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Theorem ablo4pnp 28589
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 960 . . . . 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 23599 . . . . 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 473 . . . 4  |-  ( ( G  e.  AbelOp  /\  (
( A  e.  X  /\  B  e.  X
)  /\  C  e.  X ) )  -> 
( ( A G B ) D C )  =  ( ( A D C ) G B ) )
65adantrrr 717 . . 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 6095 . 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 23594 . . . . . . 7  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
92grpocl 23510 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
1093expib 1183 . . . . . . 7  |-  ( G  e.  GrpOp  ->  ( ( A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X ) )
118, 10syl 16 . . . . . 6  |-  ( G  e.  AbelOp  ->  ( ( A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X ) )
1211anim1d 559 . . . . 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 962 . . . . 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 227 . . . 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 429 . . 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 23601 . . 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 467 . 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 23557 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  C  e.  X )  ->  ( A D C )  e.  X )
19183expib 1183 . . . . . . 7  |-  ( G  e.  GrpOp  ->  ( ( A  e.  X  /\  C  e.  X )  ->  ( A D C )  e.  X ) )
2019anim1d 559 . . . . . 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 813 . . . . . 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 962 . . . . . 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 270 . . . . 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 429 . . . 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 23559 . . . 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 467 . . 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 468 . 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 2473 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 369    /\ w3a 958    = wceq 1362    e. wcel 1755   ran crn 4828   ` cfv 5406  (class class class)co 6080   GrpOpcgr 23496    /g cgs 23499   AbelOpcablo 23591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-1st 6566  df-2nd 6567  df-grpo 23501  df-gid 23502  df-ginv 23503  df-gdiv 23504  df-ablo 23592
This theorem is referenced by: (None)
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