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Metamath Proof Explorer

Theorem grpinvval2

Description: A df-neg -like equation for inverse in terms of group subtraction. (Contributed by Mario Carneiro, 4-Oct-2015)

Ref Expression
Hypotheses grpsubcl.b 
|- B = ( Base ` G )
grpsubcl.m 
|- .- = ( -g ` G )
grpinvsub.n 
|- N = ( invg ` G )
grpinvval2.z 
|- .0. = ( 0g ` G )
Assertion grpinvval2 
|- ( ( G e. Grp /\ X e. B ) -> ( N ` X ) = ( .0. .- X ) )

Proof

Step Hyp Ref Expression
1 grpsubcl.b 
 |-  B = ( Base ` G )
2 grpsubcl.m 
 |-  .- = ( -g ` G )
3 grpinvsub.n 
 |-  N = ( invg ` G )
4 grpinvval2.z 
 |-  .0. = ( 0g ` G )
5 1 4 grpidcl 
 |-  ( G e. Grp -> .0. e. B )
6 eqid 
 |-  ( +g ` G ) = ( +g ` G )
7 1 6 3 2 grpsubval 
 |-  ( ( .0. e. B /\ X e. B ) -> ( .0. .- X ) = ( .0. ( +g ` G ) ( N ` X ) ) )
8 5 7 sylan 
 |-  ( ( G e. Grp /\ X e. B ) -> ( .0. .- X ) = ( .0. ( +g ` G ) ( N ` X ) ) )
9 1 3 grpinvcl 
 |-  ( ( G e. Grp /\ X e. B ) -> ( N ` X ) e. B )
10 1 6 4 grplid 
 |-  ( ( G e. Grp /\ ( N ` X ) e. B ) -> ( .0. ( +g ` G ) ( N ` X ) ) = ( N ` X ) )
11 9 10 syldan 
 |-  ( ( G e. Grp /\ X e. B ) -> ( .0. ( +g ` G ) ( N ` X ) ) = ( N ` X ) )
12 8 11 eqtr2d 
 |-  ( ( G e. Grp /\ X e. B ) -> ( N ` X ) = ( .0. .- X ) )