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Metamath Proof Explorer

Theorem hvaddcan2

Description: Cancellation law for vector addition. (Contributed by NM, 18-May-2005) (New usage is discouraged.)

Ref Expression
Assertion hvaddcan2 
|- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( A +h C ) = ( B +h C ) <-> A = B ) )

Proof

Step Hyp Ref Expression
1 ax-hvcom 
 |-  ( ( C e. ~H /\ A e. ~H ) -> ( C +h A ) = ( A +h C ) )
2 1 3adant3 
 |-  ( ( C e. ~H /\ A e. ~H /\ B e. ~H ) -> ( C +h A ) = ( A +h C ) )
3 ax-hvcom 
 |-  ( ( C e. ~H /\ B e. ~H ) -> ( C +h B ) = ( B +h C ) )
4 3 3adant2 
 |-  ( ( C e. ~H /\ A e. ~H /\ B e. ~H ) -> ( C +h B ) = ( B +h C ) )
5 2 4 eqeq12d 
 |-  ( ( C e. ~H /\ A e. ~H /\ B e. ~H ) -> ( ( C +h A ) = ( C +h B ) <-> ( A +h C ) = ( B +h C ) ) )
6 hvaddcan 
 |-  ( ( C e. ~H /\ A e. ~H /\ B e. ~H ) -> ( ( C +h A ) = ( C +h B ) <-> A = B ) )
7 5 6 bitr3d 
 |-  ( ( C e. ~H /\ A e. ~H /\ B e. ~H ) -> ( ( A +h C ) = ( B +h C ) <-> A = B ) )
8 7 3coml 
 |-  ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( A +h C ) = ( B +h C ) <-> A = B ) )