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Mirrors > Home > MPE Home > Th. List > islpidl | Structured version Visualization version GIF version |
Description: Property of being a principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.) |
Ref | Expression |
---|---|
lpival.p | β’ π = (LPIdealβπ ) |
lpival.k | β’ πΎ = (RSpanβπ ) |
lpival.b | β’ π΅ = (Baseβπ ) |
Ref | Expression |
---|---|
islpidl | β’ (π β Ring β (πΌ β π β βπ β π΅ πΌ = (πΎβ{π}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lpival.p | . . . 4 β’ π = (LPIdealβπ ) | |
2 | lpival.k | . . . 4 β’ πΎ = (RSpanβπ ) | |
3 | lpival.b | . . . 4 β’ π΅ = (Baseβπ ) | |
4 | 1, 2, 3 | lpival 20731 | . . 3 β’ (π β Ring β π = βͺ π β π΅ {(πΎβ{π})}) |
5 | 4 | eleq2d 2820 | . 2 β’ (π β Ring β (πΌ β π β πΌ β βͺ π β π΅ {(πΎβ{π})})) |
6 | eliun 4959 | . . 3 β’ (πΌ β βͺ π β π΅ {(πΎβ{π})} β βπ β π΅ πΌ β {(πΎβ{π})}) | |
7 | fvex 6856 | . . . . 5 β’ (πΎβ{π}) β V | |
8 | 7 | elsn2 4626 | . . . 4 β’ (πΌ β {(πΎβ{π})} β πΌ = (πΎβ{π})) |
9 | 8 | rexbii 3094 | . . 3 β’ (βπ β π΅ πΌ β {(πΎβ{π})} β βπ β π΅ πΌ = (πΎβ{π})) |
10 | 6, 9 | bitri 275 | . 2 β’ (πΌ β βͺ π β π΅ {(πΎβ{π})} β βπ β π΅ πΌ = (πΎβ{π})) |
11 | 5, 10 | bitrdi 287 | 1 β’ (π β Ring β (πΌ β π β βπ β π΅ πΌ = (πΎβ{π}))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1542 β wcel 2107 βwrex 3070 {csn 4587 βͺ ciun 4955 βcfv 6497 Basecbs 17088 Ringcrg 19969 RSpancrsp 20648 LPIdealclpidl 20727 |
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-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
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-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 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-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-iota 6449 df-fun 6499 df-fv 6505 df-lpidl 20729 |
This theorem is referenced by: lpi0 20733 lpi1 20734 lpiss 20736 lpigen 20742 ply1lpir 25559 lsmsnidl 32228 lpirlnr 41487 |
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