|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > lpirring | Structured version Visualization version GIF version | ||
| Description: Principal ideal rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.) | 
| Ref | Expression | 
|---|---|
| lpirring | ⊢ (𝑅 ∈ LPIR → 𝑅 ∈ Ring) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2737 | . . 3 ⊢ (LPIdeal‘𝑅) = (LPIdeal‘𝑅) | |
| 2 | eqid 2737 | . . 3 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
| 3 | 1, 2 | islpir 21338 | . 2 ⊢ (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ (LIdeal‘𝑅) = (LPIdeal‘𝑅))) | 
| 4 | 3 | simplbi 497 | 1 ⊢ (𝑅 ∈ LPIR → 𝑅 ∈ Ring) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 Ringcrg 20230 LIdealclidl 21216 LPIdealclpidl 21330 LPIRclpir 21331 | 
| 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-ext 2708 | 
| 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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-lpir 21333 | 
| This theorem is referenced by: lpirlnr 43129 | 
| Copyright terms: Public domain | W3C validator |