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Theorem grpsubid1 14551
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 14510 . . 3  |-  ( G  e.  Grp  ->  .0.  e.  B )
5 eqid 2283 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
6 eqid 2283 . . . 4  |-  ( inv g `  G )  =  ( inv g `  G )
7 grpsubid.m . . . 4  |-  .-  =  ( -g `  G )
82, 5, 6, 7grpsubval 14525 . . 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 14533 . . . 4  |-  ( G  e.  Grp  ->  (
( inv g `  G ) `  .0.  )  =  .0.  )
1110adantr 451 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( inv g `  G ) `  .0.  )  =  .0.  )
1211oveq2d 5874 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X ( +g  `  G ) ( ( inv g `  G
) `  .0.  )
)  =  ( X ( +g  `  G
)  .0.  ) )
132, 5, 3grprid 14513 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X ( +g  `  G )  .0.  )  =  X )
149, 12, 133eqtrd 2319 1  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X  .-  .0.  )  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   0gc0g 13400   Grpcgrp 14362   inv gcminusg 14363   -gcsg 14365
This theorem is referenced by:  odmod  14861  sylow3lem1  14938  dprdfeq0  15257  tsmsxplem1  17835  tngnm  18167  ply1divex  19522  ply1remlem  19548  lcfrlem33  31138
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491
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