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Theorem rngosn4 37932
Description: Obsolete as of 25-Jan-2020. Use rngen1zr 20778 instead. The only unital ring with one element is the zero ring. (Contributed by FL, 14-Feb-2010.) (Revised by Mario Carneiro, 30-Apr-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
on1el3.1 𝐺 = (1st𝑅)
on1el3.2 𝑋 = ran 𝐺
Assertion
Ref Expression
rngosn4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑋 ≈ 1o𝑅 = ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩))

Proof of Theorem rngosn4
StepHypRef Expression
1 en1eqsnbi 9310 . . 3 (𝐴𝑋 → (𝑋 ≈ 1o𝑋 = {𝐴}))
21adantl 481 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑋 ≈ 1o𝑋 = {𝐴}))
3 on1el3.1 . . 3 𝐺 = (1st𝑅)
4 on1el3.2 . . 3 𝑋 = ran 𝐺
53, 4rngosn3 37931 . 2 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑋 = {𝐴} ↔ 𝑅 = ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩))
62, 5bitrd 279 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑋 ≈ 1o𝑅 = ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  {csn 4626  cop 4632   class class class wbr 5143  ran crn 5686  cfv 6561  1st c1st 8012  1oc1o 8499  cen 8982  RingOpscrngo 37901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-1st 8014  df-2nd 8015  df-1o 8506  df-en 8986  df-grpo 30512  df-ablo 30564  df-rngo 37902
This theorem is referenced by:  rngosn6  37933
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