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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 ) ) )