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 ) )`