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Metamath Proof Explorer

Theorem grpsubval

Description: Group subtraction (division) operation. (Contributed by NM, 31-Mar-2014) (Revised by Mario Carneiro, 13-Dec-2014)

Ref Expression
Hypotheses grpsubval.b 
|- B = ( Base ` G )
grpsubval.p 
|- .+ = ( +g ` G )
grpsubval.i 
|- I = ( invg ` G )
grpsubval.m 
|- .- = ( -g ` G )
Assertion grpsubval 
|- ( ( X e. B /\ Y e. B ) -> ( X .- Y ) = ( X .+ ( I ` Y ) ) )

Proof

Step Hyp Ref Expression
1 grpsubval.b 
 |-  B = ( Base ` G )
2 grpsubval.p 
 |-  .+ = ( +g ` G )
3 grpsubval.i 
 |-  I = ( invg ` G )
4 grpsubval.m 
 |-  .- = ( -g ` G )
5 oveq1 
 |-  ( x = X -> ( x .+ ( I ` y ) ) = ( X .+ ( I ` y ) ) )
6 fveq2 
 |-  ( y = Y -> ( I ` y ) = ( I ` Y ) )
7 6 oveq2d 
 |-  ( y = Y -> ( X .+ ( I ` y ) ) = ( X .+ ( I ` Y ) ) )
8 1 2 3 4 grpsubfval 
 |-  .- = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) )
9 ovex 
 |-  ( X .+ ( I ` Y ) ) e. _V
10 5 7 8 9 ovmpo 
 |-  ( ( X e. B /\ Y e. B ) -> ( X .- Y ) = ( X .+ ( I ` Y ) ) )