MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpnpcan Structured version   Unicode version

Theorem grpnpcan 15999
Description: Cancellation law for subtraction (npcan 9829 analog). (Contributed by NM, 19-Apr-2014.)
Hypotheses
Ref Expression
grpsubadd.b  |-  B  =  ( Base `  G
)
grpsubadd.p  |-  .+  =  ( +g  `  G )
grpsubadd.m  |-  .-  =  ( -g `  G )
Assertion
Ref Expression
grpnpcan  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .-  Y )  .+  Y
)  =  X )

Proof of Theorem grpnpcan
StepHypRef Expression
1 grpsubadd.b . . . . . 6  |-  B  =  ( Base `  G
)
2 eqid 2441 . . . . . 6  |-  ( invg `  G )  =  ( invg `  G )
31, 2grpinvcl 15964 . . . . 5  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( ( invg `  G ) `  Y
)  e.  B )
433adant2 1014 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( invg `  G ) `  Y
)  e.  B )
5 grpsubadd.p . . . . 5  |-  .+  =  ( +g  `  G )
61, 5grpcl 15932 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  ( ( invg `  G ) `  Y
)  e.  B )  ->  ( X  .+  ( ( invg `  G ) `  Y
) )  e.  B
)
74, 6syld3an3 1272 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  (
( invg `  G ) `  Y
) )  e.  B
)
8 grpsubadd.m . . . 4  |-  .-  =  ( -g `  G )
91, 5, 2, 8grpsubval 15962 . . 3  |-  ( ( ( X  .+  (
( invg `  G ) `  Y
) )  e.  B  /\  ( ( invg `  G ) `  Y
)  e.  B )  ->  ( ( X 
.+  ( ( invg `  G ) `
 Y ) ) 
.-  ( ( invg `  G ) `
 Y ) )  =  ( ( X 
.+  ( ( invg `  G ) `
 Y ) ) 
.+  ( ( invg `  G ) `
 ( ( invg `  G ) `
 Y ) ) ) )
107, 4, 9syl2anc 661 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .+  ( ( invg `  G ) `  Y
) )  .-  (
( invg `  G ) `  Y
) )  =  ( ( X  .+  (
( invg `  G ) `  Y
) )  .+  (
( invg `  G ) `  (
( invg `  G ) `  Y
) ) ) )
111, 5, 8grppncan 15998 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  ( ( invg `  G ) `  Y
)  e.  B )  ->  ( ( X 
.+  ( ( invg `  G ) `
 Y ) ) 
.-  ( ( invg `  G ) `
 Y ) )  =  X )
124, 11syld3an3 1272 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .+  ( ( invg `  G ) `  Y
) )  .-  (
( invg `  G ) `  Y
) )  =  X )
131, 5, 2, 8grpsubval 15962 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y
)  =  ( X 
.+  ( ( invg `  G ) `
 Y ) ) )
14133adant1 1013 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y
)  =  ( X 
.+  ( ( invg `  G ) `
 Y ) ) )
1514eqcomd 2449 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  (
( invg `  G ) `  Y
) )  =  ( X  .-  Y ) )
161, 2grpinvinv 15974 . . . 4  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( ( invg `  G ) `  (
( invg `  G ) `  Y
) )  =  Y )
17163adant2 1014 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( invg `  G ) `  (
( invg `  G ) `  Y
) )  =  Y )
1815, 17oveq12d 6295 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .+  ( ( invg `  G ) `  Y
) )  .+  (
( invg `  G ) `  (
( invg `  G ) `  Y
) ) )  =  ( ( X  .-  Y )  .+  Y
) )
1910, 12, 183eqtr3rd 2491 1  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .-  Y )  .+  Y
)  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 972    = wceq 1381    e. wcel 1802   ` cfv 5574  (class class class)co 6277   Basecbs 14504   +g cplusg 14569   Grpcgrp 15922   invgcminusg 15923   -gcsg 15924
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6781  df-2nd 6782  df-0g 14711  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-grp 15926  df-minusg 15927  df-sbg 15928
This theorem is referenced by:  grpsubsub4  16000  grpnpncan  16002  grpnnncan2  16004  nsgconj  16103  conjghm  16166  conjnmz  16169  sylow2blem1  16509  ablpncan3  16696  lmodvnpcan  17432  coe1subfv  18175  ipsubdir  18544  ipsubdi  18545  mdetunilem9  18989  subgntr  20471  ghmcnp  20479  tgpt0  20483  r1pid  22426  archiabllem1a  27601  archiabllem2a  27604  ornglmulle  27661  orngrmulle  27662  kercvrlsm  30997  hbtlem5  31045
  Copyright terms: Public domain W3C validator