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Theorem rngo0cl 21943
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 21935 . 2  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
3 ring0cl.2 . . 3  |-  X  =  ran  G
4 ring0cl.3 . . 3  |-  Z  =  (GId `  G )
53, 4grpoidcl 21762 . 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 1649    e. wcel 1721   ran crn 4842   ` cfv 5417   1stc1st 6310   GrpOpcgr 21731  GIdcgi 21732   RingOpscrngo 21920
This theorem is referenced by:  rngolz  21946  rngorz  21947  rngosn6  21973  rngoueqz  21975  rngoidl  26528  0idl  26529  keridl  26536  prnc  26571  isdmn3  26578
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-fo 5423  df-fv 5425  df-ov 6047  df-1st 6312  df-2nd 6313  df-riota 6512  df-grpo 21736  df-gid 21737  df-ablo 21827  df-rngo 21921
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