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Metamath Proof Explorer

Theorem ablodiv32

Description: Swap the second and third terms in a double 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 ablodiv32 
|- ( ( G e. AbelOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A D B ) D C ) = ( ( A D C ) D B ) )

Proof

Step Hyp Ref Expression
1 abldiv.1 
 |-  X = ran G
2 abldiv.3 
 |-  D = ( /g ` G )
3 1 ablocom 
 |-  ( ( G e. AbelOp /\ B e. X /\ C e. X ) -> ( B G C ) = ( C G B ) )
4 3 3adant3r1 
 |-  ( ( G e. AbelOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( B G C ) = ( C G B ) )
5 4 oveq2d 
 |-  ( ( G e. AbelOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A D ( B G C ) ) = ( A D ( C G B ) ) )
6 1 2 ablodivdiv4 
 |-  ( ( G e. AbelOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A D B ) D C ) = ( A D ( B G C ) ) )
7 3ancomb 
 |-  ( ( A e. X /\ B e. X /\ C e. X ) <-> ( A e. X /\ C e. X /\ B e. X ) )
8 1 2 ablodivdiv4 
 |-  ( ( G e. AbelOp /\ ( A e. X /\ C e. X /\ B e. X ) ) -> ( ( A D C ) D B ) = ( A D ( C G B ) ) )
9 7 8 sylan2b 
 |-  ( ( G e. AbelOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A D C ) D B ) = ( A D ( C G B ) ) )
10 5 6 9 3eqtr4d 
 |-  ( ( G e. AbelOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A D B ) D C ) = ( ( A D C ) D B ) )