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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 | β’ π = (LPIdealβπ ) |
lpiss.u | β’ π = (LIdealβπ ) |
Ref | Expression |
---|---|
islpir | β’ (π β LPIR β (π β Ring β§ π = π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6897 | . . . 4 β’ (π = π β (LIdealβπ) = (LIdealβπ )) | |
2 | fveq2 6897 | . . . 4 β’ (π = π β (LPIdealβπ) = (LPIdealβπ )) | |
3 | 1, 2 | eqeq12d 2744 | . . 3 β’ (π = π β ((LIdealβπ) = (LPIdealβπ) β (LIdealβπ ) = (LPIdealβπ ))) |
4 | lpiss.u | . . . 4 β’ π = (LIdealβπ ) | |
5 | lpival.p | . . . 4 β’ π = (LPIdealβπ ) | |
6 | 4, 5 | eqeq12i 2746 | . . 3 β’ (π = π β (LIdealβπ ) = (LPIdealβπ )) |
7 | 3, 6 | bitr4di 289 | . 2 β’ (π = π β ((LIdealβπ) = (LPIdealβπ) β π = π)) |
8 | df-lpir 21213 | . 2 β’ LPIR = {π β Ring β£ (LIdealβπ) = (LPIdealβπ)} | |
9 | 7, 8 | elrab2 3685 | 1 β’ (π β LPIR β (π β Ring β§ π = π)) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 395 = wceq 1534 β wcel 2099 βcfv 6548 Ringcrg 20173 LIdealclidl 21102 LPIdealclpidl 21210 LPIRclpir 21211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-iota 6500 df-fv 6556 df-lpir 21213 |
This theorem is referenced by: islpir2 21220 lpirring 21221 mxidlirred 33198 lpirlnr 42541 |
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