Theorem rngosn6 37912

Description: Obsolete as of 25-Jan-2020. Use ringen1zr 20795 or srgen1zr 20233 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 = (1st ‘𝑅)
on1el3.2	⊢ 𝑋 = ran 𝐺
on1el3.3	⊢ 𝑍 = (GId‘𝐺)

Assertion

Ref	Expression
rngosn6	⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1o ↔ 𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩))

Proof of Theorem rngosn6

Step	Hyp	Ref	Expression
1		on1el3.1	. . 3 ⊢ 𝐺 = (1st ‘𝑅)
2		on1el3.2	. . 3 ⊢ 𝑋 = ran 𝐺
3		on1el3.3	. . 3 ⊢ 𝑍 = (GId‘𝐺)
4	1, 2, 3	rngo0cl 37905	. 2 ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ 𝑋)
5	1, 2	rngosn4 37911	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑍 ∈ 𝑋) → (𝑋 ≈ 1o ↔ 𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩))
6	4, 5	mpdan 687	1 ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1o ↔ 𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1536   ∈ wcel 2105  {csn 4630  ⟨cop 4636   class class class wbr 5147  ran crn 5689  ‘cfv 6562  1^st c1st 8010  1_oc1o 8497   ≈ cen 8980  GIdcgi 30518  RingOpscrngo 37880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-1st 8012  df-2nd 8013  df-1o 8504  df-en 8984  df-grpo 30521  df-gid 30522  df-ablo 30573  df-rngo 37881

This theorem is referenced by:  dvrunz  37940