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Theorem rngosn6 36414
Description: Obsolete as of 25-Jan-2020. Use ringen1zr 20763 or srgen1zr 19954 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 36407 . 2 (𝑅 ∈ RingOps → 𝑍𝑋)
51, 2rngosn4 36413 . 2 ((𝑅 ∈ RingOps ∧ 𝑍𝑋) → (𝑋 ≈ 1o𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩))
64, 5mpdan 686 1 (𝑅 ∈ RingOps → (𝑋 ≈ 1o𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wcel 2107  {csn 4591  cop 4597   class class class wbr 5110  ran crn 5639  cfv 6501  1st c1st 7924  1oc1o 8410  cen 8887  GIdcgi 29474  RingOpscrngo 36382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-1st 7926  df-2nd 7927  df-1o 8417  df-en 8891  df-grpo 29477  df-gid 29478  df-ablo 29529  df-rngo 36383
This theorem is referenced by:  dvrunz  36442
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