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Theorem grpolid 20902
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 20901 . . 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 1632    e. wcel 1696   E.wrex 2557   ran crn 4706   ` cfv 5271  (class class class)co 5874   GrpOpcgr 20869  GIdcgi 20870
This theorem is referenced by:  grpoid  20906  grpoinvid1  20913  grpoinvid2  20914  grpoinvid  20915  grpolcan  20916  grpo2grp  20917  grpoasscan1  20920  grpoinvop  20924  grpopnpcan2  20936  gxnn0suc  20947  gxcom  20952  ablonncan  20977  gxdi  20979  subgoid  20990  issubgoi  20993  ghomid  21048  rngo0lid  21083  rngolz  21084  rngorz  21085  vc0lid  21140  vcm  21143  nv0lid  21210  ghomgrpilem2  24008  ltrran2  25506  ltrinvlem  25509  rltrran  25517  addnull2  25567  grpoeqdivid  26674  keridl  26760
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-sep 4157  ax-nul 4165  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-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-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-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-ov 5877  df-riota 6320  df-grpo 20874  df-gid 20875
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