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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 6843 | . . . 4 β’ (π = π β (LIdealβπ) = (LIdealβπ )) | |
2 | fveq2 6843 | . . . 4 β’ (π = π β (LPIdealβπ) = (LPIdealβπ )) | |
3 | 1, 2 | eqeq12d 2749 | . . 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 2751 | . . 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 20730 | . 2 β’ LPIR = {π β Ring β£ (LIdealβπ) = (LPIdealβπ)} | |
9 | 7, 8 | elrab2 3649 | 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 397 = wceq 1542 β wcel 2107 βcfv 6497 Ringcrg 19969 LIdealclidl 20647 LPIdealclpidl 20727 LPIRclpir 20728 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-iota 6449 df-fv 6505 df-lpir 20730 |
This theorem is referenced by: islpir2 20737 lpirring 20738 lpirlnr 41487 |
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