Theorem leneg3d

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

	Ref	Expression
Hypotheses	leneg3d.1	|- ( ph -> A e. RR )
	leneg3d.2	|- ( ph -> B e. RR )
Assertion	leneg3d	|- ( ph -> ( -u A <_ B <-> -u B <_ A ) )

Proof

Step	Hyp	Ref	Expression
1		leneg3d.1	|- ( ph -> A e. RR )
2		leneg3d.2	|- ( ph -> B e. RR )
3	1	renegcld	|- ( ph -> -u A e. RR )
4	3 2	lenegd	|- ( ph -> ( -u A <_ B <-> -u B <_ -u -u A ) )
5	1	recnd	|- ( ph -> A e. CC )
6	5	negnegd	|- ( ph -> -u -u A = A )
7	6	breq2d	|- ( ph -> ( -u B <_ -u -u A <-> -u B <_ A ) )
8	4 7	bitrd	|- ( ph -> ( -u A <_ B <-> -u B <_ A ) )