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Theorem grpolid 20886
Description: The identity element of a group is a left identity. (Contributed by NM, 24-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpoidval.1  |-  X  =  ran  G
grpoidval.2  |-  U  =  (GId `  G )
Assertion
Ref Expression
grpolid  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( U G A )  =  A )

Proof of Theorem grpolid
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 grpoidval.1 . . . 4  |-  X  =  ran  G
2 grpoidval.2 . . . 4  |-  U  =  (GId `  G )
31, 2grpoidinv2 20885 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( ( U G A )  =  A  /\  ( A G U )  =  A )  /\  E. y  e.  X  ( (
y G A )  =  U  /\  ( A G y )  =  U ) ) )
43simpld 445 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( U G A )  =  A  /\  ( A G U )  =  A ) )
54simpld 445 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( U G A )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   ran crn 4690   ` cfv 5255  (class class class)co 5858   GrpOpcgr 20853  GIdcgi 20854
This theorem is referenced by:  grpoid  20890  grpoinvid1  20897  grpoinvid2  20898  grpoinvid  20899  grpolcan  20900  grpo2grp  20901  grpoasscan1  20904  grpoinvop  20908  grpopnpcan2  20920  gxnn0suc  20931  gxcom  20936  ablonncan  20961  gxdi  20963  subgoid  20974  issubgoi  20977  ghomid  21032  rngo0lid  21067  rngolz  21068  rngorz  21069  vc0lid  21124  vcm  21127  nv0lid  21194  ghomgrpilem2  23993  ltrran2  25403  ltrinvlem  25406  rltrran  25414  addnull2  25464  grpoeqdivid  26571  keridl  26657
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-sep 4141  ax-nul 4149  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-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-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-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-riota 6304  df-grpo 20858  df-gid 20859
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