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Theorem islpidl 21352
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 21351 . . 3 (𝑅 ∈ Ring → 𝑃 = 𝑔𝐵 {(𝐾‘{𝑔})})
54eleq2d 2824 . 2 (𝑅 ∈ Ring → (𝐼𝑃𝐼 𝑔𝐵 {(𝐾‘{𝑔})}))
6 eliun 4999 . . 3 (𝐼 𝑔𝐵 {(𝐾‘{𝑔})} ↔ ∃𝑔𝐵 𝐼 ∈ {(𝐾‘{𝑔})})
7 fvex 6919 . . . . 5 (𝐾‘{𝑔}) ∈ V
87elsn2 4669 . . . 4 (𝐼 ∈ {(𝐾‘{𝑔})} ↔ 𝐼 = (𝐾‘{𝑔}))
98rexbii 3091 . . 3 (∃𝑔𝐵 𝐼 ∈ {(𝐾‘{𝑔})} ↔ ∃𝑔𝐵 𝐼 = (𝐾‘{𝑔}))
106, 9bitri 275 . 2 (𝐼 𝑔𝐵 {(𝐾‘{𝑔})} ↔ ∃𝑔𝐵 𝐼 = (𝐾‘{𝑔}))
115, 10bitrdi 287 1 (𝑅 ∈ Ring → (𝐼𝑃 ↔ ∃𝑔𝐵 𝐼 = (𝐾‘{𝑔})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1536  wcel 2105  wrex 3067  {csn 4630   ciun 4995  cfv 6562  Basecbs 17244  Ringcrg 20250  RSpancrsp 21234  LPIdealclpidl 21347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-iota 6515  df-fun 6564  df-fv 6570  df-lpidl 21349
This theorem is referenced by:  lpi0  21353  lpi1  21354  lpiss  21356  lpigen  21362  ply1lpir  26235  lpirlidllpi  33381  lsmsnidl  33406  mxidlirred  33479  lpirlnr  43105
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