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Theorem grporid 21796
Description: The identity element of a group is a right 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
grporid  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G U )  =  A )

Proof of Theorem grporid
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 grpoidval.1 . . 3  |-  X  =  ran  G
2 grpoidval.2 . . 3  |-  U  =  (GId `  G )
31, 2grpoidinv2 21794 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( ( U G A )  =  A  /\  ( A G U )  =  A )  /\  E. x  e.  X  ( (
x G A )  =  U  /\  ( A G x )  =  U ) ) )
4 simplr 732 . 2  |-  ( ( ( ( U G A )  =  A  /\  ( A G U )  =  A )  /\  E. x  e.  X  ( (
x G A )  =  U  /\  ( A G x )  =  U ) )  -> 
( A G U )  =  A )
53, 4syl 16 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G U )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2698   ran crn 4870   ` cfv 5445  (class class class)co 6072   GrpOpcgr 21762  GIdcgi 21763
This theorem is referenced by:  grporcan  21797  grpoinvid1  21806  grpoinvid2  21807  grpoasscan2  21814  grpopncan  21827  grponpcan  21828  gxcom  21845  gxid  21849  gxnn0add  21850  gxmodid  21855  rngo0rid  21975  rngolz  21977  vc0rid  22034  vcm  22038  nv0rid  22104
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 4692
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 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-fo 5451  df-fv 5453  df-ov 6075  df-riota 6540  df-grpo 21767  df-gid 21768
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