Metamath Proof Explorer

Theorem norm-iii

Description: Theorem 3.3(iii) of Beran p. 97. (Contributed by NM, 25-Oct-1999) (New usage is discouraged.)

Ref Expression
Assertion norm-iii 
|- ( ( A e. CC /\ B e. ~H ) -> ( normh ` ( A .h B ) ) = ( ( abs ` A ) x. ( normh ` B ) ) )

Proof

Step Hyp Ref Expression
1 fvoveq1 
 |-  ( A = if ( A e. CC , A , 0 ) -> ( normh ` ( A .h B ) ) = ( normh ` ( if ( A e. CC , A , 0 ) .h B ) ) )
2 fveq2 
 |-  ( A = if ( A e. CC , A , 0 ) -> ( abs ` A ) = ( abs ` if ( A e. CC , A , 0 ) ) )
3 2 oveq1d 
 |-  ( A = if ( A e. CC , A , 0 ) -> ( ( abs ` A ) x. ( normh ` B ) ) = ( ( abs ` if ( A e. CC , A , 0 ) ) x. ( normh ` B ) ) )
4 1 3 eqeq12d 
 |-  ( A = if ( A e. CC , A , 0 ) -> ( ( normh ` ( A .h B ) ) = ( ( abs ` A ) x. ( normh ` B ) ) <-> ( normh ` ( if ( A e. CC , A , 0 ) .h B ) ) = ( ( abs ` if ( A e. CC , A , 0 ) ) x. ( normh ` B ) ) ) )
5 oveq2 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( if ( A e. CC , A , 0 ) .h B ) = ( if ( A e. CC , A , 0 ) .h if ( B e. ~H , B , 0h ) ) )
6 5 fveq2d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( normh ` ( if ( A e. CC , A , 0 ) .h B ) ) = ( normh ` ( if ( A e. CC , A , 0 ) .h 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 ) -> ( ( abs ` if ( A e. CC , A , 0 ) ) x. ( normh ` B ) ) = ( ( abs ` if ( A e. CC , A , 0 ) ) x. ( normh ` if ( B e. ~H , B , 0h ) ) ) )
9 6 8 eqeq12d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( ( normh ` ( if ( A e. CC , A , 0 ) .h B ) ) = ( ( abs ` if ( A e. CC , A , 0 ) ) x. ( normh ` B ) ) <-> ( normh ` ( if ( A e. CC , A , 0 ) .h if ( B e. ~H , B , 0h ) ) ) = ( ( abs ` if ( A e. CC , A , 0 ) ) x. ( normh ` if ( B e. ~H , B , 0h ) ) ) ) )
10 0cn 
 |-  0 e. CC
11 10 elimel 
 |-  if ( A e. CC , A , 0 ) e. CC
12 ifhvhv0 
 |-  if ( B e. ~H , B , 0h ) e. ~H
13 11 12 norm-iii-i 
 |-  ( normh ` ( if ( A e. CC , A , 0 ) .h if ( B e. ~H , B , 0h ) ) ) = ( ( abs ` if ( A e. CC , A , 0 ) ) x. ( normh ` if ( B e. ~H , B , 0h ) ) )
14 4 9 13 dedth2h 
 |-  ( ( A e. CC /\ B e. ~H ) -> ( normh ` ( A .h B ) ) = ( ( abs ` A ) x. ( normh ` B ) ) )