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Theorem grpolid 21799
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 21798 . . 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 446 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( U G A )  =  A  /\  ( A G U )  =  A ) )
54simpld 446 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( U G A )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2698   ran crn 4871   ` cfv 5446  (class class class)co 6073   GrpOpcgr 21766  GIdcgi 21767
This theorem is referenced by:  grpoid  21803  grpoinvid1  21810  grpoinvid2  21811  grpoinvid  21812  grpolcan  21813  grpo2grp  21814  grpoasscan1  21817  grpoinvop  21821  grpopnpcan2  21833  gxnn0suc  21844  gxcom  21849  ablonncan  21874  gxdi  21876  subgoid  21887  issubgoi  21890  ghomid  21945  rngo0lid  21980  rngolz  21981  rngorz  21982  vc0lid  22039  vcm  22042  nv0lid  22109  ghomgrpilem2  25089  grpoeqdivid  26537  keridl  26623
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454  df-ov 6076  df-riota 6541  df-grpo 21771  df-gid 21772
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