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Metamath Proof Explorer

Theorem grpasscan1

Description: An associative cancellation law for groups. (Contributed by Paul Chapman, 25-Feb-2008) (Revised by AV, 30-Aug-2021)

Ref Expression
Hypotheses grplcan.b 
|- B = ( Base ` G )
grplcan.p 
|- .+ = ( +g ` G )
grpasscan1.n 
|- N = ( invg ` G )
Assertion grpasscan1 
|- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .+ ( ( N ` X ) .+ Y ) ) = Y )

Proof

Step Hyp Ref Expression
1 grplcan.b 
 |-  B = ( Base ` G )
2 grplcan.p 
 |-  .+ = ( +g ` G )
3 grpasscan1.n 
 |-  N = ( invg ` G )
4 eqid 
 |-  ( 0g ` G ) = ( 0g ` G )
5 1 2 4 3 grprinv 
 |-  ( ( G e. Grp /\ X e. B ) -> ( X .+ ( N ` X ) ) = ( 0g ` G ) )
6 5 3adant3 
 |-  ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .+ ( N ` X ) ) = ( 0g ` G ) )
7 6 oveq1d 
 |-  ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( X .+ ( N ` X ) ) .+ Y ) = ( ( 0g ` G ) .+ Y ) )
8 1 3 grpinvcl 
 |-  ( ( G e. Grp /\ X e. B ) -> ( N ` X ) e. B )
9 1 2 grpass 
 |-  ( ( G e. Grp /\ ( X e. B /\ ( N ` X ) e. B /\ Y e. B ) ) -> ( ( X .+ ( N ` X ) ) .+ Y ) = ( X .+ ( ( N ` X ) .+ Y ) ) )
10 9 3exp2 
 |-  ( G e. Grp -> ( X e. B -> ( ( N ` X ) e. B -> ( Y e. B -> ( ( X .+ ( N ` X ) ) .+ Y ) = ( X .+ ( ( N ` X ) .+ Y ) ) ) ) ) )
11 10 imp 
 |-  ( ( G e. Grp /\ X e. B ) -> ( ( N ` X ) e. B -> ( Y e. B -> ( ( X .+ ( N ` X ) ) .+ Y ) = ( X .+ ( ( N ` X ) .+ Y ) ) ) ) )
12 8 11 mpd 
 |-  ( ( G e. Grp /\ X e. B ) -> ( Y e. B -> ( ( X .+ ( N ` X ) ) .+ Y ) = ( X .+ ( ( N ` X ) .+ Y ) ) ) )
13 12 3impia 
 |-  ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( X .+ ( N ` X ) ) .+ Y ) = ( X .+ ( ( N ` X ) .+ Y ) ) )
14 1 2 4 grplid 
 |-  ( ( G e. Grp /\ Y e. B ) -> ( ( 0g ` G ) .+ Y ) = Y )
15 14 3adant2 
 |-  ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( 0g ` G ) .+ Y ) = Y )
16 7 13 15 3eqtr3d 
 |-  ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .+ ( ( N ` X ) .+ Y ) ) = Y )