Metamath Proof Explorer


Theorem hvadd12

Description: Commutative/associative law. (Contributed by NM, 19-Oct-1999) (New usage is discouraged.)

Ref Expression
Assertion hvadd12
|- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( A +h ( B +h C ) ) = ( B +h ( A +h C ) ) )

Proof

Step Hyp Ref Expression
1 ax-hvcom
 |-  ( ( A e. ~H /\ B e. ~H ) -> ( A +h B ) = ( B +h A ) )
2 1 oveq1d
 |-  ( ( A e. ~H /\ B e. ~H ) -> ( ( A +h B ) +h C ) = ( ( B +h A ) +h C ) )
3 2 3adant3
 |-  ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( A +h B ) +h C ) = ( ( B +h A ) +h C ) )
4 ax-hvass
 |-  ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( A +h B ) +h C ) = ( A +h ( B +h C ) ) )
5 ax-hvass
 |-  ( ( B e. ~H /\ A e. ~H /\ C e. ~H ) -> ( ( B +h A ) +h C ) = ( B +h ( A +h C ) ) )
6 5 3com12
 |-  ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( B +h A ) +h C ) = ( B +h ( A +h C ) ) )
7 3 4 6 3eqtr3d
 |-  ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( A +h ( B +h C ) ) = ( B +h ( A +h C ) ) )