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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngosn6 | Structured version Visualization version GIF version |
Description: Obsolete as of 25-Jan-2020. Use ringen1zr 20398 or srgen1zr 20038 instead. The only unital ring with one element is the zero ring. (Contributed by FL, 15-Feb-2010.) (New usage is discouraged.) |
Ref | Expression |
---|---|
on1el3.1 | ⊢ 𝐺 = (1st ‘𝑅) |
on1el3.2 | ⊢ 𝑋 = ran 𝐺 |
on1el3.3 | ⊢ 𝑍 = (GId‘𝐺) |
Ref | Expression |
---|---|
rngosn6 | ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1o ↔ 𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩)) |
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 36782 | . 2 ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ 𝑋) |
5 | 1, 2 | rngosn4 36788 | . 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⟩}⟩)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1541 ∈ wcel 2106 {csn 4628 ⟨cop 4634 class class class wbr 5148 ran crn 5677 ‘cfv 6543 1st c1st 7972 1oc1o 8458 ≈ cen 8935 GIdcgi 29738 RingOpscrngo 36757 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-1st 7974 df-2nd 7975 df-1o 8465 df-en 8939 df-grpo 29741 df-gid 29742 df-ablo 29793 df-rngo 36758 |
This theorem is referenced by: dvrunz 36817 |
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