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Theorem grporid 21813
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 21811 . 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 733 . 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 360    = wceq 1653    e. wcel 1726   E.wrex 2708   ran crn 4882   ` cfv 5457  (class class class)co 6084   GrpOpcgr 21779  GIdcgi 21780
This theorem is referenced by:  grporcan  21814  grpoinvid1  21823  grpoinvid2  21824  grpoasscan2  21831  grpopncan  21844  grponpcan  21845  gxcom  21862  gxid  21866  gxnn0add  21867  gxmodid  21872  rngo0rid  21992  rngolz  21994  vc0rid  22051  vcm  22055  nv0rid  22121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fo 5463  df-fv 5465  df-ov 6087  df-riota 6552  df-grpo 21784  df-gid 21785
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