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Theorem grpsubrcan 15706
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 2451 . . . . . 6  |-  ( +g  `  G )  =  ( +g  `  G )
3 eqid 2451 . . . . . 6  |-  ( invg `  G )  =  ( invg `  G )
4 grpsubcl.m . . . . . 6  |-  .-  =  ( -g `  G )
51, 2, 3, 4grpsubval 15680 . . . . 5  |-  ( ( X  e.  B  /\  Z  e.  B )  ->  ( X  .-  Z
)  =  ( X ( +g  `  G
) ( ( invg `  G ) `
 Z ) ) )
653adant2 1007 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  ->  ( X  .-  Z
)  =  ( X ( +g  `  G
) ( ( invg `  G ) `
 Z ) ) )
71, 2, 3, 4grpsubval 15680 . . . . 5  |-  ( ( Y  e.  B  /\  Z  e.  B )  ->  ( Y  .-  Z
)  =  ( Y ( +g  `  G
) ( ( invg `  G ) `
 Z ) ) )
873adant1 1006 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .-  Z
)  =  ( Y ( +g  `  G
) ( ( invg `  G ) `
 Z ) ) )
96, 8eqeq12d 2472 . . 3  |-  ( ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  ->  ( ( X  .-  Z )  =  ( Y  .-  Z )  <-> 
( X ( +g  `  G ) ( ( invg `  G
) `  Z )
)  =  ( Y ( +g  `  G
) ( ( invg `  G ) `
 Z ) ) ) )
109adantl 466 . 2  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .-  Z
)  =  ( Y 
.-  Z )  <->  ( X
( +g  `  G ) ( ( invg `  G ) `  Z
) )  =  ( Y ( +g  `  G
) ( ( invg `  G ) `
 Z ) ) ) )
11 simpl 457 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  G  e.  Grp )
12 simpr1 994 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  X  e.  B )
13 simpr2 995 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Y  e.  B )
141, 3grpinvcl 15682 . . . 4  |-  ( ( G  e.  Grp  /\  Z  e.  B )  ->  ( ( invg `  G ) `  Z
)  e.  B )
15143ad2antr3 1155 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( invg `  G ) `  Z
)  e.  B )
161, 2grprcan 15670 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  ( ( invg `  G ) `  Z
)  e.  B ) )  ->  ( ( X ( +g  `  G
) ( ( invg `  G ) `
 Z ) )  =  ( Y ( +g  `  G ) ( ( invg `  G ) `  Z
) )  <->  X  =  Y ) )
1711, 12, 13, 15, 16syl13anc 1221 . 2  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X ( +g  `  G ) ( ( invg `  G
) `  Z )
)  =  ( Y ( +g  `  G
) ( ( invg `  G ) `
 Z ) )  <-> 
X  =  Y ) )
1810, 17bitrd 253 1  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .-  Z
)  =  ( Y 
.-  Z )  <->  X  =  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   ` cfv 5513  (class class class)co 6187   Basecbs 14273   +g cplusg 14337   Grpcgrp 15509   invgcminusg 15510   -gcsg 15512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-1st 6674  df-2nd 6675  df-0g 14479  df-mnd 15514  df-grp 15644  df-minusg 15645  df-sbg 15646
This theorem is referenced by:  abladdsub4  16404  ogrpsublt  26316
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