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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rngosn4 | Structured version Visualization version GIF version | ||
| Description: Obsolete as of 25-Jan-2020. Use rngen1zr 20778 instead. The only unital ring with one element is the zero ring. (Contributed by FL, 14-Feb-2010.) (Revised by Mario Carneiro, 30-Apr-2015.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| on1el3.1 | ⊢ 𝐺 = (1st ‘𝑅) |
| on1el3.2 | ⊢ 𝑋 = ran 𝐺 |
| Ref | Expression |
|---|---|
| rngosn4 | ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑋 ≈ 1o ↔ 𝑅 = 〈{〈〈𝐴, 𝐴〉, 𝐴〉}, {〈〈𝐴, 𝐴〉, 𝐴〉}〉)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | en1eqsnbi 9310 | . . 3 ⊢ (𝐴 ∈ 𝑋 → (𝑋 ≈ 1o ↔ 𝑋 = {𝐴})) | |
| 2 | 1 | adantl 481 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑋 ≈ 1o ↔ 𝑋 = {𝐴})) |
| 3 | on1el3.1 | . . 3 ⊢ 𝐺 = (1st ‘𝑅) | |
| 4 | on1el3.2 | . . 3 ⊢ 𝑋 = ran 𝐺 | |
| 5 | 3, 4 | rngosn3 37931 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑋 = {𝐴} ↔ 𝑅 = 〈{〈〈𝐴, 𝐴〉, 𝐴〉}, {〈〈𝐴, 𝐴〉, 𝐴〉}〉)) |
| 6 | 2, 5 | bitrd 279 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑋 ≈ 1o ↔ 𝑅 = 〈{〈〈𝐴, 𝐴〉, 𝐴〉}, {〈〈𝐴, 𝐴〉, 𝐴〉}〉)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {csn 4626 〈cop 4632 class class class wbr 5143 ran crn 5686 ‘cfv 6561 1st c1st 8012 1oc1o 8499 ≈ cen 8982 RingOpscrngo 37901 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-1st 8014 df-2nd 8015 df-1o 8506 df-en 8986 df-grpo 30512 df-ablo 30564 df-rngo 37902 |
| This theorem is referenced by: rngosn6 37933 |
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