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Metamath Proof Explorer

Theorem hvsub32

Description: Hilbert vector space commutative/associative law. (Contributed by Mario Carneiro, 15-May-2014) (New usage is discouraged.)

Ref Expression
Assertion hvsub32 
|- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( A -h B ) -h C ) = ( ( A -h C ) -h B ) )

Proof

Step Hyp Ref Expression
1 ax-hvcom 
 |-  ( ( B e. ~H /\ C e. ~H ) -> ( B +h C ) = ( C +h B ) )
2 1 3adant1 
 |-  ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( B +h C ) = ( C +h B ) )
3 2 oveq2d 
 |-  ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( A -h ( B +h C ) ) = ( A -h ( C +h B ) ) )
4 hvsubass 
 |-  ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( A -h B ) -h C ) = ( A -h ( B +h C ) ) )
5 hvsubass 
 |-  ( ( A e. ~H /\ C e. ~H /\ B e. ~H ) -> ( ( A -h C ) -h B ) = ( A -h ( C +h B ) ) )
6 5 3com23 
 |-  ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( A -h C ) -h B ) = ( A -h ( C +h B ) ) )
7 3 4 6 3eqtr4d 
 |-  ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( A -h B ) -h C ) = ( ( A -h C ) -h B ) )