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Metamath Proof Explorer

Theorem grpsubrcan

Description: Right cancellation law for group subtraction. (Contributed by NM, 31-Mar-2014)

Ref Expression
Hypotheses grpsubcl.b 
|- B = ( Base ` G )
grpsubcl.m 
|- .- = ( -g ` G )
Assertion grpsubrcan 
|- ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .- Z ) = ( Y .- Z ) <-> X = Y ) )

Proof

Step Hyp Ref Expression
1 grpsubcl.b 
 |-  B = ( Base ` G )
2 grpsubcl.m 
 |-  .- = ( -g ` G )
3 eqid 
 |-  ( +g ` G ) = ( +g ` G )
4 eqid 
 |-  ( invg ` G ) = ( invg ` G )
5 1 3 4 2 grpsubval 
 |-  ( ( X e. B /\ Z e. B ) -> ( X .- Z ) = ( X ( +g ` G ) ( ( invg ` G ) ` Z ) ) )
6 5 3adant2 
 |-  ( ( X e. B /\ Y e. B /\ Z e. B ) -> ( X .- Z ) = ( X ( +g ` G ) ( ( invg ` G ) ` Z ) ) )
7 1 3 4 2 grpsubval 
 |-  ( ( Y e. B /\ Z e. B ) -> ( Y .- Z ) = ( Y ( +g ` G ) ( ( invg ` G ) ` Z ) ) )
8 7 3adant1 
 |-  ( ( X e. B /\ Y e. B /\ Z e. B ) -> ( Y .- Z ) = ( Y ( +g ` G ) ( ( invg ` G ) ` Z ) ) )
9 6 8 eqeq12d 
 |-  ( ( X e. B /\ Y e. B /\ Z e. B ) -> ( ( X .- Z ) = ( Y .- Z ) <-> ( X ( +g ` G ) ( ( invg ` G ) ` Z ) ) = ( Y ( +g ` G ) ( ( invg ` G ) ` Z ) ) ) )
10 9 adantl 
 |-  ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .- Z ) = ( Y .- Z ) <-> ( X ( +g ` G ) ( ( invg ` G ) ` Z ) ) = ( Y ( +g ` G ) ( ( invg ` G ) ` Z ) ) ) )
11 simpl 
 |-  ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> G e. Grp )
12 simpr1 
 |-  ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> X e. B )
13 simpr2 
 |-  ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Y e. B )
14 1 4 grpinvcl 
 |-  ( ( G e. Grp /\ Z e. B ) -> ( ( invg ` G ) ` Z ) e. B )
15 14 3ad2antr3 
 |-  ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( invg ` G ) ` Z ) e. B )
16 1 3 grprcan 
 |-  ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ ( ( invg ` G ) ` Z ) e. B ) ) -> ( ( X ( +g ` G ) ( ( invg ` G ) ` Z ) ) = ( Y ( +g ` G ) ( ( invg ` G ) ` Z ) ) <-> X = Y ) )
17 11 12 13 15 16 syl13anc 
 |-  ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X ( +g ` G ) ( ( invg ` G ) ` Z ) ) = ( Y ( +g ` G ) ( ( invg ` G ) ` Z ) ) <-> X = Y ) )
18 10 17 bitrd 
 |-  ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .- Z ) = ( Y .- Z ) <-> X = Y ) )