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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 21174 | . . 3 β’ (π β Ring β π = βͺ π β π΅ {(πΎβ{π})}) |
5 | 4 | eleq2d 2813 | . 2 β’ (π β Ring β (πΌ β π β πΌ β βͺ π β π΅ {(πΎβ{π})})) |
6 | eliun 4994 | . . 3 β’ (πΌ β βͺ π β π΅ {(πΎβ{π})} β βπ β π΅ πΌ β {(πΎβ{π})}) | |
7 | fvex 6897 | . . . . 5 β’ (πΎβ{π}) β V | |
8 | 7 | elsn2 4662 | . . . 4 β’ (πΌ β {(πΎβ{π})} β πΌ = (πΎβ{π})) |
9 | 8 | rexbii 3088 | . . 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 1533 β wcel 2098 βwrex 3064 {csn 4623 βͺ ciun 4990 βcfv 6536 Basecbs 17150 Ringcrg 20135 RSpancrsp 21063 LPIdealclpidl 21170 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6488 df-fun 6538 df-fv 6544 df-lpidl 21172 |
This theorem is referenced by: lpi0 21176 lpi1 21177 lpiss 21179 lpigen 21185 ply1lpir 26066 lsmsnidl 33014 mxidlirred 33093 lpirlnr 42419 |
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