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Metamath Proof Explorer

Theorem normlem8

Description: Lemma used to derive properties of norm. (Contributed by NM, 30-Jun-2005) (New usage is discouraged.)

Ref Expression
Hypotheses normlem8.1 
|- A e. ~H
normlem8.2 
|- B e. ~H
normlem8.3 
|- C e. ~H
normlem8.4 
|- D e. ~H
Assertion normlem8 
|- ( ( A +h B ) .ih ( C +h D ) ) = ( ( ( A .ih C ) + ( B .ih D ) ) + ( ( A .ih D ) + ( B .ih C ) ) )

Proof

Step Hyp Ref Expression
1 normlem8.1 
 |-  A e. ~H
2 normlem8.2 
 |-  B e. ~H
3 normlem8.3 
 |-  C e. ~H
4 normlem8.4 
 |-  D e. ~H
5 his7 
 |-  ( ( A e. ~H /\ C e. ~H /\ D e. ~H ) -> ( A .ih ( C +h D ) ) = ( ( A .ih C ) + ( A .ih D ) ) )
6 1 3 4 5 mp3an 
 |-  ( A .ih ( C +h D ) ) = ( ( A .ih C ) + ( A .ih D ) )
7 his7 
 |-  ( ( B e. ~H /\ C e. ~H /\ D e. ~H ) -> ( B .ih ( C +h D ) ) = ( ( B .ih C ) + ( B .ih D ) ) )
8 2 3 4 7 mp3an 
 |-  ( B .ih ( C +h D ) ) = ( ( B .ih C ) + ( B .ih D ) )
9 6 8 oveq12i 
 |-  ( ( A .ih ( C +h D ) ) + ( B .ih ( C +h D ) ) ) = ( ( ( A .ih C ) + ( A .ih D ) ) + ( ( B .ih C ) + ( B .ih D ) ) )
10 3 4 hvaddcli 
 |-  ( C +h D ) e. ~H
11 ax-his2 
 |-  ( ( A e. ~H /\ B e. ~H /\ ( C +h D ) e. ~H ) -> ( ( A +h B ) .ih ( C +h D ) ) = ( ( A .ih ( C +h D ) ) + ( B .ih ( C +h D ) ) ) )
12 1 2 10 11 mp3an 
 |-  ( ( A +h B ) .ih ( C +h D ) ) = ( ( A .ih ( C +h D ) ) + ( B .ih ( C +h D ) ) )
13 1 3 hicli 
 |-  ( A .ih C ) e. CC
14 2 4 hicli 
 |-  ( B .ih D ) e. CC
15 1 4 hicli 
 |-  ( A .ih D ) e. CC
16 2 3 hicli 
 |-  ( B .ih C ) e. CC
17 13 14 15 16 add42i 
 |-  ( ( ( A .ih C ) + ( B .ih D ) ) + ( ( A .ih D ) + ( B .ih C ) ) ) = ( ( ( A .ih C ) + ( A .ih D ) ) + ( ( B .ih C ) + ( B .ih D ) ) )
18 9 12 17 3eqtr4i 
 |-  ( ( A +h B ) .ih ( C +h D ) ) = ( ( ( A .ih C ) + ( B .ih D ) ) + ( ( A .ih D ) + ( B .ih C ) ) )