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Theorem grpolcan 21012
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 5953 . . . . . 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 2358 . . . . . . . . . . 11  |-  (GId `  G )  =  (GId
`  G )
5 eqid 2358 . . . . . . . . . . 11  |-  ( inv `  G )  =  ( inv `  G )
63, 4, 5grpolinv 21007 . . . . . . . . . 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 5960 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  C  e.  X
)  ->  ( (
( ( inv `  G
) `  C ) G C ) G A )  =  ( (GId
`  G ) G A ) )
93, 5grpoinvcl 21005 . . . . . . . . . . . 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 20982 . . . . . . . . . 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 20998 . . . . . . . . 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 2398 . . . . . . 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 5960 . . . . . . . 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 20982 . . . . . . . . 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 20998 . . . . . . . . 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 2398 . . . . . . 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 2398 . . . 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 5953 . 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 1642    e. wcel 1710   ran crn 4772   ` cfv 5337  (class class class)co 5945   GrpOpcgr 20965  GIdcgi 20966   invcgn 20967
This theorem is referenced by:  grpo2inv  21018  rngolcan  21176  rngolz  21180  vclcan  21235  nvlcan  21296
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-riota 6391  df-grpo 20970  df-gid 20971  df-ginv 20972
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