Metamath Proof Explorer

Theorem hvaddsub12

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

Ref Expression
Assertion hvaddsub12 
|- ( ( 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 neg1cn 
 |-  -u 1 e. CC
2 hvmulcl 
 |-  ( ( -u 1 e. CC /\ C e. ~H ) -> ( -u 1 .h C ) e. ~H )
3 1 2 mpan 
 |-  ( C e. ~H -> ( -u 1 .h C ) e. ~H )
4 hvadd12 
 |-  ( ( A e. ~H /\ B e. ~H /\ ( -u 1 .h C ) e. ~H ) -> ( A +h ( B +h ( -u 1 .h C ) ) ) = ( B +h ( A +h ( -u 1 .h C ) ) ) )
5 3 4 syl3an3 
 |-  ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( A +h ( B +h ( -u 1 .h C ) ) ) = ( B +h ( A +h ( -u 1 .h C ) ) ) )
6 hvsubval 
 |-  ( ( B e. ~H /\ C e. ~H ) -> ( B -h C ) = ( B +h ( -u 1 .h C ) ) )
7 6 oveq2d 
 |-  ( ( B e. ~H /\ C e. ~H ) -> ( A +h ( B -h C ) ) = ( A +h ( B +h ( -u 1 .h C ) ) ) )
8 7 3adant1 
 |-  ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( A +h ( B -h C ) ) = ( A +h ( B +h ( -u 1 .h C ) ) ) )
9 hvsubval 
 |-  ( ( A e. ~H /\ C e. ~H ) -> ( A -h C ) = ( A +h ( -u 1 .h C ) ) )
10 9 oveq2d 
 |-  ( ( A e. ~H /\ C e. ~H ) -> ( B +h ( A -h C ) ) = ( B +h ( A +h ( -u 1 .h C ) ) ) )
11 10 3adant2 
 |-  ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( B +h ( A -h C ) ) = ( B +h ( A +h ( -u 1 .h C ) ) ) )
12 5 8 11 3eqtr4d 
 |-  ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( A +h ( B -h C ) ) = ( B +h ( A -h C ) ) )