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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 6717 | . . . 4 ⊢ (𝑟 = 𝑅 → (LIdeal‘𝑟) = (LIdeal‘𝑅)) | |
2 | fveq2 6717 | . . . 4 ⊢ (𝑟 = 𝑅 → (LPIdeal‘𝑟) = (LPIdeal‘𝑅)) | |
3 | 1, 2 | eqeq12d 2753 | . . 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 2755 | . . 3 ⊢ (𝑈 = 𝑃 ↔ (LIdeal‘𝑅) = (LPIdeal‘𝑅)) |
7 | 3, 6 | bitr4di 292 | . 2 ⊢ (𝑟 = 𝑅 → ((LIdeal‘𝑟) = (LPIdeal‘𝑟) ↔ 𝑈 = 𝑃)) |
8 | df-lpir 20282 | . 2 ⊢ LPIR = {𝑟 ∈ Ring ∣ (LIdeal‘𝑟) = (LPIdeal‘𝑟)} | |
9 | 7, 8 | elrab2 3605 | 1 ⊢ (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ 𝑈 = 𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ‘cfv 6380 Ringcrg 19562 LIdealclidl 20207 LPIdealclpidl 20279 LPIRclpir 20280 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-iota 6338 df-fv 6388 df-lpir 20282 |
This theorem is referenced by: islpir2 20289 lpirring 20290 lpirlnr 40645 |
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