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Theorem grpolcan 25362
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 6304 . . . . . 6  |-  ( ( C G A )  =  ( C G B )  ->  (
( ( inv `  G
) `  C ) G ( C G A ) )  =  ( ( ( inv `  G ) `  C
) G ( C G B ) ) )
21adantl 466 . . . . 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 2457 . . . . . . . . . . 11  |-  (GId `  G )  =  (GId
`  G )
5 eqid 2457 . . . . . . . . . . 11  |-  ( inv `  G )  =  ( inv `  G )
63, 4, 5grpolinv 25357 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  C  e.  X )  ->  (
( ( inv `  G
) `  C ) G C )  =  (GId
`  G ) )
76adantlr 714 . . . . . . . . 9  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  C  e.  X
)  ->  ( (
( inv `  G
) `  C ) G C )  =  (GId
`  G ) )
87oveq1d 6311 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  C  e.  X
)  ->  ( (
( ( inv `  G
) `  C ) G C ) G A )  =  ( (GId
`  G ) G A ) )
93, 5grpoinvcl 25355 . . . . . . . . . . . 12  |-  ( ( G  e.  GrpOp  /\  C  e.  X )  ->  (
( inv `  G
) `  C )  e.  X )
109adantrl 715 . . . . . . . . . . 11  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  C  e.  X )
)  ->  ( ( inv `  G ) `  C )  e.  X
)
11 simprr 757 . . . . . . . . . . 11  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  C  e.  X )
)  ->  C  e.  X )
12 simprl 756 . . . . . . . . . . 11  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  C  e.  X )
)  ->  A  e.  X )
1310, 11, 123jca 1176 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  C  e.  X )
)  ->  ( (
( inv `  G
) `  C )  e.  X  /\  C  e.  X  /\  A  e.  X ) )
143grpoass 25332 . . . . . . . . . 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 470 . . . . . . . . 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 648 . . . . . . . 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 25348 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
(GId `  G ) G A )  =  A )
1817adantr 465 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  C  e.  X
)  ->  ( (GId `  G ) G A )  =  A )
198, 16, 183eqtr3d 2506 . . . . . . 7  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  C  e.  X
)  ->  ( (
( inv `  G
) `  C ) G ( C G A ) )  =  A )
2019adantrl 715 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  ( B  e.  X  /\  C  e.  X ) )  -> 
( ( ( inv `  G ) `  C
) G ( C G A ) )  =  A )
2120adantr 465 . . . . 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 715 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  ( (
( inv `  G
) `  C ) G C )  =  (GId
`  G ) )
2322oveq1d 6311 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  ( (
( ( inv `  G
) `  C ) G C ) G B )  =  ( (GId
`  G ) G B ) )
249adantrl 715 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  ( ( inv `  G ) `  C )  e.  X
)
25 simprr 757 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  C  e.  X )
26 simprl 756 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  B  e.  X )
2724, 25, 263jca 1176 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  ( (
( inv `  G
) `  C )  e.  X  /\  C  e.  X  /\  B  e.  X ) )
283grpoass 25332 . . . . . . . . 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 470 . . . . . . . 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 25348 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  (
(GId `  G ) G B )  =  B )
3130adantrr 716 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  ( (GId `  G ) G B )  =  B )
3223, 29, 313eqtr3d 2506 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  ( (
( inv `  G
) `  C ) G ( C G B ) )  =  B )
3332adantlr 714 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  ( B  e.  X  /\  C  e.  X ) )  -> 
( ( ( inv `  G ) `  C
) G ( C G B ) )  =  B )
3433adantr 465 . . . . 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 2506 . . . 4  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  ( B  e.  X  /\  C  e.  X ) )  /\  ( C G A )  =  ( C G B ) )  ->  A  =  B )
3635exp53 617 . . 3  |-  ( G  e.  GrpOp  ->  ( A  e.  X  ->  ( B  e.  X  ->  ( C  e.  X  ->  ( ( C G A )  =  ( C G B )  ->  A  =  B )
) ) ) )
37363imp2 1211 . 2  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( C G A )  =  ( C G B )  ->  A  =  B ) )
38 oveq2 6304 . 2  |-  ( A  =  B  ->  ( C G A )  =  ( C G B ) )
3937, 38impbid1 203 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   ran crn 5009   ` cfv 5594  (class class class)co 6296   GrpOpcgr 25315  GIdcgi 25316   invcgn 25317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-grpo 25320  df-gid 25321  df-ginv 25322
This theorem is referenced by:  grpo2inv  25368  rngolcan  25526  rngolz  25530  vclcan  25585  nvlcan  25646
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