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Theorem grpopncan 21680
Description: Cancellation law for group division. (pncan 9236 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpdivf.1  |-  X  =  ran  G
grpdivf.3  |-  D  =  (  /g  `  G
)
Assertion
Ref Expression
grpopncan  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A G B ) D B )  =  A )

Proof of Theorem grpopncan
StepHypRef Expression
1 simp1 957 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  G  e.  GrpOp )
2 simp2 958 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  A  e.  X )
3 simp3 959 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  B  e.  X )
4 grpdivf.1 . . . 4  |-  X  =  ran  G
5 grpdivf.3 . . . 4  |-  D  =  (  /g  `  G
)
64, 5grpomuldivass 21678 . . 3  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  B  e.  X )
)  ->  ( ( A G B ) D B )  =  ( A G ( B D B ) ) )
71, 2, 3, 3, 6syl13anc 1186 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A G B ) D B )  =  ( A G ( B D B ) ) )
8 eqid 2380 . . . . 5  |-  (GId `  G )  =  (GId
`  G )
94, 5, 8grpodivid 21679 . . . 4  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  ( B D B )  =  (GId `  G )
)
109oveq2d 6029 . . 3  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  ( A G ( B D B ) )  =  ( A G (GId
`  G ) ) )
11103adant2 976 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G ( B D B ) )  =  ( A G (GId
`  G ) ) )
124, 8grporid 21649 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G (GId `  G
) )  =  A )
13123adant3 977 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G (GId `  G
) )  =  A )
147, 11, 133eqtrd 2416 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A G B ) D B )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   ran crn 4812   ` cfv 5387  (class class class)co 6013   GrpOpcgr 21615  GIdcgi 21616    /g cgs 21618
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 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-grpo 21620  df-gid 21621  df-ginv 21622  df-gdiv 21623
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