Mathbox for Jeff Madsen < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rngo0cl Structured version   Visualization version   GIF version

Theorem rngo0cl 32888
 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 = (1st𝑅)
ring0cl.2 𝑋 = ran 𝐺
ring0cl.3 𝑍 = (GId‘𝐺)
Assertion
Ref Expression
rngo0cl (𝑅 ∈ RingOps → 𝑍𝑋)

Proof of Theorem rngo0cl
StepHypRef Expression
1 ring0cl.1 . . 3 𝐺 = (1st𝑅)
21rngogrpo 32879 . 2 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3 ring0cl.2 . . 3 𝑋 = ran 𝐺
4 ring0cl.3 . . 3 𝑍 = (GId‘𝐺)
53, 4grpoidcl 26752 . 2 (𝐺 ∈ GrpOp → 𝑍𝑋)
62, 5syl 17 1 (𝑅 ∈ RingOps → 𝑍𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ran crn 5039  ‘cfv 5804  1st c1st 7057  GrpOpcgr 26727  GIdcgi 26728  RingOpscrngo 32863 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-riota 6511  df-ov 6552  df-1st 7059  df-2nd 7060  df-grpo 26731  df-gid 26732  df-ablo 26783  df-rngo 32864 This theorem is referenced by:  rngolz  32891  rngorz  32892  rngosn6  32895  rngoueqz  32909  rngoidl  32993  0idl  32994  keridl  33001  prnc  33036  isdmn3  33043
 Copyright terms: Public domain W3C validator