Theorem rngosn6 37432

Description: Obsolete as of 25-Jan-2020. Use ringen1zr 20673 or srgen1zr 20163 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 37425	. 2 ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ 𝑋)
5	1, 2	rngosn4 37431	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑍 ∈ 𝑋) → (𝑋 ≈ 1o ↔ 𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩))
6	4, 5	mpdan 685	1 ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1o ↔ 𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098  {csn 4632  ⟨cop 4638   class class class wbr 5152  ran crn 5683  ‘cfv 6553  1^st c1st 7997  1_oc1o 8486   ≈ cen 8967  GIdcgi 30320  RingOpscrngo 37400

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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-1st 7999  df-2nd 8000  df-1o 8493  df-en 8971  df-grpo 30323  df-gid 30324  df-ablo 30375  df-rngo 37401

This theorem is referenced by:  dvrunz  37460