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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 2777 | . . 3 ⊢ (LPIdeal‘𝑅) = (LPIdeal‘𝑅) | |
2 | eqid 2777 | . . 3 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
3 | 1, 2 | islpir 19646 | . 2 ⊢ (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ (LIdeal‘𝑅) = (LPIdeal‘𝑅))) |
4 | 3 | simplbi 493 | 1 ⊢ (𝑅 ∈ LPIR → 𝑅 ∈ Ring) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2106 ‘cfv 6135 Ringcrg 18934 LIdealclidl 19567 LPIdealclpidl 19638 LPIRclpir 19639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-rex 3095 df-rab 3098 df-v 3399 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-iota 6099 df-fv 6143 df-lpir 19641 |
This theorem is referenced by: lpirlnr 38638 |
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