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Theorem grpsubid1 15591
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 15546 . . 3  |-  ( G  e.  Grp  ->  .0.  e.  B )
5 eqid 2433 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
6 eqid 2433 . . . 4  |-  ( invg `  G )  =  ( invg `  G )
7 grpsubid.m . . . 4  |-  .-  =  ( -g `  G )
82, 5, 6, 7grpsubval 15561 . . 3  |-  ( ( X  e.  B  /\  .0.  e.  B )  -> 
( X  .-  .0.  )  =  ( X
( +g  `  G ) ( ( invg `  G ) `  .0.  ) ) )
91, 4, 8syl2anr 475 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X  .-  .0.  )  =  ( X
( +g  `  G ) ( ( invg `  G ) `  .0.  ) ) )
103, 6grpinvid 15569 . . . 4  |-  ( G  e.  Grp  ->  (
( invg `  G ) `  .0.  )  =  .0.  )
1110adantr 462 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( invg `  G ) `  .0.  )  =  .0.  )
1211oveq2d 6096 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X ( +g  `  G ) ( ( invg `  G
) `  .0.  )
)  =  ( X ( +g  `  G
)  .0.  ) )
132, 5, 3grprid 15549 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X ( +g  `  G )  .0.  )  =  X )
149, 12, 133eqtrd 2469 1  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X  .-  .0.  )  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1362    e. wcel 1755   ` cfv 5406  (class class class)co 6080   Basecbs 14157   +g cplusg 14221   0gc0g 14361   Grpcgrp 15393   invgcminusg 15394   -gcsg 15396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-1st 6566  df-2nd 6567  df-0g 14363  df-mnd 15398  df-grp 15525  df-minusg 15526  df-sbg 15527
This theorem is referenced by:  odmod  16029  sylow3lem1  16106  dprdfeq0  16486  dprdfeq0OLD  16493  tsmsxplem1  19569  tngnm  20079  ply1divex  21493  ply1remlem  21519  qqhcn  26274  lcfrlem33  34793
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