )
Assertion	leneg2d	|- ( ph -> ( A <_ -u B <-> B <_ -u A ) )

Proof

Step	Hyp	Ref	Expression
1		leneg2d.1	|- ( ph -> A e. RR )
2		leneg2d.2	|- ( ph -> B e. RR )
3	2	renegcld	|- ( ph -> -u B e. RR )
4	1 3	lenegd	|- ( ph -> ( A <_ -u B <-> -u -u B <_ -u A ) )
5	2	recnd	|- ( ph -> B e. CC )
6	5	negnegd	|- ( ph -> -u -u B = B )
7	6	breq1d	|- ( ph -> ( -u -u B <_ -u A <-> B <_ -u A ) )
8	4 7	bitrd	|- ( ph -> ( A <_ -u B <-> B <_ -u A ) )