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Theorem galcan 15843
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 6120 . . 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 15831 . . . . . . . 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 994 . . . . . . 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 2443 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
8 eqid 2443 . . . . . . . 8  |-  ( 0g
`  G )  =  ( 0g `  G
)
9 eqid 2443 . . . . . . . 8  |-  ( invg `  G )  =  ( invg `  G )
106, 7, 8, 9grplinv 15605 . . . . . . 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 6127 . . . . 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 15604 . . . . . . 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 995 . . . . . 6  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  B  e.  Y )
166, 7gaass 15836 . . . . . 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 1220 . . . . 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 15833 . . . . . 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 2483 . . . 4  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( (
( invg `  G ) `  A
)  .(+)  ( A  .(+)  B ) )  =  B )
2111oveq1d 6127 . . . . 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 996 . . . . . 6  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  C  e.  Y )
236, 7gaass 15836 . . . . . 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 1220 . . . . 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 15833 . . . . . 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 2483 . . . 4  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
)  ->  ( (
( invg `  G ) `  A
)  .(+)  ( A  .(+)  C ) )  =  C )
2820, 27eqeq12d 2457 . . 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 6120 . 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 965    = wceq 1369    e. wcel 1756   ` cfv 5439  (class class class)co 6112   Basecbs 14195   +g cplusg 14259   0gc0g 14399   Grpcgrp 15431   invgcminusg 15432    GrpAct cga 15828
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 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
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 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-map 7237  df-0g 14401  df-mnd 15436  df-grp 15566  df-minusg 15567  df-ga 15829
This theorem is referenced by:  gacan  15844
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