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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 2739 | . . 3 ⊢ (LPIdeal‘𝑅) = (LPIdeal‘𝑅) | |
2 | eqid 2739 | . . 3 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
3 | 1, 2 | islpir 20154 | . 2 ⊢ (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ (LIdeal‘𝑅) = (LPIdeal‘𝑅))) |
4 | 3 | simplbi 501 | 1 ⊢ (𝑅 ∈ LPIR → 𝑅 ∈ Ring) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ‘cfv 6350 Ringcrg 19429 LIdealclidl 20074 LPIdealclpidl 20146 LPIRclpir 20147 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2711 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-ex 1787 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-rab 3063 df-v 3402 df-un 3858 df-in 3860 df-ss 3870 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-iota 6308 df-fv 6358 df-lpir 20149 |
This theorem is referenced by: lpirlnr 40555 |
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