Description: Absolute value of differences around common element. (Contributed by NM, 2-Oct-1999)
Ref | Expression | ||
---|---|---|---|
Hypotheses | absvalsqi.1 | |- A e. CC |
|
abssub.2 | |- B e. CC |
||
abs3dif.3 | |- C e. CC |
||
Assertion | abs3difi | |- ( abs ` ( A - B ) ) <_ ( ( abs ` ( A - C ) ) + ( abs ` ( C - B ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | absvalsqi.1 | |- A e. CC |
|
2 | abssub.2 | |- B e. CC |
|
3 | abs3dif.3 | |- 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 | mp3an | |- ( abs ` ( A - B ) ) <_ ( ( abs ` ( A - C ) ) + ( abs ` ( C - B ) ) ) |