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Theorem islpidl 20732
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 20731 . . 3 (𝑅 ∈ Ring β†’ 𝑃 = βˆͺ 𝑔 ∈ 𝐡 {(πΎβ€˜{𝑔})})
54eleq2d 2820 . 2 (𝑅 ∈ Ring β†’ (𝐼 ∈ 𝑃 ↔ 𝐼 ∈ βˆͺ 𝑔 ∈ 𝐡 {(πΎβ€˜{𝑔})}))
6 eliun 4959 . . 3 (𝐼 ∈ βˆͺ 𝑔 ∈ 𝐡 {(πΎβ€˜{𝑔})} ↔ βˆƒπ‘” ∈ 𝐡 𝐼 ∈ {(πΎβ€˜{𝑔})})
7 fvex 6856 . . . . 5 (πΎβ€˜{𝑔}) ∈ V
87elsn2 4626 . . . 4 (𝐼 ∈ {(πΎβ€˜{𝑔})} ↔ 𝐼 = (πΎβ€˜{𝑔}))
98rexbii 3094 . . 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 1542   ∈ wcel 2107  βˆƒwrex 3070  {csn 4587  βˆͺ ciun 4955  β€˜cfv 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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