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Theorem gacan 16157
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 16144 . . . . . . . 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 1002 . . . . . . 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 2467 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
6 eqid 2467 . . . . . . . 8  |-  ( 0g
`  G )  =  ( 0g `  G
)
7 gacan.2 . . . . . . . 8  |-  N  =  ( invg `  G )
84, 5, 6, 7grprinv 15911 . . . . . . 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 6300 . . . . 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 15909 . . . . . . 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 1004 . . . . . 6  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  C  e.  Y )
154, 5gaass 16149 . . . . . 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 1230 . . . . 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 16146 . . . . . 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 2516 . . . 4  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( A  .(+) 
( ( N `  A )  .(+)  C ) )  =  C )
2019eqeq2d 2481 . . 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 1003 . . . 4  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  B  e.  Y )
224gaf 16147 . . . . . 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 6432 . . . 4  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( ( N `  A )  .(+)  C )  e.  Y
)
254galcan 16156 . . . 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 1230 . . 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 2476 . 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 973    = wceq 1379    e. wcel 1767    X. cxp 4997   -->wf 5584   ` cfv 5588  (class class class)co 6285   Basecbs 14493   +g cplusg 14558   0gc0g 14698   Grpcgrp 15730   invgcminusg 15731    GrpAct cga 16141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-map 7423  df-0g 14700  df-mnd 15735  df-grp 15871  df-minusg 15872  df-ga 16142
This theorem is referenced by:  gapm  16158  gaorber  16160  gastacl  16161  gastacos  16162
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