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Theorem rngo0cl 21991
Description: A ring has an additive identity element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
ring0cl.1  |-  G  =  ( 1st `  R
)
ring0cl.2  |-  X  =  ran  G
ring0cl.3  |-  Z  =  (GId `  G )
Assertion
Ref Expression
rngo0cl  |-  ( R  e.  RingOps  ->  Z  e.  X
)

Proof of Theorem rngo0cl
StepHypRef Expression
1 ring0cl.1 . . 3  |-  G  =  ( 1st `  R
)
21rngogrpo 21983 . 2  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
3 ring0cl.2 . . 3  |-  X  =  ran  G
4 ring0cl.3 . . 3  |-  Z  =  (GId `  G )
53, 4grpoidcl 21810 . 2  |-  ( G  e.  GrpOp  ->  Z  e.  X )
62, 5syl 16 1  |-  ( R  e.  RingOps  ->  Z  e.  X
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   ran crn 4882   ` cfv 5457   1stc1st 6350   GrpOpcgr 21779  GIdcgi 21780   RingOpscrngo 21968
This theorem is referenced by:  rngolz  21994  rngorz  21995  rngosn6  22021  rngoueqz  22023  rngoidl  26648  0idl  26649  keridl  26656  prnc  26691  isdmn3  26698
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-pow 4380  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-1st 6352  df-2nd 6353  df-riota 6552  df-grpo 21784  df-gid 21785  df-ablo 21875  df-rngo 21969
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