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Theorem grpsubrcan 14862
Description: Right cancellation law for group subtraction. (Contributed by NM, 31-Mar-2014.)
Hypotheses
Ref Expression
grpsubcl.b  |-  B  =  ( Base `  G
)
grpsubcl.m  |-  .-  =  ( -g `  G )
Assertion
Ref Expression
grpsubrcan  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .-  Z
)  =  ( Y 
.-  Z )  <->  X  =  Y ) )

Proof of Theorem grpsubrcan
StepHypRef Expression
1 grpsubcl.b . . . . . 6  |-  B  =  ( Base `  G
)
2 eqid 2435 . . . . . 6  |-  ( +g  `  G )  =  ( +g  `  G )
3 eqid 2435 . . . . . 6  |-  ( inv g `  G )  =  ( inv g `  G )
4 grpsubcl.m . . . . . 6  |-  .-  =  ( -g `  G )
51, 2, 3, 4grpsubval 14840 . . . . 5  |-  ( ( X  e.  B  /\  Z  e.  B )  ->  ( X  .-  Z
)  =  ( X ( +g  `  G
) ( ( inv g `  G ) `
 Z ) ) )
653adant2 976 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  ->  ( X  .-  Z
)  =  ( X ( +g  `  G
) ( ( inv g `  G ) `
 Z ) ) )
71, 2, 3, 4grpsubval 14840 . . . . 5  |-  ( ( Y  e.  B  /\  Z  e.  B )  ->  ( Y  .-  Z
)  =  ( Y ( +g  `  G
) ( ( inv g `  G ) `
 Z ) ) )
873adant1 975 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .-  Z
)  =  ( Y ( +g  `  G
) ( ( inv g `  G ) `
 Z ) ) )
96, 8eqeq12d 2449 . . 3  |-  ( ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  ->  ( ( X  .-  Z )  =  ( Y  .-  Z )  <-> 
( X ( +g  `  G ) ( ( inv g `  G
) `  Z )
)  =  ( Y ( +g  `  G
) ( ( inv g `  G ) `
 Z ) ) ) )
109adantl 453 . 2  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .-  Z
)  =  ( Y 
.-  Z )  <->  ( X
( +g  `  G ) ( ( inv g `  G ) `  Z
) )  =  ( Y ( +g  `  G
) ( ( inv g `  G ) `
 Z ) ) ) )
11 simpl 444 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  G  e.  Grp )
12 simpr1 963 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  X  e.  B )
13 simpr2 964 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Y  e.  B )
141, 3grpinvcl 14842 . . . 4  |-  ( ( G  e.  Grp  /\  Z  e.  B )  ->  ( ( inv g `  G ) `  Z
)  e.  B )
15143ad2antr3 1124 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( inv g `  G ) `  Z
)  e.  B )
161, 2grprcan 14830 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  ( ( inv g `  G ) `  Z
)  e.  B ) )  ->  ( ( X ( +g  `  G
) ( ( inv g `  G ) `
 Z ) )  =  ( Y ( +g  `  G ) ( ( inv g `  G ) `  Z
) )  <->  X  =  Y ) )
1711, 12, 13, 15, 16syl13anc 1186 . 2  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X ( +g  `  G ) ( ( inv g `  G
) `  Z )
)  =  ( Y ( +g  `  G
) ( ( inv g `  G ) `
 Z ) )  <-> 
X  =  Y ) )
1810, 17bitrd 245 1  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .-  Z
)  =  ( Y 
.-  Z )  <->  X  =  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   ` cfv 5446  (class class class)co 6073   Basecbs 13461   +g cplusg 13521   Grpcgrp 14677   inv gcminusg 14678   -gcsg 14680
This theorem is referenced by:  abladdsub4  15430
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-0g 13719  df-mnd 14682  df-grp 14804  df-minusg 14805  df-sbg 14806
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