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Metamath Proof Explorer

Theorem hvsub4

Description: Hilbert vector space addition/subtraction law. (Contributed by NM, 17-Oct-1999) (New usage is discouraged.)

Ref Expression
Assertion hvsub4 
|- ( ( ( A e. ~H /\ B e. ~H ) /\ ( C e. ~H /\ D e. ~H ) ) -> ( ( A +h B ) -h ( C +h D ) ) = ( ( A -h C ) +h ( B -h D ) ) )

Proof

Step Hyp Ref Expression
1 hvaddcl 
 |-  ( ( A e. ~H /\ B e. ~H ) -> ( A +h B ) e. ~H )
2 hvaddcl 
 |-  ( ( C e. ~H /\ D e. ~H ) -> ( C +h D ) e. ~H )
3 hvsubval 
 |-  ( ( ( A +h B ) e. ~H /\ ( C +h D ) e. ~H ) -> ( ( A +h B ) -h ( C +h D ) ) = ( ( A +h B ) +h ( -u 1 .h ( C +h D ) ) ) )
4 1 2 3 syl2an 
 |-  ( ( ( A e. ~H /\ B e. ~H ) /\ ( C e. ~H /\ D e. ~H ) ) -> ( ( A +h B ) -h ( C +h D ) ) = ( ( A +h B ) +h ( -u 1 .h ( C +h D ) ) ) )
5 hvsubval 
 |-  ( ( A e. ~H /\ C e. ~H ) -> ( A -h C ) = ( A +h ( -u 1 .h C ) ) )
6 5 ad2ant2r 
 |-  ( ( ( A e. ~H /\ B e. ~H ) /\ ( C e. ~H /\ D e. ~H ) ) -> ( A -h C ) = ( A +h ( -u 1 .h C ) ) )
7 hvsubval 
 |-  ( ( B e. ~H /\ D e. ~H ) -> ( B -h D ) = ( B +h ( -u 1 .h D ) ) )
8 7 ad2ant2l 
 |-  ( ( ( A e. ~H /\ B e. ~H ) /\ ( C e. ~H /\ D e. ~H ) ) -> ( B -h D ) = ( B +h ( -u 1 .h D ) ) )
9 6 8 oveq12d 
 |-  ( ( ( A e. ~H /\ B e. ~H ) /\ ( C e. ~H /\ D e. ~H ) ) -> ( ( A -h C ) +h ( B -h D ) ) = ( ( A +h ( -u 1 .h C ) ) +h ( B +h ( -u 1 .h D ) ) ) )
10 neg1cn 
 |-  -u 1 e. CC
11 ax-hvdistr1 
 |-  ( ( -u 1 e. CC /\ C e. ~H /\ D e. ~H ) -> ( -u 1 .h ( C +h D ) ) = ( ( -u 1 .h C ) +h ( -u 1 .h D ) ) )
12 10 11 mp3an1 
 |-  ( ( C e. ~H /\ D e. ~H ) -> ( -u 1 .h ( C +h D ) ) = ( ( -u 1 .h C ) +h ( -u 1 .h D ) ) )
13 12 adantl 
 |-  ( ( ( A e. ~H /\ B e. ~H ) /\ ( C e. ~H /\ D e. ~H ) ) -> ( -u 1 .h ( C +h D ) ) = ( ( -u 1 .h C ) +h ( -u 1 .h D ) ) )
14 13 oveq2d 
 |-  ( ( ( A e. ~H /\ B e. ~H ) /\ ( C e. ~H /\ D e. ~H ) ) -> ( ( A +h B ) +h ( -u 1 .h ( C +h D ) ) ) = ( ( A +h B ) +h ( ( -u 1 .h C ) +h ( -u 1 .h D ) ) ) )
15 hvmulcl 
 |-  ( ( -u 1 e. CC /\ C e. ~H ) -> ( -u 1 .h C ) e. ~H )
16 10 15 mpan 
 |-  ( C e. ~H -> ( -u 1 .h C ) e. ~H )
17 16 anim2i 
 |-  ( ( A e. ~H /\ C e. ~H ) -> ( A e. ~H /\ ( -u 1 .h C ) e. ~H ) )
18 hvmulcl 
 |-  ( ( -u 1 e. CC /\ D e. ~H ) -> ( -u 1 .h D ) e. ~H )
19 10 18 mpan 
 |-  ( D e. ~H -> ( -u 1 .h D ) e. ~H )
20 19 anim2i 
 |-  ( ( B e. ~H /\ D e. ~H ) -> ( B e. ~H /\ ( -u 1 .h D ) e. ~H ) )
21 17 20 anim12i 
 |-  ( ( ( A e. ~H /\ C e. ~H ) /\ ( B e. ~H /\ D e. ~H ) ) -> ( ( A e. ~H /\ ( -u 1 .h C ) e. ~H ) /\ ( B e. ~H /\ ( -u 1 .h D ) e. ~H ) ) )
22 21 an4s 
 |-  ( ( ( A e. ~H /\ B e. ~H ) /\ ( C e. ~H /\ D e. ~H ) ) -> ( ( A e. ~H /\ ( -u 1 .h C ) e. ~H ) /\ ( B e. ~H /\ ( -u 1 .h D ) e. ~H ) ) )
23 hvadd4 
 |-  ( ( ( A e. ~H /\ ( -u 1 .h C ) e. ~H ) /\ ( B e. ~H /\ ( -u 1 .h D ) e. ~H ) ) -> ( ( A +h ( -u 1 .h C ) ) +h ( B +h ( -u 1 .h D ) ) ) = ( ( A +h B ) +h ( ( -u 1 .h C ) +h ( -u 1 .h D ) ) ) )
24 22 23 syl 
 |-  ( ( ( A e. ~H /\ B e. ~H ) /\ ( C e. ~H /\ D e. ~H ) ) -> ( ( A +h ( -u 1 .h C ) ) +h ( B +h ( -u 1 .h D ) ) ) = ( ( A +h B ) +h ( ( -u 1 .h C ) +h ( -u 1 .h D ) ) ) )
25 14 24 eqtr4d 
 |-  ( ( ( A e. ~H /\ B e. ~H ) /\ ( C e. ~H /\ D e. ~H ) ) -> ( ( A +h B ) +h ( -u 1 .h ( C +h D ) ) ) = ( ( A +h ( -u 1 .h C ) ) +h ( B +h ( -u 1 .h D ) ) ) )
26 9 25 eqtr4d 
 |-  ( ( ( A e. ~H /\ B e. ~H ) /\ ( C e. ~H /\ D e. ~H ) ) -> ( ( A -h C ) +h ( B -h D ) ) = ( ( A +h B ) +h ( -u 1 .h ( C +h D ) ) ) )
27 4 26 eqtr4d 
 |-  ( ( ( A e. ~H /\ B e. ~H ) /\ ( C e. ~H /\ D e. ~H ) ) -> ( ( A +h B ) -h ( C +h D ) ) = ( ( A -h C ) +h ( B -h D ) ) )