Metamath Proof Explorer

Theorem hvsubsub4i

Description: Hilbert vector space addition law. (Contributed by NM, 31-Oct-1999) (New usage is discouraged.)

Ref Expression
Hypotheses hvass.1 
|- A e. ~H
hvass.2 
|- B e. ~H
hvass.3 
|- C e. ~H
hvadd4.4 
|- D e. ~H
Assertion hvsubsub4i 
|- ( ( A -h B ) -h ( C -h D ) ) = ( ( A -h C ) -h ( B -h D ) )

Proof

Step Hyp Ref Expression
1 hvass.1 
 |-  A e. ~H
2 hvass.2 
 |-  B e. ~H
3 hvass.3 
 |-  C e. ~H
4 hvadd4.4 
 |-  D e. ~H
5 neg1cn 
 |-  -u 1 e. CC
6 5 2 hvmulcli 
 |-  ( -u 1 .h B ) e. ~H
7 5 3 hvmulcli 
 |-  ( -u 1 .h C ) e. ~H
8 5 4 hvmulcli 
 |-  ( -u 1 .h D ) e. ~H
9 5 8 hvmulcli 
 |-  ( -u 1 .h ( -u 1 .h D ) ) e. ~H
10 1 6 7 9 hvadd4i 
 |-  ( ( A +h ( -u 1 .h B ) ) +h ( ( -u 1 .h C ) +h ( -u 1 .h ( -u 1 .h D ) ) ) ) = ( ( A +h ( -u 1 .h C ) ) +h ( ( -u 1 .h B ) +h ( -u 1 .h ( -u 1 .h D ) ) ) )
11 5 3 8 hvdistr1i 
 |-  ( -u 1 .h ( C +h ( -u 1 .h D ) ) ) = ( ( -u 1 .h C ) +h ( -u 1 .h ( -u 1 .h D ) ) )
12 11 oveq2i 
 |-  ( ( A +h ( -u 1 .h B ) ) +h ( -u 1 .h ( C +h ( -u 1 .h D ) ) ) ) = ( ( A +h ( -u 1 .h B ) ) +h ( ( -u 1 .h C ) +h ( -u 1 .h ( -u 1 .h D ) ) ) )
13 5 2 8 hvdistr1i 
 |-  ( -u 1 .h ( B +h ( -u 1 .h D ) ) ) = ( ( -u 1 .h B ) +h ( -u 1 .h ( -u 1 .h D ) ) )
14 13 oveq2i 
 |-  ( ( A +h ( -u 1 .h C ) ) +h ( -u 1 .h ( B +h ( -u 1 .h D ) ) ) ) = ( ( A +h ( -u 1 .h C ) ) +h ( ( -u 1 .h B ) +h ( -u 1 .h ( -u 1 .h D ) ) ) )
15 10 12 14 3eqtr4i 
 |-  ( ( A +h ( -u 1 .h B ) ) +h ( -u 1 .h ( C +h ( -u 1 .h D ) ) ) ) = ( ( A +h ( -u 1 .h C ) ) +h ( -u 1 .h ( B +h ( -u 1 .h D ) ) ) )
16 1 6 hvaddcli 
 |-  ( A +h ( -u 1 .h B ) ) e. ~H
17 3 8 hvaddcli 
 |-  ( C +h ( -u 1 .h D ) ) e. ~H
18 16 17 hvsubvali 
 |-  ( ( A +h ( -u 1 .h B ) ) -h ( C +h ( -u 1 .h D ) ) ) = ( ( A +h ( -u 1 .h B ) ) +h ( -u 1 .h ( C +h ( -u 1 .h D ) ) ) )
19 1 7 hvaddcli 
 |-  ( A +h ( -u 1 .h C ) ) e. ~H
20 2 8 hvaddcli 
 |-  ( B +h ( -u 1 .h D ) ) e. ~H
21 19 20 hvsubvali 
 |-  ( ( A +h ( -u 1 .h C ) ) -h ( B +h ( -u 1 .h D ) ) ) = ( ( A +h ( -u 1 .h C ) ) +h ( -u 1 .h ( B +h ( -u 1 .h D ) ) ) )
22 15 18 21 3eqtr4i 
 |-  ( ( A +h ( -u 1 .h B ) ) -h ( C +h ( -u 1 .h D ) ) ) = ( ( A +h ( -u 1 .h C ) ) -h ( B +h ( -u 1 .h D ) ) )
23 1 2 hvsubvali 
 |-  ( A -h B ) = ( A +h ( -u 1 .h B ) )
24 3 4 hvsubvali 
 |-  ( C -h D ) = ( C +h ( -u 1 .h D ) )
25 23 24 oveq12i 
 |-  ( ( A -h B ) -h ( C -h D ) ) = ( ( A +h ( -u 1 .h B ) ) -h ( C +h ( -u 1 .h D ) ) )
26 1 3 hvsubvali 
 |-  ( A -h C ) = ( A +h ( -u 1 .h C ) )
27 2 4 hvsubvali 
 |-  ( B -h D ) = ( B +h ( -u 1 .h D ) )
28 26 27 oveq12i 
 |-  ( ( A -h C ) -h ( B -h D ) ) = ( ( A +h ( -u 1 .h C ) ) -h ( B +h ( -u 1 .h D ) ) )
29 22 25 28 3eqtr4i 
 |-  ( ( A -h B ) -h ( C -h D ) ) = ( ( A -h C ) -h ( B -h D ) )