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Theorem rngosn6 37306
Description: Obsolete as of 25-Jan-2020. Use ringen1zr 20626 or srgen1zr 20118 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 37299 . 2 (𝑅 ∈ RingOps → 𝑍𝑋)
51, 2rngosn4 37305 . 2 ((𝑅 ∈ RingOps ∧ 𝑍𝑋) → (𝑋 ≈ 1o𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩))
64, 5mpdan 684 1 (𝑅 ∈ RingOps → (𝑋 ≈ 1o𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wcel 2098  {csn 4623  cop 4629   class class class wbr 5141  ran crn 5670  cfv 6536  1st c1st 7969  1oc1o 8457  cen 8935  GIdcgi 30247  RingOpscrngo 37274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-1st 7971  df-2nd 7972  df-1o 8464  df-en 8939  df-grpo 30250  df-gid 30251  df-ablo 30302  df-rngo 37275
This theorem is referenced by:  dvrunz  37334
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