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Theorem grpolid 25035
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 25034 . . 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 1379    e. wcel 1767   E.wrex 2818   ran crn 5006   ` cfv 5594  (class class class)co 6295   GrpOpcgr 25002  GIdcgi 25003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fo 5600  df-fv 5602  df-riota 6256  df-ov 6298  df-grpo 25007  df-gid 25008
This theorem is referenced by:  grpoid  25039  grpoinvid1  25046  grpoinvid2  25047  grpoinvid  25048  grpolcan  25049  grpo2grp  25050  grpoasscan1  25053  grpoinvop  25057  grpopnpcan2  25069  gxnn0suc  25080  gxcom  25085  ablonncan  25110  gxdi  25112  subgoid  25123  issubgoi  25126  ghomid  25181  rngo0lid  25216  rngolz  25217  rngorz  25218  vc0lid  25275  vcm  25278  nv0lid  25345  ghomgrpilem2  28842  grpoeqdivid  30261  keridl  30347
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