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Theorem galcan 16130
Description: The action of a particular group element is left-cancelable. (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 13-Jan-2015.)
Hypothesis
Ref Expression
galcan.1  |-  X  =  ( Base `  G
)
Assertion
Ref Expression
galcan  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( ( A  .(+)  B )  =  ( A  .(+)  C )  <-> 
B  =  C ) )

Proof of Theorem galcan
StepHypRef Expression
1 oveq2 6283 . . 3  |-  ( ( A  .(+)  B )  =  ( A  .(+)  C )  ->  ( (
( invg `  G ) `  A
)  .(+)  ( A  .(+)  B ) )  =  ( ( ( invg `  G ) `  A
)  .(+)  ( A  .(+)  C ) ) )
2 simpl 457 . . . . . . . 8  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  .(+)  e.  ( G  GrpAct  Y ) )
3 gagrp 16118 . . . . . . . 8  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  G  e.  Grp )
42, 3syl 16 . . . . . . 7  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  G  e.  Grp )
5 simpr1 997 . . . . . . 7  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  A  e.  X )
6 galcan.1 . . . . . . . 8  |-  X  =  ( Base `  G
)
7 eqid 2460 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
8 eqid 2460 . . . . . . . 8  |-  ( 0g
`  G )  =  ( 0g `  G
)
9 eqid 2460 . . . . . . . 8  |-  ( invg `  G )  =  ( invg `  G )
106, 7, 8, 9grplinv 15890 . . . . . . 7  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( ( invg `  G ) `
 A ) ( +g  `  G ) A )  =  ( 0g `  G ) )
114, 5, 10syl2anc 661 . . . . . 6  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( (
( invg `  G ) `  A
) ( +g  `  G
) A )  =  ( 0g `  G
) )
1211oveq1d 6290 . . . . 5  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( (
( ( invg `  G ) `  A
) ( +g  `  G
) A )  .(+)  B )  =  ( ( 0g `  G ) 
.(+)  B ) )
136, 9grpinvcl 15889 . . . . . . 7  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( invg `  G ) `  A
)  e.  X )
144, 5, 13syl2anc 661 . . . . . 6  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( ( invg `  G ) `
 A )  e.  X )
15 simpr2 998 . . . . . 6  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  B  e.  Y )
166, 7gaass 16123 . . . . . 6  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  (
( ( invg `  G ) `  A
)  e.  X  /\  A  e.  X  /\  B  e.  Y )
)  ->  ( (
( ( invg `  G ) `  A
) ( +g  `  G
) A )  .(+)  B )  =  ( ( ( invg `  G ) `  A
)  .(+)  ( A  .(+)  B ) ) )
172, 14, 5, 15, 16syl13anc 1225 . . . . 5  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( (
( ( invg `  G ) `  A
) ( +g  `  G
) A )  .(+)  B )  =  ( ( ( invg `  G ) `  A
)  .(+)  ( A  .(+)  B ) ) )
188gagrpid 16120 . . . . . 6  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  B  e.  Y )  ->  (
( 0g `  G
)  .(+)  B )  =  B )
192, 15, 18syl2anc 661 . . . . 5  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( ( 0g `  G )  .(+)  B )  =  B )
2012, 17, 193eqtr3d 2509 . . . 4  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( (
( invg `  G ) `  A
)  .(+)  ( A  .(+)  B ) )  =  B )
2111oveq1d 6290 . . . . 5  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( (
( ( invg `  G ) `  A
) ( +g  `  G
) A )  .(+)  C )  =  ( ( 0g `  G ) 
.(+)  C ) )
22 simpr3 999 . . . . . 6  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  C  e.  Y )
236, 7gaass 16123 . . . . . 6  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  (
( ( invg `  G ) `  A
)  e.  X  /\  A  e.  X  /\  C  e.  Y )
)  ->  ( (
( ( invg `  G ) `  A
) ( +g  `  G
) A )  .(+)  C )  =  ( ( ( invg `  G ) `  A
)  .(+)  ( A  .(+)  C ) ) )
242, 14, 5, 22, 23syl13anc 1225 . . . . 5  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( (
( ( invg `  G ) `  A
) ( +g  `  G
) A )  .(+)  C )  =  ( ( ( invg `  G ) `  A
)  .(+)  ( A  .(+)  C ) ) )
258gagrpid 16120 . . . . . 6  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  C  e.  Y )  ->  (
( 0g `  G
)  .(+)  C )  =  C )
262, 22, 25syl2anc 661 . . . . 5  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( ( 0g `  G )  .(+)  C )  =  C )
2721, 24, 263eqtr3d 2509 . . . 4  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( (
( invg `  G ) `  A
)  .(+)  ( A  .(+)  C ) )  =  C )
2820, 27eqeq12d 2482 . . 3  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( (
( ( invg `  G ) `  A
)  .(+)  ( A  .(+)  B ) )  =  ( ( ( invg `  G ) `  A
)  .(+)  ( A  .(+)  C ) )  <->  B  =  C ) )
291, 28syl5ib 219 . 2  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( ( A  .(+)  B )  =  ( A  .(+)  C )  ->  B  =  C ) )
30 oveq2 6283 . 2  |-  ( B  =  C  ->  ( A  .(+)  B )  =  ( A  .(+)  C ) )
3129, 30impbid1 203 1  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( ( A  .(+)  B )  =  ( A  .(+)  C )  <-> 
B  =  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   ` cfv 5579  (class class class)co 6275   Basecbs 14479   +g cplusg 14544   0gc0g 14684   Grpcgrp 15716   invgcminusg 15717    GrpAct cga 16115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-map 7412  df-0g 14686  df-mnd 15721  df-grp 15851  df-minusg 15852  df-ga 16116
This theorem is referenced by:  gacan  16131
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