Description: Triangle inequality for absolute value. Proposition 10-3.7(h) of Gleason p. 133. (Contributed by Mario Carneiro, 29-May-2016)
Ref | Expression | ||
---|---|---|---|
Hypotheses | abscld.1 | |- ( ph -> A e. CC ) |
|
abssubd.2 | |- ( ph -> B e. CC ) |
||
Assertion | abstrid | |- ( ph -> ( abs ` ( A + B ) ) <_ ( ( abs ` A ) + ( abs ` B ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abscld.1 | |- ( ph -> A e. CC ) |
|
2 | abssubd.2 | |- ( ph -> B e. CC ) |
|
3 | abstri | |- ( ( A e. CC /\ B e. CC ) -> ( abs ` ( A + B ) ) <_ ( ( abs ` A ) + ( abs ` B ) ) ) |
|
4 | 1 2 3 | syl2anc | |- ( ph -> ( abs ` ( A + B ) ) <_ ( ( abs ` A ) + ( abs ` B ) ) ) |