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Mirrors > Home > MPE Home > Th. List > islpir | Structured version Visualization version GIF version |
Description: Principal ideal rings are where all ideals are principal. (Contributed by Stefan O'Rear, 3-Jan-2015.) |
Ref | Expression |
---|---|
lpival.p | ⊢ 𝑃 = (LPIdeal‘𝑅) |
lpiss.u | ⊢ 𝑈 = (LIdeal‘𝑅) |
Ref | Expression |
---|---|
islpir | ⊢ (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ 𝑈 = 𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6670 | . . . 4 ⊢ (𝑟 = 𝑅 → (LIdeal‘𝑟) = (LIdeal‘𝑅)) | |
2 | fveq2 6670 | . . . 4 ⊢ (𝑟 = 𝑅 → (LPIdeal‘𝑟) = (LPIdeal‘𝑅)) | |
3 | 1, 2 | eqeq12d 2837 | . . 3 ⊢ (𝑟 = 𝑅 → ((LIdeal‘𝑟) = (LPIdeal‘𝑟) ↔ (LIdeal‘𝑅) = (LPIdeal‘𝑅))) |
4 | lpiss.u | . . . 4 ⊢ 𝑈 = (LIdeal‘𝑅) | |
5 | lpival.p | . . . 4 ⊢ 𝑃 = (LPIdeal‘𝑅) | |
6 | 4, 5 | eqeq12i 2836 | . . 3 ⊢ (𝑈 = 𝑃 ↔ (LIdeal‘𝑅) = (LPIdeal‘𝑅)) |
7 | 3, 6 | syl6bbr 291 | . 2 ⊢ (𝑟 = 𝑅 → ((LIdeal‘𝑟) = (LPIdeal‘𝑟) ↔ 𝑈 = 𝑃)) |
8 | df-lpir 20017 | . 2 ⊢ LPIR = {𝑟 ∈ Ring ∣ (LIdeal‘𝑟) = (LPIdeal‘𝑟)} | |
9 | 7, 8 | elrab2 3683 | 1 ⊢ (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ 𝑈 = 𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6355 Ringcrg 19297 LIdealclidl 19942 LPIdealclpidl 20014 LPIRclpir 20015 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 df-lpir 20017 |
This theorem is referenced by: islpir2 20024 lpirring 20025 lpirlnr 39737 |
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