Metamath Proof Explorer


Theorem abstrid

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 ) ) )

Proof

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 ) ) )