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Metamath Proof Explorer

Theorem ablodivdiv

Description: Law for double group division. (Contributed by NM, 29-Feb-2008) (New usage is discouraged.)

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

Proof

Step Hyp Ref Expression
1 abldiv.1 
 |-  X = ran G
2 abldiv.3 
 |-  D = ( /g ` G )
3 ablogrpo 
 |-  ( G e. AbelOp -> G e. GrpOp )
4 1 2 grpodivdiv 
 |-  ( ( G e. GrpOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A D ( B D C ) ) = ( A G ( C D B ) ) )
5 3 4 sylan 
 |-  ( ( G e. AbelOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A D ( B D C ) ) = ( A G ( C D B ) ) )
6 3ancomb 
 |-  ( ( A e. X /\ B e. X /\ C e. X ) <-> ( A e. X /\ C e. X /\ B e. X ) )
7 1 2 grpomuldivass 
 |-  ( ( G e. GrpOp /\ ( A e. X /\ C e. X /\ B e. X ) ) -> ( ( A G C ) D B ) = ( A G ( C D B ) ) )
8 3 7 sylan 
 |-  ( ( G e. AbelOp /\ ( A e. X /\ C e. X /\ B e. X ) ) -> ( ( A G C ) D B ) = ( A G ( C D B ) ) )
9 1 2 ablomuldiv 
 |-  ( ( G e. AbelOp /\ ( A e. X /\ C e. X /\ B e. X ) ) -> ( ( A G C ) D B ) = ( ( A D B ) G C ) )
10 8 9 eqtr3d 
 |-  ( ( G e. AbelOp /\ ( A e. X /\ C e. X /\ B e. X ) ) -> ( A G ( C D B ) ) = ( ( A D B ) G C ) )
11 6 10 sylan2b 
 |-  ( ( G e. AbelOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A G ( C D B ) ) = ( ( A D B ) G C ) )
12 5 11 eqtrd 
 |-  ( ( G e. AbelOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A D ( B D C ) ) = ( ( A D B ) G C ) )