Metamath Proof Explorer

Theorem grpomuldivass

Description: Associative-type law for multiplication and division. (Contributed by NM, 15-Feb-2008) (New usage is discouraged.)

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

Proof

Step Hyp Ref Expression
1 grpdivf.1 
 |-  X = ran G
2 grpdivf.3 
 |-  D = ( /g ` G )
3 simpr1 
 |-  ( ( G e. GrpOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> A e. X )
4 simpr2 
 |-  ( ( G e. GrpOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> B e. X )
5 eqid 
 |-  ( inv ` G ) = ( inv ` G )
6 1 5 grpoinvcl 
 |-  ( ( G e. GrpOp /\ C e. X ) -> ( ( inv ` G ) ` C ) e. X )
7 6 3ad2antr3 
 |-  ( ( G e. GrpOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( inv ` G ) ` C ) e. X )
8 3 4 7 3jca 
 |-  ( ( G e. GrpOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A e. X /\ B e. X /\ ( ( inv ` G ) ` C ) e. X ) )
9 1 grpoass 
 |-  ( ( G e. GrpOp /\ ( A e. X /\ B e. X /\ ( ( inv ` G ) ` C ) e. X ) ) -> ( ( A G B ) G ( ( inv ` G ) ` C ) ) = ( A G ( B G ( ( inv ` G ) ` C ) ) ) )
10 8 9 syldan 
 |-  ( ( G e. GrpOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A G B ) G ( ( inv ` G ) ` C ) ) = ( A G ( B G ( ( inv ` G ) ` C ) ) ) )
11 simpl 
 |-  ( ( G e. GrpOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> G e. GrpOp )
12 1 grpocl 
 |-  ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( A G B ) e. X )
13 12 3adant3r3 
 |-  ( ( G e. GrpOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A G B ) e. X )
14 simpr3 
 |-  ( ( G e. GrpOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> C e. X )
15 1 5 2 grpodivval 
 |-  ( ( G e. GrpOp /\ ( A G B ) e. X /\ C e. X ) -> ( ( A G B ) D C ) = ( ( A G B ) G ( ( inv ` G ) ` C ) ) )
16 11 13 14 15 syl3anc 
 |-  ( ( G e. GrpOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A G B ) D C ) = ( ( A G B ) G ( ( inv ` G ) ` C ) ) )
17 1 5 2 grpodivval 
 |-  ( ( G e. GrpOp /\ B e. X /\ C e. X ) -> ( B D C ) = ( B G ( ( inv ` G ) ` C ) ) )
18 17 3adant3r1 
 |-  ( ( G e. GrpOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( B D C ) = ( B G ( ( inv ` G ) ` C ) ) )
19 18 oveq2d 
 |-  ( ( G e. GrpOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A G ( B D C ) ) = ( A G ( B G ( ( inv ` G ) ` C ) ) ) )
20 10 16 19 3eqtr4d 
 |-  ( ( G e. GrpOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A G B ) D C ) = ( A G ( B D C ) ) )