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Theorem gacan 14759
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 14746 . . . . . . . 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 2283 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
6 eqid 2283 . . . . . . . 8  |-  ( 0g
`  G )  =  ( 0g `  G
)
7 gacan.2 . . . . . . . 8  |-  N  =  ( inv g `  G )
84, 5, 6, 7grprinv 14529 . . . . . . 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 5873 . . . . 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 14527 . . . . . . 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 14751 . . . . . 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 14748 . . . . . 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 2323 . . . 4  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( A  .(+) 
( ( N `  A )  .(+)  C ) )  =  C )
2019eqeq2d 2294 . . 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 14749 . . . . . 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 5990 . . . . 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 14758 . . . 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 2285 . 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 1623    e. wcel 1684    X. cxp 4687   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   0gc0g 13400   Grpcgrp 14362   inv gcminusg 14363    GrpAct cga 14743
This theorem is referenced by:  gapm  14760  gaorber  14762  gastacl  14763  gastacos  14764
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-map 6774  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-ga 14744
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