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Theorem gacan 15084
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 15071 . . . . . . . 8  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  G  e.  Grp )
21adantr 453 . . . . . . 7  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  G  e.  Grp )
3 simpr1 964 . . . . . . 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 2438 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
6 eqid 2438 . . . . . . . 8  |-  ( 0g
`  G )  =  ( 0g `  G
)
7 gacan.2 . . . . . . . 8  |-  N  =  ( inv g `  G )
84, 5, 6, 7grprinv 14854 . . . . . . 7  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( A ( +g  `  G ) ( N `
 A ) )  =  ( 0g `  G ) )
92, 3, 8syl2anc 644 . . . . . 6  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( A
( +g  `  G ) ( N `  A
) )  =  ( 0g `  G ) )
109oveq1d 6098 . . . . 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 445 . . . . . 6  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  .(+)  e.  ( G  GrpAct  Y ) )
124, 7grpinvcl 14852 . . . . . . 7  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( N `  A
)  e.  X )
132, 3, 12syl2anc 644 . . . . . 6  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( N `  A )  e.  X
)
14 simpr3 966 . . . . . 6  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  C  e.  Y )
154, 5gaass 15076 . . . . . 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 1187 . . . . 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 15073 . . . . . 6  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  C  e.  Y )  ->  (
( 0g `  G
)  .(+)  C )  =  C )
1811, 14, 17syl2anc 644 . . . . 5  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( ( 0g `  G )  .(+)  C )  =  C )
1910, 16, 183eqtr3d 2478 . . . 4  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( A  .(+) 
( ( N `  A )  .(+)  C ) )  =  C )
2019eqeq2d 2449 . . 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 965 . . . 4  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  B  e.  Y )
224gaf 15074 . . . . . 6  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  .(+)  : ( X  X.  Y ) --> Y )
2322adantr 453 . . . . 5  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  .(+)  : ( X  X.  Y ) --> Y )
2423, 13, 14fovrnd 6220 . . . 4  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( ( N `  A )  .(+)  C )  e.  Y
)
254galcan 15083 . . . 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 1187 . . 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 248 . 2  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( ( A  .(+)  B )  =  C  <->  B  =  (
( N `  A
)  .(+)  C ) ) )
28 eqcom 2440 . 2  |-  ( B  =  ( ( N `
 A )  .(+)  C )  <->  ( ( N `
 A )  .(+)  C )  =  B )
2927, 28syl6bb 254 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    X. cxp 4878   -->wf 5452   ` cfv 5456  (class class class)co 6083   Basecbs 13471   +g cplusg 13531   0gc0g 13725   Grpcgrp 14687   inv gcminusg 14688    GrpAct cga 15068
This theorem is referenced by:  gapm  15085  gaorber  15087  gastacl  15088  gastacos  15089
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-riota 6551  df-map 7022  df-0g 13729  df-mnd 14692  df-grp 14814  df-minusg 14815  df-ga 15069
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