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Theorem grpsubid1 15729
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 15684 . . 3  |-  ( G  e.  Grp  ->  .0.  e.  B )
5 eqid 2454 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
6 eqid 2454 . . . 4  |-  ( invg `  G )  =  ( invg `  G )
7 grpsubid.m . . . 4  |-  .-  =  ( -g `  G )
82, 5, 6, 7grpsubval 15699 . . 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 15707 . . . 4  |-  ( G  e.  Grp  ->  (
( invg `  G ) `  .0.  )  =  .0.  )
1110adantr 465 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( invg `  G ) `  .0.  )  =  .0.  )
1211oveq2d 6215 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X ( +g  `  G ) ( ( invg `  G
) `  .0.  )
)  =  ( X ( +g  `  G
)  .0.  ) )
132, 5, 3grprid 15687 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X ( +g  `  G )  .0.  )  =  X )
149, 12, 133eqtrd 2499 1  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X  .-  .0.  )  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   ` cfv 5525  (class class class)co 6199   Basecbs 14291   +g cplusg 14356   0gc0g 14496   Grpcgrp 15528   invgcminusg 15529   -gcsg 15531
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 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-1st 6686  df-2nd 6687  df-0g 14498  df-mnd 15533  df-grp 15663  df-minusg 15664  df-sbg 15665
This theorem is referenced by:  odmod  16169  sylow3lem1  16246  dprdfeq0  16633  dprdfeq0OLD  16640  tsmsxplem1  19858  tngnm  20368  ply1divex  21740  ply1remlem  21766  qqhcn  26564  telescgsum  30965  cp0mat  31317  lcfrlem33  35543
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