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Metamath Proof Explorer

Theorem hvadd4

Description: Hilbert vector space addition law. (Contributed by NM, 16-Oct-1999) (New usage is discouraged.)

Ref Expression
Assertion hvadd4 
|- ( ( ( 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 hvadd32 
 |-  ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( A +h B ) +h C ) = ( ( A +h C ) +h B ) )
2 1 oveq1d 
 |-  ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( ( A +h B ) +h C ) +h D ) = ( ( ( A +h C ) +h B ) +h D ) )
3 2 3expa 
 |-  ( ( ( A e. ~H /\ B e. ~H ) /\ C e. ~H ) -> ( ( ( A +h B ) +h C ) +h D ) = ( ( ( A +h C ) +h B ) +h D ) )
4 3 adantrr 
 |-  ( ( ( 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 ) )
5 hvaddcl 
 |-  ( ( A e. ~H /\ B e. ~H ) -> ( A +h B ) e. ~H )
6 ax-hvass 
 |-  ( ( ( A +h B ) e. ~H /\ C e. ~H /\ D e. ~H ) -> ( ( ( A +h B ) +h C ) +h D ) = ( ( A +h B ) +h ( C +h D ) ) )
7 6 3expb 
 |-  ( ( ( A +h B ) e. ~H /\ ( C e. ~H /\ D e. ~H ) ) -> ( ( ( A +h B ) +h C ) +h D ) = ( ( A +h B ) +h ( C +h D ) ) )
8 5 7 sylan 
 |-  ( ( ( A e. ~H /\ B e. ~H ) /\ ( C e. ~H /\ D e. ~H ) ) -> ( ( ( A +h B ) +h C ) +h D ) = ( ( A +h B ) +h ( C +h D ) ) )
9 hvaddcl 
 |-  ( ( A e. ~H /\ C e. ~H ) -> ( A +h C ) e. ~H )
10 ax-hvass 
 |-  ( ( ( A +h C ) e. ~H /\ B e. ~H /\ D e. ~H ) -> ( ( ( A +h C ) +h B ) +h D ) = ( ( A +h C ) +h ( B +h D ) ) )
11 10 3expb 
 |-  ( ( ( A +h C ) e. ~H /\ ( B e. ~H /\ D e. ~H ) ) -> ( ( ( A +h C ) +h B ) +h D ) = ( ( A +h C ) +h ( B +h D ) ) )
12 9 11 sylan 
 |-  ( ( ( A e. ~H /\ C e. ~H ) /\ ( B e. ~H /\ D e. ~H ) ) -> ( ( ( A +h C ) +h B ) +h D ) = ( ( A +h C ) +h ( B +h D ) ) )
13 12 an4s 
 |-  ( ( ( A e. ~H /\ B e. ~H ) /\ ( C e. ~H /\ D e. ~H ) ) -> ( ( ( A +h C ) +h B ) +h D ) = ( ( A +h C ) +h ( B +h D ) ) )
14 4 8 13 3eqtr3d 
 |-  ( ( ( 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 ) ) )