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Theorem ablodivdiv4 21917
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
ablodivdiv4  |-  ( ( 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 ablodivdiv4
StepHypRef Expression
1 ablogrpo 21910 . . 3  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
2 simpl 445 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  G  e.  GrpOp
)
3 abldiv.1 . . . . . 6  |-  X  =  ran  G
4 abldiv.3 . . . . . 6  |-  D  =  (  /g  `  G
)
53, 4grpodivcl 21873 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  e.  X )
653adant3r3 1165 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A D B )  e.  X
)
7 simpr3 966 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  C  e.  X )
8 eqid 2443 . . . . 5  |-  ( inv `  G )  =  ( inv `  G )
93, 8, 4grpodivval 21869 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( A D B )  e.  X  /\  C  e.  X )  ->  (
( A D B ) D C )  =  ( ( A D B ) G ( ( inv `  G
) `  C )
) )
102, 6, 7, 9syl3anc 1185 . . 3  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A D B ) D C )  =  ( ( A D B ) G ( ( inv `  G ) `
 C ) ) )
111, 10sylan 459 . 2  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A D B ) D C )  =  ( ( A D B ) G ( ( inv `  G ) `
 C ) ) )
12 simpr1 964 . . . 4  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  A  e.  X )
13 simpr2 965 . . . 4  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  B  e.  X )
14 simp3 960 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  ->  C  e.  X )
153, 8grpoinvcl 21852 . . . . 5  |-  ( ( G  e.  GrpOp  /\  C  e.  X )  ->  (
( inv `  G
) `  C )  e.  X )
161, 14, 15syl2an 465 . . . 4  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( inv `  G ) `  C )  e.  X
)
1712, 13, 163jca 1135 . . 3  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A  e.  X  /\  B  e.  X  /\  ( ( inv `  G ) `
 C )  e.  X ) )
183, 4ablodivdiv 21916 . . 3  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  ( ( inv `  G
) `  C )  e.  X ) )  -> 
( A D ( B D ( ( inv `  G ) `
 C ) ) )  =  ( ( A D B ) G ( ( inv `  G ) `  C
) ) )
1917, 18syldan 458 . 2  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A D ( B D ( ( inv `  G
) `  C )
) )  =  ( ( A D B ) G ( ( inv `  G ) `
 C ) ) )
203, 8, 4grpodivinv 21870 . . . . 5  |-  ( ( G  e.  GrpOp  /\  B  e.  X  /\  C  e.  X )  ->  ( B D ( ( inv `  G ) `  C
) )  =  ( B G C ) )
211, 20syl3an1 1218 . . . 4  |-  ( ( G  e.  AbelOp  /\  B  e.  X  /\  C  e.  X )  ->  ( B D ( ( inv `  G ) `  C
) )  =  ( B G C ) )
22213adant3r1 1163 . . 3  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B D ( ( inv `  G ) `  C
) )  =  ( B G C ) )
2322oveq2d 6133 . 2  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A D ( B D ( ( inv `  G
) `  C )
) )  =  ( A D ( B G C ) ) )
2411, 19, 233eqtr2d 2481 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 360    /\ w3a 937    = wceq 1654    e. wcel 1728   ran crn 4914   ` cfv 5489  (class class class)co 6117   GrpOpcgr 21812   invcgn 21814    /g cgs 21815   AbelOpcablo 21907
This theorem is referenced by:  ablodiv32  21918  ablonnncan  21919  ablo4pnp  26597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-rep 4351  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-reu 2719  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-iun 4124  df-br 4244  df-opab 4298  df-mpt 4299  df-id 4533  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-1st 6385  df-2nd 6386  df-riota 6585  df-grpo 21817  df-gid 21818  df-ginv 21819  df-gdiv 21820  df-ablo 21908
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