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Theorem grpolid 23859
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 23858 . . 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 459 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( U G A )  =  A  /\  ( A G U )  =  A ) )
54simpld 459 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( U G A )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   E.wrex 2800   ran crn 4950   ` cfv 5527  (class class class)co 6201   GrpOpcgr 23826  GIdcgi 23827
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-fo 5533  df-fv 5535  df-riota 6162  df-ov 6204  df-grpo 23831  df-gid 23832
This theorem is referenced by:  grpoid  23863  grpoinvid1  23870  grpoinvid2  23871  grpoinvid  23872  grpolcan  23873  grpo2grp  23874  grpoasscan1  23877  grpoinvop  23881  grpopnpcan2  23893  gxnn0suc  23904  gxcom  23909  ablonncan  23934  gxdi  23936  subgoid  23947  issubgoi  23950  ghomid  24005  rngo0lid  24040  rngolz  24041  rngorz  24042  vc0lid  24099  vcm  24102  nv0lid  24169  ghomgrpilem2  27450  grpoeqdivid  28895  keridl  28981
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