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Theorem islpidl 20016
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 20015 . . 3 (𝑅 ∈ Ring → 𝑃 = 𝑔𝐵 {(𝐾‘{𝑔})})
54eleq2d 2878 . 2 (𝑅 ∈ Ring → (𝐼𝑃𝐼 𝑔𝐵 {(𝐾‘{𝑔})}))
6 eliun 4888 . . 3 (𝐼 𝑔𝐵 {(𝐾‘{𝑔})} ↔ ∃𝑔𝐵 𝐼 ∈ {(𝐾‘{𝑔})})
7 fvex 6662 . . . . 5 (𝐾‘{𝑔}) ∈ V
87elsn2 4567 . . . 4 (𝐼 ∈ {(𝐾‘{𝑔})} ↔ 𝐼 = (𝐾‘{𝑔}))
98rexbii 3213 . . 3 (∃𝑔𝐵 𝐼 ∈ {(𝐾‘{𝑔})} ↔ ∃𝑔𝐵 𝐼 = (𝐾‘{𝑔}))
106, 9bitri 278 . 2 (𝐼 𝑔𝐵 {(𝐾‘{𝑔})} ↔ ∃𝑔𝐵 𝐼 = (𝐾‘{𝑔}))
115, 10syl6bb 290 1 (𝑅 ∈ Ring → (𝐼𝑃 ↔ ∃𝑔𝐵 𝐼 = (𝐾‘{𝑔})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wcel 2112  wrex 3110  {csn 4528   ciun 4884  cfv 6328  Basecbs 16479  Ringcrg 19294  RSpancrsp 19940  LPIdealclpidl 20011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6287  df-fun 6330  df-fv 6336  df-lpidl 20013
This theorem is referenced by:  lpi0  20017  lpi1  20018  lpiss  20020  lpigen  20026  ply1lpir  24783  lsmsnidl  31010  lpirlnr  40054
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