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

Theorem gacan 14808
Description: Group inverses cancel in a group action. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
Hypotheses
Ref Expression
galcan.1  |-  X  =  ( Base `  G
)
gacan.2  |-  N  =  ( inv g `  G )
Assertion
Ref Expression
gacan  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( ( A  .(+)  B )  =  C  <->  ( ( N `
 A )  .(+)  C )  =  B ) )

Proof of Theorem gacan
StepHypRef Expression
1 gagrp 14795 . . . . . . . 8  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  G  e.  Grp )
21adantr 451 . . . . . . 7  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  G  e.  Grp )
3 simpr1 961 . . . . . . 7  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  A  e.  X )
4 galcan.1 . . . . . . . 8  |-  X  =  ( Base `  G
)
5 eqid 2316 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
6 eqid 2316 . . . . . . . 8  |-  ( 0g
`  G )  =  ( 0g `  G
)
7 gacan.2 . . . . . . . 8  |-  N  =  ( inv g `  G )
84, 5, 6, 7grprinv 14578 . . . . . . 7  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( A ( +g  `  G ) ( N `
 A ) )  =  ( 0g `  G ) )
92, 3, 8syl2anc 642 . . . . . 6  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( A
( +g  `  G ) ( N `  A
) )  =  ( 0g `  G ) )
109oveq1d 5915 . . . . 5  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( ( A ( +g  `  G
) ( N `  A ) )  .(+)  C )  =  ( ( 0g `  G ) 
.(+)  C ) )
11 simpl 443 . . . . . 6  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  .(+)  e.  ( G  GrpAct  Y ) )
124, 7grpinvcl 14576 . . . . . . 7  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( N `  A
)  e.  X )
132, 3, 12syl2anc 642 . . . . . 6  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( N `  A )  e.  X
)
14 simpr3 963 . . . . . 6  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  C  e.  Y )
154, 5gaass 14800 . . . . . 6  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  ( N `  A )  e.  X  /\  C  e.  Y ) )  -> 
( ( A ( +g  `  G ) ( N `  A
) )  .(+)  C )  =  ( A  .(+)  ( ( N `  A
)  .(+)  C ) ) )
1611, 3, 13, 14, 15syl13anc 1184 . . . . 5  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( ( A ( +g  `  G
) ( N `  A ) )  .(+)  C )  =  ( A 
.(+)  ( ( N `
 A )  .(+)  C ) ) )
176gagrpid 14797 . . . . . 6  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  C  e.  Y )  ->  (
( 0g `  G
)  .(+)  C )  =  C )
1811, 14, 17syl2anc 642 . . . . 5  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( ( 0g `  G )  .(+)  C )  =  C )
1910, 16, 183eqtr3d 2356 . . . 4  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( A  .(+) 
( ( N `  A )  .(+)  C ) )  =  C )
2019eqeq2d 2327 . . 3  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( ( A  .(+)  B )  =  ( A  .(+)  ( ( N `  A ) 
.(+)  C ) )  <->  ( A  .(+) 
B )  =  C ) )
21 simpr2 962 . . . 4  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  B  e.  Y )
224gaf 14798 . . . . . 6  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  .(+)  : ( X  X.  Y ) --> Y )
2322adantr 451 . . . . 5  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  .(+)  : ( X  X.  Y ) --> Y )
24 fovrn 6032 . . . . 5  |-  ( ( 
.(+)  : ( X  X.  Y ) --> Y  /\  ( N `  A )  e.  X  /\  C  e.  Y )  ->  (
( N `  A
)  .(+)  C )  e.  Y )
2523, 13, 14, 24syl3anc 1182 . . . 4  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( ( N `  A )  .(+)  C )  e.  Y
)
264galcan 14807 . . . 4  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  ( ( N `  A )  .(+)  C )  e.  Y ) )  ->  ( ( A 
.(+)  B )  =  ( A  .(+)  ( ( N `  A )  .(+)  C ) )  <->  B  =  ( ( N `  A )  .(+)  C ) ) )
2711, 3, 21, 25, 26syl13anc 1184 . . 3  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( ( A  .(+)  B )  =  ( A  .(+)  ( ( N `  A ) 
.(+)  C ) )  <->  B  =  ( ( N `  A )  .(+)  C ) ) )
2820, 27bitr3d 246 . 2  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( ( A  .(+)  B )  =  C  <->  B  =  (
( N `  A
)  .(+)  C ) ) )
29 eqcom 2318 . 2  |-  ( B  =  ( ( N `
 A )  .(+)  C )  <->  ( ( N `
 A )  .(+)  C )  =  B )
3028, 29syl6bb 252 1  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( ( A  .(+)  B )  =  C  <->  ( ( N `
 A )  .(+)  C )  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701    X. cxp 4724   -->wf 5288   ` cfv 5292  (class class class)co 5900   Basecbs 13195   +g cplusg 13255   0gc0g 13449   Grpcgrp 14411   inv gcminusg 14412    GrpAct cga 14792
This theorem is referenced by:  gapm  14809  gaorber  14811  gastacl  14812  gastacos  14813
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-riota 6346  df-map 6817  df-0g 13453  df-mnd 14416  df-grp 14538  df-minusg 14539  df-ga 14793
  Copyright terms: Public domain W3C validator