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Theorem grpolcan 21826
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 6092 . . . . . 6  |-  ( ( C G A )  =  ( C G B )  ->  (
( ( inv `  G
) `  C ) G ( C G A ) )  =  ( ( ( inv `  G ) `  C
) G ( C G B ) ) )
21adantl 454 . . . . 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 2438 . . . . . . . . . . 11  |-  (GId `  G )  =  (GId
`  G )
5 eqid 2438 . . . . . . . . . . 11  |-  ( inv `  G )  =  ( inv `  G )
63, 4, 5grpolinv 21821 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  C  e.  X )  ->  (
( ( inv `  G
) `  C ) G C )  =  (GId
`  G ) )
76adantlr 697 . . . . . . . . 9  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  C  e.  X
)  ->  ( (
( inv `  G
) `  C ) G C )  =  (GId
`  G ) )
87oveq1d 6099 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  C  e.  X
)  ->  ( (
( ( inv `  G
) `  C ) G C ) G A )  =  ( (GId
`  G ) G A ) )
93, 5grpoinvcl 21819 . . . . . . . . . . . 12  |-  ( ( G  e.  GrpOp  /\  C  e.  X )  ->  (
( inv `  G
) `  C )  e.  X )
109adantrl 698 . . . . . . . . . . 11  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  C  e.  X )
)  ->  ( ( inv `  G ) `  C )  e.  X
)
11 simprr 735 . . . . . . . . . . 11  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  C  e.  X )
)  ->  C  e.  X )
12 simprl 734 . . . . . . . . . . 11  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  C  e.  X )
)  ->  A  e.  X )
1310, 11, 123jca 1135 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  C  e.  X )
)  ->  ( (
( inv `  G
) `  C )  e.  X  /\  C  e.  X  /\  A  e.  X ) )
143grpoass 21796 . . . . . . . . . 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 458 . . . . . . . . 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 631 . . . . . . . 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 21812 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
(GId `  G ) G A )  =  A )
1817adantr 453 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  C  e.  X
)  ->  ( (GId `  G ) G A )  =  A )
198, 16, 183eqtr3d 2478 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  C  e.  X
)  ->  ( (
( inv `  G
) `  C ) G ( C G A ) )  =  A )
2019adantrl 698 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  ( B  e.  X  /\  C  e.  X ) )  -> 
( ( ( inv `  G ) `  C
) G ( C G A ) )  =  A )
2120adantr 453 . . . . 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 698 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  ( (
( inv `  G
) `  C ) G C )  =  (GId
`  G ) )
2322oveq1d 6099 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  ( (
( ( inv `  G
) `  C ) G C ) G B )  =  ( (GId
`  G ) G B ) )
249adantrl 698 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  ( ( inv `  G ) `  C )  e.  X
)
25 simprr 735 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  C  e.  X )
26 simprl 734 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  B  e.  X )
2724, 25, 263jca 1135 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  ( (
( inv `  G
) `  C )  e.  X  /\  C  e.  X  /\  B  e.  X ) )
283grpoass 21796 . . . . . . . . 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 458 . . . . . . . 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 21812 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  (
(GId `  G ) G B )  =  B )
3130adantrr 699 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  ( (GId `  G ) G B )  =  B )
3223, 29, 313eqtr3d 2478 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  ( (
( inv `  G
) `  C ) G ( C G B ) )  =  B )
3332adantlr 697 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  ( B  e.  X  /\  C  e.  X ) )  -> 
( ( ( inv `  G ) `  C
) G ( C G B ) )  =  B )
3433adantr 453 . . . . 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 2478 . . . 4  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  ( B  e.  X  /\  C  e.  X ) )  /\  ( C G A )  =  ( C G B ) )  ->  A  =  B )
3635exp53 602 . . 3  |-  ( G  e.  GrpOp  ->  ( A  e.  X  ->  ( B  e.  X  ->  ( C  e.  X  ->  ( ( C G A )  =  ( C G B )  ->  A  =  B )
) ) ) )
37363imp2 1169 . 2  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( C G A )  =  ( C G B )  ->  A  =  B ) )
38 oveq2 6092 . 2  |-  ( A  =  B  ->  ( C G A )  =  ( C G B ) )
3937, 38impbid1 196 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   ran crn 4882   ` cfv 5457  (class class class)co 6084   GrpOpcgr 21779  GIdcgi 21780   invcgn 21781
This theorem is referenced by:  grpo2inv  21832  rngolcan  21990  rngolz  21994  vclcan  22049  nvlcan  22110
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 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pr 4406  ax-un 4704
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 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-riota 6552  df-grpo 21784  df-gid 21785  df-ginv 21786
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