MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpoinvid Structured version   Unicode version

Theorem grpoinvid 21822
Description: The inverse of the identity element of a group. (Contributed by NM, 4-Dec-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpinvid.1  |-  U  =  (GId `  G )
grpinvid.2  |-  N  =  ( inv `  G
)
Assertion
Ref Expression
grpoinvid  |-  ( G  e.  GrpOp  ->  ( N `  U )  =  U )

Proof of Theorem grpoinvid
StepHypRef Expression
1 eqid 2438 . . . 4  |-  ran  G  =  ran  G
2 grpinvid.1 . . . 4  |-  U  =  (GId `  G )
31, 2grpoidcl 21807 . . 3  |-  ( G  e.  GrpOp  ->  U  e.  ran  G )
41, 2grpolid 21809 . . 3  |-  ( ( G  e.  GrpOp  /\  U  e.  ran  G )  -> 
( U G U )  =  U )
53, 4mpdan 651 . 2  |-  ( G  e.  GrpOp  ->  ( U G U )  =  U )
6 grpinvid.2 . . . 4  |-  N  =  ( inv `  G
)
71, 2, 6grpoinvid1 21820 . . 3  |-  ( ( G  e.  GrpOp  /\  U  e.  ran  G  /\  U  e.  ran  G )  -> 
( ( N `  U )  =  U  <-> 
( U G U )  =  U ) )
83, 3, 7mpd3an23 1282 . 2  |-  ( G  e.  GrpOp  ->  ( ( N `  U )  =  U  <->  ( U G U )  =  U ) )
95, 8mpbird 225 1  |-  ( G  e.  GrpOp  ->  ( N `  U )  =  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726   ran crn 4881   ` cfv 5456  (class class class)co 6083   GrpOpcgr 21776  GIdcgi 21777   invcgn 21778
This theorem is referenced by:  gxnn0neg  21853  gxinv  21860  gxid  21863
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-riota 6551  df-grpo 21781  df-gid 21782  df-ginv 21783
  Copyright terms: Public domain W3C validator