MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpolcan Structured version   Unicode version

Theorem grpolcan 23732
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 6111 . . . . . 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 2443 . . . . . . . . . . 11  |-  (GId `  G )  =  (GId
`  G )
5 eqid 2443 . . . . . . . . . . 11  |-  ( inv `  G )  =  ( inv `  G )
63, 4, 5grpolinv 23727 . . . . . . . . . 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 6118 . . . . . . . 8  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  C  e.  X
)  ->  ( (
( ( inv `  G
) `  C ) G C ) G A )  =  ( (GId
`  G ) G A ) )
93, 5grpoinvcl 23725 . . . . . . . . . . . 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 756 . . . . . . . . . . 11  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  C  e.  X )
)  ->  C  e.  X )
12 simprl 755 . . . . . . . . . . 11  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  C  e.  X )
)  ->  A  e.  X )
1310, 11, 123jca 1168 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  C  e.  X )
)  ->  ( (
( inv `  G
) `  C )  e.  X  /\  C  e.  X  /\  A  e.  X ) )
143grpoass 23702 . . . . . . . . . 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 23718 . . . . . . . . 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 2483 . . . . . . 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 6118 . . . . . . . 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 756 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  C  e.  X )
26 simprl 755 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  B  e.  X )
2724, 25, 263jca 1168 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  ( (
( inv `  G
) `  C )  e.  X  /\  C  e.  X  /\  B  e.  X ) )
283grpoass 23702 . . . . . . . . 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 23718 . . . . . . . . 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 2483 . . . . . . 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 2483 . . . 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 1202 . 2  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( C G A )  =  ( C G B )  ->  A  =  B ) )
38 oveq2 6111 . 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 965    = wceq 1369    e. wcel 1756   ran crn 4853   ` cfv 5430  (class class class)co 6103   GrpOpcgr 23685  GIdcgi 23686   invcgn 23687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-grpo 23690  df-gid 23691  df-ginv 23692
This theorem is referenced by:  grpo2inv  23738  rngolcan  23896  rngolz  23900  vclcan  23955  nvlcan  24016
  Copyright terms: Public domain W3C validator