Description: Absolute value of differences around common element. (Contributed by Mario Carneiro, 29-May-2016)
Ref | Expression | ||
---|---|---|---|
Hypotheses | abscld.1 | |- ( ph -> A e. CC ) |
|
abssubd.2 | |- ( ph -> B e. CC ) |
||
abs3difd.3 | |- ( ph -> C e. CC ) |
||
Assertion | abs3difd | |- ( ph -> ( abs ` ( A - B ) ) <_ ( ( abs ` ( A - C ) ) + ( abs ` ( C - B ) ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abscld.1 | |- ( ph -> A e. CC ) |
|
2 | abssubd.2 | |- ( ph -> B e. CC ) |
|
3 | abs3difd.3 | |- ( ph -> C e. CC ) |
|
4 | abs3dif | |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( abs ` ( A - B ) ) <_ ( ( abs ` ( A - C ) ) + ( abs ` ( C - B ) ) ) ) |
|
5 | 1 2 3 4 | syl3anc | |- ( ph -> ( abs ` ( A - B ) ) <_ ( ( abs ` ( A - C ) ) + ( abs ` ( C - B ) ) ) ) |