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Metamath Proof Explorer

Theorem grpsubadd0sub

Description: Subtraction expressed as addition of the difference of the identity element and the subtrahend. (Contributed by AV, 9-Nov-2019)

Ref Expression
Hypotheses grpsubid.b 
|- B = ( Base ` G )
grpsubid.o 
|- .0. = ( 0g ` G )
grpsubid.m 
|- .- = ( -g ` G )
grpsubadd0sub.p 
|- .+ = ( +g ` G )
Assertion grpsubadd0sub 
|- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .- Y ) = ( X .+ ( .0. .- Y ) ) )

Proof

Step Hyp Ref Expression
1 grpsubid.b 
 |-  B = ( Base ` G )
2 grpsubid.o 
 |-  .0. = ( 0g ` G )
3 grpsubid.m 
 |-  .- = ( -g ` G )
4 grpsubadd0sub.p 
 |-  .+ = ( +g ` G )
5 eqid 
 |-  ( invg ` G ) = ( invg ` G )
6 1 4 5 3 grpsubval 
 |-  ( ( X e. B /\ Y e. B ) -> ( X .- Y ) = ( X .+ ( ( invg ` G ) ` Y ) ) )
7 6 3adant1 
 |-  ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .- Y ) = ( X .+ ( ( invg ` G ) ` Y ) ) )
8 1 3 5 2 grpinvval2 
 |-  ( ( G e. Grp /\ Y e. B ) -> ( ( invg ` G ) ` Y ) = ( .0. .- Y ) )
9 8 3adant2 
 |-  ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( invg ` G ) ` Y ) = ( .0. .- Y ) )
10 9 oveq2d 
 |-  ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .+ ( ( invg ` G ) ` Y ) ) = ( X .+ ( .0. .- Y ) ) )
11 7 10 eqtrd 
 |-  ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .- Y ) = ( X .+ ( .0. .- Y ) ) )