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Theorem grpodivdiv 21829
Description: Double group division. (Contributed by NM, 24-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpdivf.1  |-  X  =  ran  G
grpdivf.3  |-  D  =  (  /g  `  G
)
Assertion
Ref Expression
grpodivdiv  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A D ( B D C ) )  =  ( A G ( C D B ) ) )

Proof of Theorem grpodivdiv
StepHypRef Expression
1 simpl 444 . . 3  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  G  e.  GrpOp
)
2 simpr1 963 . . 3  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  A  e.  X )
3 grpdivf.1 . . . . 5  |-  X  =  ran  G
4 grpdivf.3 . . . . 5  |-  D  =  (  /g  `  G
)
53, 4grpodivcl 21828 . . . 4  |-  ( ( G  e.  GrpOp  /\  B  e.  X  /\  C  e.  X )  ->  ( B D C )  e.  X )
653adant3r1 1162 . . 3  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B D C )  e.  X
)
7 eqid 2436 . . . 4  |-  ( inv `  G )  =  ( inv `  G )
83, 7, 4grpodivval 21824 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( B D C )  e.  X )  ->  ( A D ( B D C ) )  =  ( A G ( ( inv `  G
) `  ( B D C ) ) ) )
91, 2, 6, 8syl3anc 1184 . 2  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A D ( B D C ) )  =  ( A G ( ( inv `  G
) `  ( B D C ) ) ) )
103, 7, 4grpoinvdiv 21826 . . . 4  |-  ( ( G  e.  GrpOp  /\  B  e.  X  /\  C  e.  X )  ->  (
( inv `  G
) `  ( B D C ) )  =  ( C D B ) )
11103adant3r1 1162 . . 3  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( inv `  G ) `  ( B D C ) )  =  ( C D B ) )
1211oveq2d 6090 . 2  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A G ( ( inv `  G ) `  ( B D C ) ) )  =  ( A G ( C D B ) ) )
139, 12eqtrd 2468 1  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A D ( B D C ) )  =  ( A G ( C D B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   ran crn 4872   ` cfv 5447  (class class class)co 6074   GrpOpcgr 21767   invcgn 21769    /g cgs 21770
This theorem is referenced by:  ablodivdiv  21871
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-reu 2705  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-riota 6542  df-grpo 21772  df-gid 21773  df-ginv 21774  df-gdiv 21775
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