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Theorem grporid 23879
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 23877 . 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 754 . 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 369    = wceq 1370    e. wcel 1758   E.wrex 2800   ran crn 4952   ` cfv 5529  (class class class)co 6203   GrpOpcgr 23845  GIdcgi 23846
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 4524  ax-nul 4532  ax-pr 4642  ax-un 6485
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 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fo 5535  df-fv 5537  df-riota 6164  df-ov 6206  df-grpo 23850  df-gid 23851
This theorem is referenced by:  grporcan  23880  grpoinvid1  23889  grpoinvid2  23890  grpoasscan2  23897  grpopncan  23910  grponpcan  23911  gxcom  23928  gxid  23932  gxnn0add  23933  gxmodid  23938  rngo0rid  24058  rngolz  24060  vc0rid  24117  vcm  24121  nv0rid  24187
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