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Metamath Proof Explorer

Theorem normlem9at

Description: Lemma used to derive properties of norm. Part of Remark 3.4(B) of Beran p. 98. (Contributed by NM, 10-May-2005) (New usage is discouraged.)

Ref Expression
Assertion normlem9at 
|- ( ( A e. ~H /\ B e. ~H ) -> ( ( A -h B ) .ih ( A -h B ) ) = ( ( ( A .ih A ) + ( B .ih B ) ) - ( ( A .ih B ) + ( B .ih A ) ) ) )

Proof

Step Hyp Ref Expression
1 oveq1 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( A -h B ) = ( if ( A e. ~H , A , 0h ) -h B ) )
2 1 1 oveq12d 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( ( A -h B ) .ih ( A -h B ) ) = ( ( if ( A e. ~H , A , 0h ) -h B ) .ih ( if ( A e. ~H , A , 0h ) -h B ) ) )
3 id 
 |-  ( A = if ( A e. ~H , A , 0h ) -> A = if ( A e. ~H , A , 0h ) )
4 3 3 oveq12d 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( A .ih A ) = ( if ( A e. ~H , A , 0h ) .ih if ( A e. ~H , A , 0h ) ) )
5 4 oveq1d 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( ( A .ih A ) + ( B .ih B ) ) = ( ( if ( A e. ~H , A , 0h ) .ih if ( A e. ~H , A , 0h ) ) + ( B .ih B ) ) )
6 oveq1 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( A .ih B ) = ( if ( A e. ~H , A , 0h ) .ih B ) )
7 oveq2 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( B .ih A ) = ( B .ih if ( A e. ~H , A , 0h ) ) )
8 6 7 oveq12d 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( ( A .ih B ) + ( B .ih A ) ) = ( ( if ( A e. ~H , A , 0h ) .ih B ) + ( B .ih if ( A e. ~H , A , 0h ) ) ) )
9 5 8 oveq12d 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( ( ( A .ih A ) + ( B .ih B ) ) - ( ( A .ih B ) + ( B .ih A ) ) ) = ( ( ( if ( A e. ~H , A , 0h ) .ih if ( A e. ~H , A , 0h ) ) + ( B .ih B ) ) - ( ( if ( A e. ~H , A , 0h ) .ih B ) + ( B .ih if ( A e. ~H , A , 0h ) ) ) ) )
10 2 9 eqeq12d 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( ( ( A -h B ) .ih ( A -h B ) ) = ( ( ( A .ih A ) + ( B .ih B ) ) - ( ( A .ih B ) + ( B .ih A ) ) ) <-> ( ( if ( A e. ~H , A , 0h ) -h B ) .ih ( if ( A e. ~H , A , 0h ) -h B ) ) = ( ( ( if ( A e. ~H , A , 0h ) .ih if ( A e. ~H , A , 0h ) ) + ( B .ih B ) ) - ( ( if ( A e. ~H , A , 0h ) .ih B ) + ( B .ih if ( A e. ~H , A , 0h ) ) ) ) ) )
11 oveq2 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( if ( A e. ~H , A , 0h ) -h B ) = ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) )
12 11 11 oveq12d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( ( if ( A e. ~H , A , 0h ) -h B ) .ih ( if ( A e. ~H , A , 0h ) -h B ) ) = ( ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) .ih ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) )
13 id 
 |-  ( B = if ( B e. ~H , B , 0h ) -> B = if ( B e. ~H , B , 0h ) )
14 13 13 oveq12d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( B .ih B ) = ( if ( B e. ~H , B , 0h ) .ih if ( B e. ~H , B , 0h ) ) )
15 14 oveq2d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( ( if ( A e. ~H , A , 0h ) .ih if ( A e. ~H , A , 0h ) ) + ( B .ih B ) ) = ( ( if ( A e. ~H , A , 0h ) .ih if ( A e. ~H , A , 0h ) ) + ( if ( B e. ~H , B , 0h ) .ih if ( B e. ~H , B , 0h ) ) ) )
16 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 ) ) )
17 oveq1 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( B .ih if ( A e. ~H , A , 0h ) ) = ( if ( B e. ~H , B , 0h ) .ih if ( A e. ~H , A , 0h ) ) )
18 16 17 oveq12d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( ( if ( A e. ~H , A , 0h ) .ih B ) + ( B .ih if ( A e. ~H , A , 0h ) ) ) = ( ( if ( A e. ~H , A , 0h ) .ih if ( B e. ~H , B , 0h ) ) + ( if ( B e. ~H , B , 0h ) .ih if ( A e. ~H , A , 0h ) ) ) )
19 15 18 oveq12d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( ( ( if ( A e. ~H , A , 0h ) .ih if ( A e. ~H , A , 0h ) ) + ( B .ih B ) ) - ( ( if ( A e. ~H , A , 0h ) .ih B ) + ( B .ih if ( A e. ~H , A , 0h ) ) ) ) = ( ( ( if ( A e. ~H , A , 0h ) .ih if ( A e. ~H , A , 0h ) ) + ( if ( B e. ~H , B , 0h ) .ih if ( B e. ~H , B , 0h ) ) ) - ( ( if ( A e. ~H , A , 0h ) .ih if ( B e. ~H , B , 0h ) ) + ( if ( B e. ~H , B , 0h ) .ih if ( A e. ~H , A , 0h ) ) ) ) )
20 12 19 eqeq12d 
 |-  ( B = if ( B e. ~H , B , 0h ) -> ( ( ( if ( A e. ~H , A , 0h ) -h B ) .ih ( if ( A e. ~H , A , 0h ) -h B ) ) = ( ( ( if ( A e. ~H , A , 0h ) .ih if ( A e. ~H , A , 0h ) ) + ( B .ih B ) ) - ( ( if ( A e. ~H , A , 0h ) .ih B ) + ( B .ih if ( A e. ~H , A , 0h ) ) ) ) <-> ( ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) .ih ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) = ( ( ( if ( A e. ~H , A , 0h ) .ih if ( A e. ~H , A , 0h ) ) + ( if ( B e. ~H , B , 0h ) .ih if ( B e. ~H , B , 0h ) ) ) - ( ( if ( A e. ~H , A , 0h ) .ih if ( B e. ~H , B , 0h ) ) + ( if ( B e. ~H , B , 0h ) .ih if ( A e. ~H , A , 0h ) ) ) ) ) )
21 ifhvhv0 
 |-  if ( A e. ~H , A , 0h ) e. ~H
22 ifhvhv0 
 |-  if ( B e. ~H , B , 0h ) e. ~H
23 21 22 21 22 normlem9 
 |-  ( ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) .ih ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) ) = ( ( ( if ( A e. ~H , A , 0h ) .ih if ( A e. ~H , A , 0h ) ) + ( if ( B e. ~H , B , 0h ) .ih if ( B e. ~H , B , 0h ) ) ) - ( ( if ( A e. ~H , A , 0h ) .ih if ( B e. ~H , B , 0h ) ) + ( if ( B e. ~H , B , 0h ) .ih if ( A e. ~H , A , 0h ) ) ) )
24 10 20 23 dedth2h 
 |-  ( ( A e. ~H /\ B e. ~H ) -> ( ( A -h B ) .ih ( A -h B ) ) = ( ( ( A .ih A ) + ( B .ih B ) ) - ( ( A .ih B ) + ( B .ih A ) ) ) )