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Theorem grpsubid1 14567
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 19 . . 3  |-  ( X  e.  B  ->  X  e.  B )
2 grpsubid.b . . . 4  |-  B  =  ( Base `  G
)
3 grpsubid.o . . . 4  |-  .0.  =  ( 0g `  G )
42, 3grpidcl 14526 . . 3  |-  ( G  e.  Grp  ->  .0.  e.  B )
5 eqid 2296 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
6 eqid 2296 . . . 4  |-  ( inv g `  G )  =  ( inv g `  G )
7 grpsubid.m . . . 4  |-  .-  =  ( -g `  G )
82, 5, 6, 7grpsubval 14541 . . 3  |-  ( ( X  e.  B  /\  .0.  e.  B )  -> 
( X  .-  .0.  )  =  ( X
( +g  `  G ) ( ( inv g `  G ) `  .0.  ) ) )
91, 4, 8syl2anr 464 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X  .-  .0.  )  =  ( X
( +g  `  G ) ( ( inv g `  G ) `  .0.  ) ) )
103, 6grpinvid 14549 . . . 4  |-  ( G  e.  Grp  ->  (
( inv g `  G ) `  .0.  )  =  .0.  )
1110adantr 451 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( inv g `  G ) `  .0.  )  =  .0.  )
1211oveq2d 5890 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X ( +g  `  G ) ( ( inv g `  G
) `  .0.  )
)  =  ( X ( +g  `  G
)  .0.  ) )
132, 5, 3grprid 14529 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X ( +g  `  G )  .0.  )  =  X )
149, 12, 133eqtrd 2332 1  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X  .-  .0.  )  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   0gc0g 13416   Grpcgrp 14378   inv gcminusg 14379   -gcsg 14381
This theorem is referenced by:  odmod  14877  sylow3lem1  14954  dprdfeq0  15273  tsmsxplem1  17851  tngnm  18183  ply1divex  19538  ply1remlem  19564  lcfrlem33  32387
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507
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