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Theorem islpidl 19614
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 19613 . . 3 (𝑅 ∈ Ring → 𝑃 = 𝑔𝐵 {(𝐾‘{𝑔})})
54eleq2d 2892 . 2 (𝑅 ∈ Ring → (𝐼𝑃𝐼 𝑔𝐵 {(𝐾‘{𝑔})}))
6 eliun 4746 . . 3 (𝐼 𝑔𝐵 {(𝐾‘{𝑔})} ↔ ∃𝑔𝐵 𝐼 ∈ {(𝐾‘{𝑔})})
7 fvex 6450 . . . . 5 (𝐾‘{𝑔}) ∈ V
87elsn2 4434 . . . 4 (𝐼 ∈ {(𝐾‘{𝑔})} ↔ 𝐼 = (𝐾‘{𝑔}))
98rexbii 3251 . . 3 (∃𝑔𝐵 𝐼 ∈ {(𝐾‘{𝑔})} ↔ ∃𝑔𝐵 𝐼 = (𝐾‘{𝑔}))
106, 9bitri 267 . 2 (𝐼 𝑔𝐵 {(𝐾‘{𝑔})} ↔ ∃𝑔𝐵 𝐼 = (𝐾‘{𝑔}))
115, 10syl6bb 279 1 (𝑅 ∈ Ring → (𝐼𝑃 ↔ ∃𝑔𝐵 𝐼 = (𝐾‘{𝑔})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1656  wcel 2164  wrex 3118  {csn 4399   ciun 4742  cfv 6127  Basecbs 16229  Ringcrg 18908  RSpancrsp 19539  LPIdealclpidl 19609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-iota 6090  df-fun 6129  df-fv 6135  df-lpidl 19611
This theorem is referenced by:  lpi0  19615  lpi1  19616  lpiss  19618  lpigen  19624  ply1lpir  24344  lpirlnr  38529
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