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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 6663 | . . . 4 ⊢ (𝑟 = 𝑅 → (LIdeal‘𝑟) = (LIdeal‘𝑅)) | |
2 | fveq2 6663 | . . . 4 ⊢ (𝑟 = 𝑅 → (LPIdeal‘𝑟) = (LPIdeal‘𝑅)) | |
3 | 1, 2 | eqeq12d 2836 | . . 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 2835 | . . 3 ⊢ (𝑈 = 𝑃 ↔ (LIdeal‘𝑅) = (LPIdeal‘𝑅)) |
7 | 3, 6 | syl6bbr 291 | . 2 ⊢ (𝑟 = 𝑅 → ((LIdeal‘𝑟) = (LPIdeal‘𝑟) ↔ 𝑈 = 𝑃)) |
8 | df-lpir 20012 | . 2 ⊢ LPIR = {𝑟 ∈ Ring ∣ (LIdeal‘𝑟) = (LPIdeal‘𝑟)} | |
9 | 7, 8 | elrab2 3679 | 1 ⊢ (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ 𝑈 = 𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ‘cfv 6348 Ringcrg 19292 LIdealclidl 19937 LPIdealclpidl 20009 LPIRclpir 20010 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-iota 6307 df-fv 6356 df-lpir 20012 |
This theorem is referenced by: islpir2 20019 lpirring 20020 lpirlnr 39793 |
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