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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 21221 | . . 3 β’ (π β Ring β π = βͺ π β π΅ {(πΎβ{π})}) |
5 | 4 | eleq2d 2815 | . 2 β’ (π β Ring β (πΌ β π β πΌ β βͺ π β π΅ {(πΎβ{π})})) |
6 | eliun 5004 | . . 3 β’ (πΌ β βͺ π β π΅ {(πΎβ{π})} β βπ β π΅ πΌ β {(πΎβ{π})}) | |
7 | fvex 6915 | . . . . 5 β’ (πΎβ{π}) β V | |
8 | 7 | elsn2 4672 | . . . 4 β’ (πΌ β {(πΎβ{π})} β πΌ = (πΎβ{π})) |
9 | 8 | rexbii 3091 | . . 3 β’ (βπ β π΅ πΌ β {(πΎβ{π})} β βπ β π΅ πΌ = (πΎβ{π})) |
10 | 6, 9 | bitri 274 | . 2 β’ (πΌ β βͺ π β π΅ {(πΎβ{π})} β βπ β π΅ πΌ = (πΎβ{π})) |
11 | 5, 10 | bitrdi 286 | 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 3067 {csn 4632 βͺ ciun 5000 βcfv 6553 Basecbs 17187 Ringcrg 20180 RSpancrsp 21110 LPIdealclpidl 21217 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-iota 6505 df-fun 6555 df-fv 6561 df-lpidl 21219 |
This theorem is referenced by: lpi0 21223 lpi1 21224 lpiss 21226 lpigen 21232 ply1lpir 26136 lsmsnidl 33133 mxidlirred 33210 lpirlnr 42572 |
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