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Theorem grpolcan 20900
Description: Left cancellation law for groups. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
grplcan.1  |-  X  =  ran  G
Assertion
Ref Expression
grpolcan  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( C G A )  =  ( C G B )  <->  A  =  B
) )

Proof of Theorem grpolcan
StepHypRef Expression
1 oveq2 5866 . . . . . 6  |-  ( ( C G A )  =  ( C G B )  ->  (
( ( inv `  G
) `  C ) G ( C G A ) )  =  ( ( ( inv `  G ) `  C
) G ( C G B ) ) )
21adantl 452 . . . . 5  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  ( B  e.  X  /\  C  e.  X ) )  /\  ( C G A )  =  ( C G B ) )  -> 
( ( ( inv `  G ) `  C
) G ( C G A ) )  =  ( ( ( inv `  G ) `
 C ) G ( C G B ) ) )
3 grplcan.1 . . . . . . . . . . 11  |-  X  =  ran  G
4 eqid 2283 . . . . . . . . . . 11  |-  (GId `  G )  =  (GId
`  G )
5 eqid 2283 . . . . . . . . . . 11  |-  ( inv `  G )  =  ( inv `  G )
63, 4, 5grpolinv 20895 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  C  e.  X )  ->  (
( ( inv `  G
) `  C ) G C )  =  (GId
`  G ) )
76adantlr 695 . . . . . . . . 9  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  C  e.  X
)  ->  ( (
( inv `  G
) `  C ) G C )  =  (GId
`  G ) )
87oveq1d 5873 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  C  e.  X
)  ->  ( (
( ( inv `  G
) `  C ) G C ) G A )  =  ( (GId
`  G ) G A ) )
93, 5grpoinvcl 20893 . . . . . . . . . . . 12  |-  ( ( G  e.  GrpOp  /\  C  e.  X )  ->  (
( inv `  G
) `  C )  e.  X )
109adantrl 696 . . . . . . . . . . 11  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  C  e.  X )
)  ->  ( ( inv `  G ) `  C )  e.  X
)
11 simprr 733 . . . . . . . . . . 11  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  C  e.  X )
)  ->  C  e.  X )
12 simprl 732 . . . . . . . . . . 11  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  C  e.  X )
)  ->  A  e.  X )
1310, 11, 123jca 1132 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  C  e.  X )
)  ->  ( (
( inv `  G
) `  C )  e.  X  /\  C  e.  X  /\  A  e.  X ) )
143grpoass 20870 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  (
( ( inv `  G
) `  C )  e.  X  /\  C  e.  X  /\  A  e.  X ) )  -> 
( ( ( ( inv `  G ) `
 C ) G C ) G A )  =  ( ( ( inv `  G
) `  C ) G ( C G A ) ) )
1513, 14syldan 456 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  C  e.  X )
)  ->  ( (
( ( inv `  G
) `  C ) G C ) G A )  =  ( ( ( inv `  G
) `  C ) G ( C G A ) ) )
1615anassrs 629 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  C  e.  X
)  ->  ( (
( ( inv `  G
) `  C ) G C ) G A )  =  ( ( ( inv `  G
) `  C ) G ( C G A ) ) )
173, 4grpolid 20886 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
(GId `  G ) G A )  =  A )
1817adantr 451 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  C  e.  X
)  ->  ( (GId `  G ) G A )  =  A )
198, 16, 183eqtr3d 2323 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  C  e.  X
)  ->  ( (
( inv `  G
) `  C ) G ( C G A ) )  =  A )
2019adantrl 696 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  ( B  e.  X  /\  C  e.  X ) )  -> 
( ( ( inv `  G ) `  C
) G ( C G A ) )  =  A )
2120adantr 451 . . . . 5  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  ( B  e.  X  /\  C  e.  X ) )  /\  ( C G A )  =  ( C G B ) )  -> 
( ( ( inv `  G ) `  C
) G ( C G A ) )  =  A )
226adantrl 696 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  ( (
( inv `  G
) `  C ) G C )  =  (GId
`  G ) )
2322oveq1d 5873 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  ( (
( ( inv `  G
) `  C ) G C ) G B )  =  ( (GId
`  G ) G B ) )
249adantrl 696 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  ( ( inv `  G ) `  C )  e.  X
)
25 simprr 733 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  C  e.  X )
26 simprl 732 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  B  e.  X )
2724, 25, 263jca 1132 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  ( (
( inv `  G
) `  C )  e.  X  /\  C  e.  X  /\  B  e.  X ) )
283grpoass 20870 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  (
( ( inv `  G
) `  C )  e.  X  /\  C  e.  X  /\  B  e.  X ) )  -> 
( ( ( ( inv `  G ) `
 C ) G C ) G B )  =  ( ( ( inv `  G
) `  C ) G ( C G B ) ) )
2927, 28syldan 456 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  ( (
( ( inv `  G
) `  C ) G C ) G B )  =  ( ( ( inv `  G
) `  C ) G ( C G B ) ) )
303, 4grpolid 20886 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  (
(GId `  G ) G B )  =  B )
3130adantrr 697 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  ( (GId `  G ) G B )  =  B )
3223, 29, 313eqtr3d 2323 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  ( (
( inv `  G
) `  C ) G ( C G B ) )  =  B )
3332adantlr 695 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  ( B  e.  X  /\  C  e.  X ) )  -> 
( ( ( inv `  G ) `  C
) G ( C G B ) )  =  B )
3433adantr 451 . . . . 5  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  ( B  e.  X  /\  C  e.  X ) )  /\  ( C G A )  =  ( C G B ) )  -> 
( ( ( inv `  G ) `  C
) G ( C G B ) )  =  B )
352, 21, 343eqtr3d 2323 . . . 4  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  ( B  e.  X  /\  C  e.  X ) )  /\  ( C G A )  =  ( C G B ) )  ->  A  =  B )
3635exp53 600 . . 3  |-  ( G  e.  GrpOp  ->  ( A  e.  X  ->  ( B  e.  X  ->  ( C  e.  X  ->  ( ( C G A )  =  ( C G B )  ->  A  =  B )
) ) ) )
37363imp2 1166 . 2  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( C G A )  =  ( C G B )  ->  A  =  B ) )
38 oveq2 5866 . 2  |-  ( A  =  B  ->  ( C G A )  =  ( C G B ) )
3937, 38impbid1 194 1  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( C G A )  =  ( C G B )  <->  A  =  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ran crn 4690   ` cfv 5255  (class class class)co 5858   GrpOpcgr 20853  GIdcgi 20854   invcgn 20855
This theorem is referenced by:  grpo2inv  20906  rngolcan  21064  rngolz  21068  vclcan  21121  nvlcan  21182  grpodlcan  25376  ltrooo  25404  rltrooo  25415  veclcan  25476  mulinvsca  25480
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-riota 6304  df-grpo 20858  df-gid 20859  df-ginv 20860
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