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Theorem grpsubid1 15602
Description: Subtraction of the identity from a group element. (Contributed by Mario Carneiro, 14-Jan-2015.)
Hypotheses
Ref Expression
grpsubid.b  |-  B  =  ( Base `  G
)
grpsubid.o  |-  .0.  =  ( 0g `  G )
grpsubid.m  |-  .-  =  ( -g `  G )
Assertion
Ref Expression
grpsubid1  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X  .-  .0.  )  =  X )

Proof of Theorem grpsubid1
StepHypRef Expression
1 id 22 . . 3  |-  ( X  e.  B  ->  X  e.  B )
2 grpsubid.b . . . 4  |-  B  =  ( Base `  G
)
3 grpsubid.o . . . 4  |-  .0.  =  ( 0g `  G )
42, 3grpidcl 15557 . . 3  |-  ( G  e.  Grp  ->  .0.  e.  B )
5 eqid 2438 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
6 eqid 2438 . . . 4  |-  ( invg `  G )  =  ( invg `  G )
7 grpsubid.m . . . 4  |-  .-  =  ( -g `  G )
82, 5, 6, 7grpsubval 15572 . . 3  |-  ( ( X  e.  B  /\  .0.  e.  B )  -> 
( X  .-  .0.  )  =  ( X
( +g  `  G ) ( ( invg `  G ) `  .0.  ) ) )
91, 4, 8syl2anr 478 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X  .-  .0.  )  =  ( X
( +g  `  G ) ( ( invg `  G ) `  .0.  ) ) )
103, 6grpinvid 15580 . . . 4  |-  ( G  e.  Grp  ->  (
( invg `  G ) `  .0.  )  =  .0.  )
1110adantr 465 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( invg `  G ) `  .0.  )  =  .0.  )
1211oveq2d 6102 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X ( +g  `  G ) ( ( invg `  G
) `  .0.  )
)  =  ( X ( +g  `  G
)  .0.  ) )
132, 5, 3grprid 15560 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X ( +g  `  G )  .0.  )  =  X )
149, 12, 133eqtrd 2474 1  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X  .-  .0.  )  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   ` cfv 5413  (class class class)co 6086   Basecbs 14166   +g cplusg 14230   0gc0g 14370   Grpcgrp 15402   invgcminusg 15403   -gcsg 15405
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-1st 6572  df-2nd 6573  df-0g 14372  df-mnd 15407  df-grp 15536  df-minusg 15537  df-sbg 15538
This theorem is referenced by:  odmod  16040  sylow3lem1  16117  dprdfeq0  16502  dprdfeq0OLD  16509  tsmsxplem1  19707  tngnm  20217  ply1divex  21588  ply1remlem  21614  qqhcn  26389  lcfrlem33  35113
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