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Metamath Proof Explorer

Theorem bcs

Description: Bunjakovaskij-Cauchy-Schwarz inequality. Remark 3.4 of Beran p. 98. (Contributed by NM, 16-Feb-2006) (New usage is discouraged.)

Ref Expression
Assertion bcs 
|- ( ( A e. ~H /\ B e. ~H ) -> ( abs ` ( A .ih B ) ) <_ ( ( normh ` A ) x. ( normh ` B ) ) )

Proof

Step Hyp Ref Expression
1 fvoveq1 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( abs ` ( A .ih B ) ) = ( abs ` ( if ( A e. ~H , A , 0h ) .ih B ) ) )
2 fveq2 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( normh ` A ) = ( normh ` if ( A e. ~H , A , 0h ) ) )
3 2 oveq1d 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( ( normh ` A ) x. ( normh ` B ) ) = ( ( normh ` if ( A e. ~H , A , 0h ) ) x. ( normh ` B ) ) )
4 1 3 breq12d 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( ( abs ` ( A .ih B ) ) <_ ( ( normh ` A ) x. ( normh ` B ) ) <-> ( abs ` ( if ( A e. ~H , A , 0h ) .ih B ) ) <_ ( ( normh ` if ( A e. ~H , A , 0h ) ) x. ( normh ` B ) ) ) )
5 oveq2 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( if ( A e. ~H , A , 0h ) .ih B ) = ( if ( A e. ~H , A , 0h ) .ih if ( B e. ~H , B , 0h ) ) )
6 5 fveq2d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( abs ` ( if ( A e. ~H , A , 0h ) .ih B ) ) = ( abs ` ( if ( A e. ~H , A , 0h ) .ih if ( B e. ~H , B , 0h ) ) ) )
7 fveq2 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( normh ` B ) = ( normh ` if ( B e. ~H , B , 0h ) ) )
8 7 oveq2d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( ( normh ` if ( A e. ~H , A , 0h ) ) x. ( normh ` B ) ) = ( ( normh ` if ( A e. ~H , A , 0h ) ) x. ( normh ` if ( B e. ~H , B , 0h ) ) ) )
9 6 8 breq12d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( ( abs ` ( if ( A e. ~H , A , 0h ) .ih B ) ) <_ ( ( normh ` if ( A e. ~H , A , 0h ) ) x. ( normh ` B ) ) <-> ( abs ` ( if ( A e. ~H , A , 0h ) .ih if ( B e. ~H , B , 0h ) ) ) <_ ( ( normh ` if ( A e. ~H , A , 0h ) ) x. ( normh ` if ( B e. ~H , B , 0h ) ) ) ) )
10 ifhvhv0 
 |-  if ( A e. ~H , A , 0h ) e. ~H
11 ifhvhv0 
 |-  if ( B e. ~H , B , 0h ) e. ~H
12 10 11 bcsiHIL 
 |-  ( abs ` ( if ( A e. ~H , A , 0h ) .ih if ( B e. ~H , B , 0h ) ) ) <_ ( ( normh ` if ( A e. ~H , A , 0h ) ) x. ( normh ` if ( B e. ~H , B , 0h ) ) )
13 4 9 12 dedth2h 
 |-  ( ( A e. ~H /\ B e. ~H ) -> ( abs ` ( A .ih B ) ) <_ ( ( normh ` A ) x. ( normh ` B ) ) )