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Theorem islpidl 21175
Description: Property of being a principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
Hypotheses
Ref Expression
lpival.p 𝑃 = (LPIdealβ€˜π‘…)
lpival.k 𝐾 = (RSpanβ€˜π‘…)
lpival.b 𝐡 = (Baseβ€˜π‘…)
Assertion
Ref Expression
islpidl (𝑅 ∈ Ring β†’ (𝐼 ∈ 𝑃 ↔ βˆƒπ‘” ∈ 𝐡 𝐼 = (πΎβ€˜{𝑔})))
Distinct variable groups:   𝑅,𝑔   𝑃,𝑔   𝐡,𝑔   𝑔,𝐾   𝑔,𝐼

Proof of Theorem islpidl
StepHypRef Expression
1 lpival.p . . . 4 𝑃 = (LPIdealβ€˜π‘…)
2 lpival.k . . . 4 𝐾 = (RSpanβ€˜π‘…)
3 lpival.b . . . 4 𝐡 = (Baseβ€˜π‘…)
41, 2, 3lpival 21174 . . 3 (𝑅 ∈ Ring β†’ 𝑃 = βˆͺ 𝑔 ∈ 𝐡 {(πΎβ€˜{𝑔})})
54eleq2d 2813 . 2 (𝑅 ∈ Ring β†’ (𝐼 ∈ 𝑃 ↔ 𝐼 ∈ βˆͺ 𝑔 ∈ 𝐡 {(πΎβ€˜{𝑔})}))
6 eliun 4994 . . 3 (𝐼 ∈ βˆͺ 𝑔 ∈ 𝐡 {(πΎβ€˜{𝑔})} ↔ βˆƒπ‘” ∈ 𝐡 𝐼 ∈ {(πΎβ€˜{𝑔})})
7 fvex 6897 . . . . 5 (πΎβ€˜{𝑔}) ∈ V
87elsn2 4662 . . . 4 (𝐼 ∈ {(πΎβ€˜{𝑔})} ↔ 𝐼 = (πΎβ€˜{𝑔}))
98rexbii 3088 . . 3 (βˆƒπ‘” ∈ 𝐡 𝐼 ∈ {(πΎβ€˜{𝑔})} ↔ βˆƒπ‘” ∈ 𝐡 𝐼 = (πΎβ€˜{𝑔}))
106, 9bitri 275 . 2 (𝐼 ∈ βˆͺ 𝑔 ∈ 𝐡 {(πΎβ€˜{𝑔})} ↔ βˆƒπ‘” ∈ 𝐡 𝐼 = (πΎβ€˜{𝑔}))
115, 10bitrdi 287 1 (𝑅 ∈ Ring β†’ (𝐼 ∈ 𝑃 ↔ βˆƒπ‘” ∈ 𝐡 𝐼 = (πΎβ€˜{𝑔})))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3064  {csn 4623  βˆͺ ciun 4990  β€˜cfv 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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