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Metamath Proof Explorer

Theorem abs3lemd

Description: Lemma involving absolute value of differences. (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 )
abs3lemd.4 
|- ( ph -> D e. RR )
abs3lemd.5 
|- ( ph -> ( abs ` ( A - C ) ) < ( D / 2 ) )
abs3lemd.6 
|- ( ph -> ( abs ` ( C - B ) ) < ( D / 2 ) )
Assertion abs3lemd 
|- ( ph -> ( abs ` ( A - B ) ) < D )

Proof

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 abs3lemd.4 
 |-  ( ph -> D e. RR )
5 abs3lemd.5 
 |-  ( ph -> ( abs ` ( A - C ) ) < ( D / 2 ) )
6 abs3lemd.6 
 |-  ( ph -> ( abs ` ( C - B ) ) < ( D / 2 ) )
7 abs3lem 
 |-  ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. RR ) ) -> ( ( ( abs ` ( A - C ) ) < ( D / 2 ) /\ ( abs ` ( C - B ) ) < ( D / 2 ) ) -> ( abs ` ( A - B ) ) < D ) )
8 1 2 3 4 7 syl22anc 
 |-  ( ph -> ( ( ( abs ` ( A - C ) ) < ( D / 2 ) /\ ( abs ` ( C - B ) ) < ( D / 2 ) ) -> ( abs ` ( A - B ) ) < D ) )
9 5 6 8 mp2and 
 |-  ( ph -> ( abs ` ( A - B ) ) < D )