Description: Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016)
Ref | Expression | ||
---|---|---|---|
Hypotheses | leidd.1 | |- ( ph -> A e. RR ) |
|
ltnegd.2 | |- ( ph -> B e. RR ) |
||
ltadd1d.3 | |- ( ph -> C e. RR ) |
||
subled.4 | |- ( ph -> ( A - B ) <_ C ) |
||
Assertion | subled | |- ( ph -> ( A - C ) <_ B ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leidd.1 | |- ( ph -> A e. RR ) |
|
2 | ltnegd.2 | |- ( ph -> B e. RR ) |
|
3 | ltadd1d.3 | |- ( ph -> C e. RR ) |
|
4 | subled.4 | |- ( ph -> ( A - B ) <_ C ) |
|
5 | suble | |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A - B ) <_ C <-> ( A - C ) <_ B ) ) |
|
6 | 1 2 3 5 | syl3anc | |- ( ph -> ( ( A - B ) <_ C <-> ( A - C ) <_ B ) ) |
7 | 4 6 | mpbid | |- ( ph -> ( A - C ) <_ B ) |