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 ) )`
` |-  ( ( G e. AbelOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( B G C ) = ( C G B ) )`
` |-  ( ( G e. AbelOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A D ( B G C ) ) = ( A D ( C G B ) ) )`
` |-  ( ( G e. AbelOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A D B ) D C ) = ( A D ( B G C ) ) )`
` |-  ( ( A e. X /\ B e. X /\ C e. X ) <-> ( A e. X /\ C e. X /\ B e. X ) )`
` |-  ( ( G e. AbelOp /\ ( A e. X /\ C e. X /\ B e. X ) ) -> ( ( A D C ) D B ) = ( A D ( C G B ) ) )`
` |-  ( ( G e. AbelOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A D C ) D B ) = ( A D ( C G B ) ) )`
` |-  ( ( G e. AbelOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A D B ) D C ) = ( ( A D C ) D B ) )`