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Theorem gacan 15823
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  =  ( invg `  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 15810 . . . . . . . 8  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  G  e.  Grp )
21adantr 465 . . . . . . 7  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  G  e.  Grp )
3 simpr1 994 . . . . . . 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 2443 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
6 eqid 2443 . . . . . . . 8  |-  ( 0g
`  G )  =  ( 0g `  G
)
7 gacan.2 . . . . . . . 8  |-  N  =  ( invg `  G )
84, 5, 6, 7grprinv 15585 . . . . . . 7  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( A ( +g  `  G ) ( N `
 A ) )  =  ( 0g `  G ) )
92, 3, 8syl2anc 661 . . . . . 6  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( A
( +g  `  G ) ( N `  A
) )  =  ( 0g `  G ) )
109oveq1d 6106 . . . . 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 457 . . . . . 6  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  .(+)  e.  ( G  GrpAct  Y ) )
124, 7grpinvcl 15583 . . . . . . 7  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( N `  A
)  e.  X )
132, 3, 12syl2anc 661 . . . . . 6  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( N `  A )  e.  X
)
14 simpr3 996 . . . . . 6  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  C  e.  Y )
154, 5gaass 15815 . . . . . 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 1220 . . . . 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 15812 . . . . . 6  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  C  e.  Y )  ->  (
( 0g `  G
)  .(+)  C )  =  C )
1811, 14, 17syl2anc 661 . . . . 5  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( ( 0g `  G )  .(+)  C )  =  C )
1910, 16, 183eqtr3d 2483 . . . 4  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( A  .(+) 
( ( N `  A )  .(+)  C ) )  =  C )
2019eqeq2d 2454 . . 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 995 . . . 4  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  B  e.  Y )
224gaf 15813 . . . . . 6  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  .(+)  : ( X  X.  Y ) --> Y )
2322adantr 465 . . . . 5  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  .(+)  : ( X  X.  Y ) --> Y )
2423, 13, 14fovrnd 6235 . . . 4  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( ( N `  A )  .(+)  C )  e.  Y
)
254galcan 15822 . . . 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 ) ) )
2611, 3, 21, 24, 25syl13anc 1220 . . 3  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( ( A  .(+)  B )  =  ( A  .(+)  ( ( N `  A ) 
.(+)  C ) )  <->  B  =  ( ( N `  A )  .(+)  C ) ) )
2720, 26bitr3d 255 . 2  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( ( A  .(+)  B )  =  C  <->  B  =  (
( N `  A
)  .(+)  C ) ) )
28 eqcom 2445 . 2  |-  ( B  =  ( ( N `
 A )  .(+)  C )  <->  ( ( N `
 A )  .(+)  C )  =  B )
2927, 28syl6bb 261 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    X. cxp 4838   -->wf 5414   ` cfv 5418  (class class class)co 6091   Basecbs 14174   +g cplusg 14238   0gc0g 14378   Grpcgrp 15410   invgcminusg 15411    GrpAct cga 15807
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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
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 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-map 7216  df-0g 14380  df-mnd 15415  df-grp 15545  df-minusg 15546  df-ga 15808
This theorem is referenced by:  gapm  15824  gaorber  15826  gastacl  15827  gastacos  15828
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