Metamath Proof Explorer

Theorem grpodivdiv

Description: Double group division. (Contributed by NM, 24-Feb-2008) (New usage is discouraged.)

Ref Expression
Hypotheses grpdivf.1 
|- X = ran G
grpdivf.3 
|- D = ( /g ` G )
Assertion grpodivdiv 
|- ( ( G e. GrpOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A D ( B D C ) ) = ( A G ( C D B ) ) )

Proof

Step Hyp Ref Expression
1 grpdivf.1 
 |-  X = ran G
2 grpdivf.3 
 |-  D = ( /g ` G )
3 simpl 
 |-  ( ( G e. GrpOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> G e. GrpOp )
4 simpr1 
 |-  ( ( G e. GrpOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> A e. X )
5 1 2 grpodivcl 
 |-  ( ( G e. GrpOp /\ B e. X /\ C e. X ) -> ( B D C ) e. X )
6 5 3adant3r1 
 |-  ( ( G e. GrpOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( B D C ) e. X )
7 eqid 
 |-  ( inv ` G ) = ( inv ` G )
8 1 7 2 grpodivval 
 |-  ( ( G e. GrpOp /\ A e. X /\ ( B D C ) e. X ) -> ( A D ( B D C ) ) = ( A G ( ( inv ` G ) ` ( B D C ) ) ) )
9 3 4 6 8 syl3anc 
 |-  ( ( G e. GrpOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A D ( B D C ) ) = ( A G ( ( inv ` G ) ` ( B D C ) ) ) )
10 1 7 2 grpoinvdiv 
 |-  ( ( G e. GrpOp /\ B e. X /\ C e. X ) -> ( ( inv ` G ) ` ( B D C ) ) = ( C D B ) )
11 10 3adant3r1 
 |-  ( ( G e. GrpOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( inv ` G ) ` ( B D C ) ) = ( C D B ) )
12 11 oveq2d 
 |-  ( ( G e. GrpOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A G ( ( inv ` G ) ` ( B D C ) ) ) = ( A G ( C D B ) ) )
13 9 12 eqtrd 
 |-  ( ( G e. GrpOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A D ( B D C ) ) = ( A G ( C D B ) ) )