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Metamath Proof Explorer

Theorem grpasscan2

Description: An associative cancellation law for groups. (Contributed by Paul Chapman, 17-Apr-2009) (Revised by AV, 30-Aug-2021)

Ref Expression
Hypotheses grplcan.b 
|- B = ( Base ` G )
grplcan.p 
|- .+ = ( +g ` G )
grpasscan1.n 
|- N = ( invg ` G )
Assertion grpasscan2 
|- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( X .+ ( N ` Y ) ) .+ Y ) = X )

Proof

Step Hyp Ref Expression
1 grplcan.b 
 |-  B = ( Base ` G )
2 grplcan.p 
 |-  .+ = ( +g ` G )
3 grpasscan1.n 
 |-  N = ( invg ` G )
4 simp1 
 |-  ( ( G e. Grp /\ X e. B /\ Y e. B ) -> G e. Grp )
5 simp2 
 |-  ( ( G e. Grp /\ X e. B /\ Y e. B ) -> X e. B )
6 1 3 grpinvcl 
 |-  ( ( G e. Grp /\ Y e. B ) -> ( N ` Y ) e. B )
7 6 3adant2 
 |-  ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( N ` Y ) e. B )
8 simp3 
 |-  ( ( G e. Grp /\ X e. B /\ Y e. B ) -> Y e. B )
9 1 2 grpass 
 |-  ( ( G e. Grp /\ ( X e. B /\ ( N ` Y ) e. B /\ Y e. B ) ) -> ( ( X .+ ( N ` Y ) ) .+ Y ) = ( X .+ ( ( N ` Y ) .+ Y ) ) )
10 4 5 7 8 9 syl13anc 
 |-  ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( X .+ ( N ` Y ) ) .+ Y ) = ( X .+ ( ( N ` Y ) .+ Y ) ) )
11 eqid 
 |-  ( 0g ` G ) = ( 0g ` G )
12 1 2 11 3 grplinv 
 |-  ( ( G e. Grp /\ Y e. B ) -> ( ( N ` Y ) .+ Y ) = ( 0g ` G ) )
13 12 3adant2 
 |-  ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( N ` Y ) .+ Y ) = ( 0g ` G ) )
14 13 oveq2d 
 |-  ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .+ ( ( N ` Y ) .+ Y ) ) = ( X .+ ( 0g ` G ) ) )
15 1 2 11 grprid 
 |-  ( ( G e. Grp /\ X e. B ) -> ( X .+ ( 0g ` G ) ) = X )
16 15 3adant3 
 |-  ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .+ ( 0g ` G ) ) = X )
17 10 14 16 3eqtrd 
 |-  ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( X .+ ( N ` Y ) ) .+ Y ) = X )