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Theorem grporid 25420
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 25418 . 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 753 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   E.wrex 2805   ran crn 4989   ` cfv 5570  (class class class)co 6270   GrpOpcgr 25386  GIdcgi 25387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fo 5576  df-fv 5578  df-riota 6232  df-ov 6273  df-grpo 25391  df-gid 25392
This theorem is referenced by:  grporcan  25421  grpoinvid1  25430  grpoinvid2  25431  grpoasscan2  25438  grpopncan  25451  grponpcan  25452  gxcom  25469  gxid  25473  gxnn0add  25474  gxmodid  25479  rngo0rid  25599  rngolz  25601  vc0rid  25658  vcm  25662  nv0rid  25728
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