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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 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.) |
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 36407 | . 2 ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ 𝑋) |
5 | 1, 2 | rngosn4 36413 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑍 ∈ 𝑋) → (𝑋 ≈ 1o ↔ 𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩)) |
6 | 4, 5 | mpdan 686 | 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 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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