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Theorem rngo0cl 21171
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 21163 . 2  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
3 ring0cl.2 . . 3  |-  X  =  ran  G
4 ring0cl.3 . . 3  |-  Z  =  (GId `  G )
53, 4grpoidcl 20990 . 2  |-  ( G  e.  GrpOp  ->  Z  e.  X )
62, 5syl 15 1  |-  ( R  e.  RingOps  ->  Z  e.  X
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1642    e. wcel 1710   ran crn 4769   ` cfv 5334   1stc1st 6204   GrpOpcgr 20959  GIdcgi 20960   RingOpscrngo 21148
This theorem is referenced by:  rngolz  21174  rngorz  21175  rngosn6  21201  rngoueqz  21203  rngoidl  25972  0idl  25973  keridl  25980  prnc  26015  isdmn3  26022
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-fo 5340  df-fv 5342  df-ov 5945  df-1st 6206  df-2nd 6207  df-riota 6388  df-grpo 20964  df-gid 20965  df-ablo 21055  df-rngo 21149
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