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Theorem ablodivdiv 21728
Description: Law for double group division. (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
abldiv.1  |-  X  =  ran  G
abldiv.3  |-  D  =  (  /g  `  G
)
Assertion
Ref Expression
ablodivdiv  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A D ( B D C ) )  =  ( ( A D B ) G C ) )

Proof of Theorem ablodivdiv
StepHypRef Expression
1 ablogrpo 21722 . . 3  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
2 abldiv.1 . . . 4  |-  X  =  ran  G
3 abldiv.3 . . . 4  |-  D  =  (  /g  `  G
)
42, 3grpodivdiv 21686 . . 3  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A D ( B D C ) )  =  ( A G ( C D B ) ) )
51, 4sylan 458 . 2  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A D ( B D C ) )  =  ( A G ( C D B ) ) )
6 3ancomb 945 . . 3  |-  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  <->  ( A  e.  X  /\  C  e.  X  /\  B  e.  X )
)
72, 3grpomuldivass 21687 . . . . 5  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  C  e.  X  /\  B  e.  X )
)  ->  ( ( A G C ) D B )  =  ( A G ( C D B ) ) )
81, 7sylan 458 . . . 4  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  C  e.  X  /\  B  e.  X )
)  ->  ( ( A G C ) D B )  =  ( A G ( C D B ) ) )
92, 3ablomuldiv 21727 . . . 4  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  C  e.  X  /\  B  e.  X )
)  ->  ( ( A G C ) D B )  =  ( ( A D B ) G C ) )
108, 9eqtr3d 2423 . . 3  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  C  e.  X  /\  B  e.  X )
)  ->  ( A G ( C D B ) )  =  ( ( A D B ) G C ) )
116, 10sylan2b 462 . 2  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A G ( C D B ) )  =  ( ( A D B ) G C ) )
125, 11eqtrd 2421 1  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A D ( B D C ) )  =  ( ( A D B ) G C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   ran crn 4821   ` cfv 5396  (class class class)co 6022   GrpOpcgr 21624    /g cgs 21627   AbelOpcablo 21719
This theorem is referenced by:  ablodivdiv4  21729  ablonncan  21732
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-grpo 21629  df-gid 21630  df-ginv 21631  df-gdiv 21632  df-ablo 21720
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