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Theorem ablo4pnp 32171
Description: A commutative/associative law for Abelian groups. (Contributed by Jeff Madsen, 11-Jun-2010.)
Hypotheses
Ref Expression
abl4pnp.1  |-  X  =  ran  G
abl4pnp.2  |-  D  =  (  /g  `  G
)
Assertion
Ref Expression
ablo4pnp  |-  ( ( G  e.  AbelOp  /\  (
( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  F  e.  X ) ) )  ->  ( ( A G B ) D ( C G F ) )  =  ( ( A D C ) G ( B D F ) ) )

Proof of Theorem ablo4pnp
StepHypRef Expression
1 df-3an 986 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  <->  ( ( A  e.  X  /\  B  e.  X
)  /\  C  e.  X ) )
2 abl4pnp.1 . . . . . 6  |-  X  =  ran  G
3 abl4pnp.2 . . . . . 6  |-  D  =  (  /g  `  G
)
42, 3ablomuldiv 26010 . . . . 5  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G B ) D C )  =  ( ( A D C ) G B ) )
51, 4sylan2br 479 . . . 4  |-  ( ( G  e.  AbelOp  /\  (
( A  e.  X  /\  B  e.  X
)  /\  C  e.  X ) )  -> 
( ( A G B ) D C )  =  ( ( A D C ) G B ) )
65adantrrr 730 . . 3  |-  ( ( G  e.  AbelOp  /\  (
( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  F  e.  X ) ) )  ->  ( ( A G B ) D C )  =  ( ( A D C ) G B ) )
76oveq1d 6303 . 2  |-  ( ( G  e.  AbelOp  /\  (
( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  F  e.  X ) ) )  ->  ( ( ( A G B ) D C ) D F )  =  ( ( ( A D C ) G B ) D F ) )
8 ablogrpo 26005 . . . . . . 7  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
92grpocl 25921 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
1093expib 1210 . . . . . . 7  |-  ( G  e.  GrpOp  ->  ( ( A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X ) )
118, 10syl 17 . . . . . 6  |-  ( G  e.  AbelOp  ->  ( ( A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X ) )
1211anim1d 567 . . . . 5  |-  ( G  e.  AbelOp  ->  ( ( ( A  e.  X  /\  B  e.  X )  /\  ( C  e.  X  /\  F  e.  X
) )  ->  (
( A G B )  e.  X  /\  ( C  e.  X  /\  F  e.  X
) ) ) )
13 3anass 988 . . . . 5  |-  ( ( ( A G B )  e.  X  /\  C  e.  X  /\  F  e.  X )  <->  ( ( A G B )  e.  X  /\  ( C  e.  X  /\  F  e.  X
) ) )
1412, 13syl6ibr 231 . . . 4  |-  ( G  e.  AbelOp  ->  ( ( ( A  e.  X  /\  B  e.  X )  /\  ( C  e.  X  /\  F  e.  X
) )  ->  (
( A G B )  e.  X  /\  C  e.  X  /\  F  e.  X )
) )
1514imp 431 . . 3  |-  ( ( G  e.  AbelOp  /\  (
( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  F  e.  X ) ) )  ->  ( ( A G B )  e.  X  /\  C  e.  X  /\  F  e.  X ) )
162, 3ablodivdiv4 26012 . . 3  |-  ( ( G  e.  AbelOp  /\  (
( A G B )  e.  X  /\  C  e.  X  /\  F  e.  X )
)  ->  ( (
( A G B ) D C ) D F )  =  ( ( A G B ) D ( C G F ) ) )
1715, 16syldan 473 . 2  |-  ( ( G  e.  AbelOp  /\  (
( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  F  e.  X ) ) )  ->  ( ( ( A G B ) D C ) D F )  =  ( ( A G B ) D ( C G F ) ) )
182, 3grpodivcl 25968 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  C  e.  X )  ->  ( A D C )  e.  X )
19183expib 1210 . . . . . . 7  |-  ( G  e.  GrpOp  ->  ( ( A  e.  X  /\  C  e.  X )  ->  ( A D C )  e.  X ) )
2019anim1d 567 . . . . . 6  |-  ( G  e.  GrpOp  ->  ( (
( A  e.  X  /\  C  e.  X
)  /\  ( B  e.  X  /\  F  e.  X ) )  -> 
( ( A D C )  e.  X  /\  ( B  e.  X  /\  F  e.  X
) ) ) )
21 an4 832 . . . . . 6  |-  ( ( ( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  F  e.  X ) )  <->  ( ( A  e.  X  /\  C  e.  X )  /\  ( B  e.  X  /\  F  e.  X
) ) )
22 3anass 988 . . . . . 6  |-  ( ( ( A D C )  e.  X  /\  B  e.  X  /\  F  e.  X )  <->  ( ( A D C )  e.  X  /\  ( B  e.  X  /\  F  e.  X
) ) )
2320, 21, 223imtr4g 274 . . . . 5  |-  ( G  e.  GrpOp  ->  ( (
( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  F  e.  X ) )  -> 
( ( A D C )  e.  X  /\  B  e.  X  /\  F  e.  X
) ) )
2423imp 431 . . . 4  |-  ( ( G  e.  GrpOp  /\  (
( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  F  e.  X ) ) )  ->  ( ( A D C )  e.  X  /\  B  e.  X  /\  F  e.  X ) )
252, 3grpomuldivass 25970 . . . 4  |-  ( ( G  e.  GrpOp  /\  (
( A D C )  e.  X  /\  B  e.  X  /\  F  e.  X )
)  ->  ( (
( A D C ) G B ) D F )  =  ( ( A D C ) G ( B D F ) ) )
2624, 25syldan 473 . . 3  |-  ( ( G  e.  GrpOp  /\  (
( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  F  e.  X ) ) )  ->  ( ( ( A D C ) G B ) D F )  =  ( ( A D C ) G ( B D F ) ) )
278, 26sylan 474 . 2  |-  ( ( G  e.  AbelOp  /\  (
( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  F  e.  X ) ) )  ->  ( ( ( A D C ) G B ) D F )  =  ( ( A D C ) G ( B D F ) ) )
287, 17, 273eqtr3d 2492 1  |-  ( ( G  e.  AbelOp  /\  (
( A  e.  X  /\  B  e.  X
)  /\  ( C  e.  X  /\  F  e.  X ) ) )  ->  ( ( A G B ) D ( C G F ) )  =  ( ( A D C ) G ( B D F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 984    = wceq 1443    e. wcel 1886   ran crn 4834   ` cfv 5581  (class class class)co 6288   GrpOpcgr 25907    /g cgs 25910   AbelOpcablo 26002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-1st 6790  df-2nd 6791  df-grpo 25912  df-gid 25913  df-ginv 25914  df-gdiv 25915  df-ablo 26003
This theorem is referenced by: (None)
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