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Theorem islpidl 19067
 Description: Property of being a principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
Hypotheses
Ref Expression
lpival.p 𝑃 = (LPIdeal‘𝑅)
lpival.k 𝐾 = (RSpan‘𝑅)
lpival.b 𝐵 = (Base‘𝑅)
Assertion
Ref Expression
islpidl (𝑅 ∈ Ring → (𝐼𝑃 ↔ ∃𝑔𝐵 𝐼 = (𝐾‘{𝑔})))
Distinct variable groups:   𝑅,𝑔   𝑃,𝑔   𝐵,𝑔   𝑔,𝐾   𝑔,𝐼

Proof of Theorem islpidl
StepHypRef Expression
1 lpival.p . . . 4 𝑃 = (LPIdeal‘𝑅)
2 lpival.k . . . 4 𝐾 = (RSpan‘𝑅)
3 lpival.b . . . 4 𝐵 = (Base‘𝑅)
41, 2, 3lpival 19066 . . 3 (𝑅 ∈ Ring → 𝑃 = 𝑔𝐵 {(𝐾‘{𝑔})})
54eleq2d 2673 . 2 (𝑅 ∈ Ring → (𝐼𝑃𝐼 𝑔𝐵 {(𝐾‘{𝑔})}))
6 eliun 4460 . . 3 (𝐼 𝑔𝐵 {(𝐾‘{𝑔})} ↔ ∃𝑔𝐵 𝐼 ∈ {(𝐾‘{𝑔})})
7 fvex 6113 . . . . 5 (𝐾‘{𝑔}) ∈ V
87elsn2 4158 . . . 4 (𝐼 ∈ {(𝐾‘{𝑔})} ↔ 𝐼 = (𝐾‘{𝑔}))
98rexbii 3023 . . 3 (∃𝑔𝐵 𝐼 ∈ {(𝐾‘{𝑔})} ↔ ∃𝑔𝐵 𝐼 = (𝐾‘{𝑔}))
106, 9bitri 263 . 2 (𝐼 𝑔𝐵 {(𝐾‘{𝑔})} ↔ ∃𝑔𝐵 𝐼 = (𝐾‘{𝑔}))
115, 10syl6bb 275 1 (𝑅 ∈ Ring → (𝐼𝑃 ↔ ∃𝑔𝐵 𝐼 = (𝐾‘{𝑔})))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  {csn 4125  ∪ ciun 4455  ‘cfv 5804  Basecbs 15695  Ringcrg 18370  RSpancrsp 18992  LPIdealclpidl 19062 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-lpidl 19064 This theorem is referenced by:  lpi0  19068  lpi1  19069  lpiss  19071  lpigen  19077  ply1lpir  23742  lpirlnr  36706
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