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 ) ) |
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 ) ) |