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Theorem islpidl 21222
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 21221 . . 3 (𝑅 ∈ Ring β†’ 𝑃 = βˆͺ 𝑔 ∈ 𝐡 {(πΎβ€˜{𝑔})})
54eleq2d 2815 . 2 (𝑅 ∈ Ring β†’ (𝐼 ∈ 𝑃 ↔ 𝐼 ∈ βˆͺ 𝑔 ∈ 𝐡 {(πΎβ€˜{𝑔})}))
6 eliun 5004 . . 3 (𝐼 ∈ βˆͺ 𝑔 ∈ 𝐡 {(πΎβ€˜{𝑔})} ↔ βˆƒπ‘” ∈ 𝐡 𝐼 ∈ {(πΎβ€˜{𝑔})})
7 fvex 6915 . . . . 5 (πΎβ€˜{𝑔}) ∈ V
87elsn2 4672 . . . 4 (𝐼 ∈ {(πΎβ€˜{𝑔})} ↔ 𝐼 = (πΎβ€˜{𝑔}))
98rexbii 3091 . . 3 (βˆƒπ‘” ∈ 𝐡 𝐼 ∈ {(πΎβ€˜{𝑔})} ↔ βˆƒπ‘” ∈ 𝐡 𝐼 = (πΎβ€˜{𝑔}))
106, 9bitri 274 . 2 (𝐼 ∈ βˆͺ 𝑔 ∈ 𝐡 {(πΎβ€˜{𝑔})} ↔ βˆƒπ‘” ∈ 𝐡 𝐼 = (πΎβ€˜{𝑔}))
115, 10bitrdi 286 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 3067  {csn 4632  βˆͺ ciun 5000  β€˜cfv 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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