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Theorem ablodivdiv4 25158
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 25151 . . 3  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
2 simpl 457 . . . 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 25114 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  e.  X )
653adant3r3 1206 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A D B )  e.  X
)
7 simpr3 1003 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  C  e.  X )
8 eqid 2441 . . . . 5  |-  ( inv `  G )  =  ( inv `  G )
93, 8, 4grpodivval 25110 . . . 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 1227 . . 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 471 . 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 1001 . . . 4  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  A  e.  X )
13 simpr2 1002 . . . 4  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  B  e.  X )
14 simp3 997 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  ->  C  e.  X )
153, 8grpoinvcl 25093 . . . . 5  |-  ( ( G  e.  GrpOp  /\  C  e.  X )  ->  (
( inv `  G
) `  C )  e.  X )
161, 14, 15syl2an 477 . . . 4  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( inv `  G ) `  C )  e.  X
)
1712, 13, 163jca 1175 . . 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 25157 . . 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 470 . 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 25111 . . . . 5  |-  ( ( G  e.  GrpOp  /\  B  e.  X  /\  C  e.  X )  ->  ( B D ( ( inv `  G ) `  C
) )  =  ( B G C ) )
211, 20syl3an1 1260 . . . 4  |-  ( ( G  e.  AbelOp  /\  B  e.  X  /\  C  e.  X )  ->  ( B D ( ( inv `  G ) `  C
) )  =  ( B G C ) )
22213adant3r1 1204 . . 3  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B D ( ( inv `  G ) `  C
) )  =  ( B G C ) )
2322oveq2d 6293 . 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 2488 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802   ran crn 4986   ` cfv 5574  (class class class)co 6277   GrpOpcgr 25053   invcgn 25055    /g cgs 25056   AbelOpcablo 25148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6781  df-2nd 6782  df-grpo 25058  df-gid 25059  df-ginv 25060  df-gdiv 25061  df-ablo 25149
This theorem is referenced by:  ablodiv32  25159  ablonnncan  25160  ablo4pnp  30310
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