Mathbox for Jeff Madsen < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rngosn6 Structured version   Visualization version   GIF version

Theorem rngosn6 34050
 Description: Obsolete as of 25-Jan-2020. Use ringen1zr 19491 or srgen1zr 18737 instead. The only unital ring with one element is the zero ring. (Contributed by FL, 15-Feb-2010.) (New usage is discouraged.)
Hypotheses
Ref Expression
on1el3.1 𝐺 = (1st𝑅)
on1el3.2 𝑋 = ran 𝐺
on1el3.3 𝑍 = (GId‘𝐺)
Assertion
Ref Expression
rngosn6 (𝑅 ∈ RingOps → (𝑋 ≈ 1𝑜𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩))

Proof of Theorem rngosn6
StepHypRef Expression
1 on1el3.1 . . 3 𝐺 = (1st𝑅)
2 on1el3.2 . . 3 𝑋 = ran 𝐺
3 on1el3.3 . . 3 𝑍 = (GId‘𝐺)
41, 2, 3rngo0cl 34043 . 2 (𝑅 ∈ RingOps → 𝑍𝑋)
51, 2rngosn4 34049 . 2 ((𝑅 ∈ RingOps ∧ 𝑍𝑋) → (𝑋 ≈ 1𝑜𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩))
64, 5mpdan 659 1 (𝑅 ∈ RingOps → (𝑋 ≈ 1𝑜𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   = wceq 1630   ∈ wcel 2144  {csn 4314  ⟨cop 4320   class class class wbr 4784  ran crn 5250  ‘cfv 6031  1st c1st 7312  1𝑜c1o 7705   ≈ cen 8105  GIdcgi 27678  RingOpscrngo 34018 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-om 7212  df-1st 7314  df-2nd 7315  df-1o 7712  df-er 7895  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-grpo 27681  df-gid 27682  df-ablo 27733  df-rngo 34019 This theorem is referenced by:  dvrunz  34078
 Copyright terms: Public domain W3C validator