Theorem islpidl 20013

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 → (𝐼 ∈ 𝑃 ↔ ∃𝑔 ∈ 𝐵 𝐼 = (𝐾‘{𝑔})))

Distinct variable groups:   𝑅,𝑔   𝑃,𝑔   𝐵,𝑔   𝑔,𝐾   𝑔,𝐼

Proof of Theorem islpidl

Step	Hyp	Ref	Expression
1		lpival.p	. . . 4 ⊢ 𝑃 = (LPIdeal‘𝑅)
2		lpival.k	. . . 4 ⊢ 𝐾 = (RSpan‘𝑅)
3		lpival.b	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
4	1, 2, 3	lpival 20012	. . 3 ⊢ (𝑅 ∈ Ring → 𝑃 = ∪ 𝑔 ∈ 𝐵 {(𝐾‘{𝑔})})
5	4	eleq2d 2898	. 2 ⊢ (𝑅 ∈ Ring → (𝐼 ∈ 𝑃 ↔ 𝐼 ∈ ∪ 𝑔 ∈ 𝐵 {(𝐾‘{𝑔})}))
6		eliun 4915	. . 3 ⊢ (𝐼 ∈ ∪ 𝑔 ∈ 𝐵 {(𝐾‘{𝑔})} ↔ ∃𝑔 ∈ 𝐵 𝐼 ∈ {(𝐾‘{𝑔})})
7		fvex 6677	. . . . 5 ⊢ (𝐾‘{𝑔}) ∈ V
8	7	elsn2 4597	. . . 4 ⊢ (𝐼 ∈ {(𝐾‘{𝑔})} ↔ 𝐼 = (𝐾‘{𝑔}))
9	8	rexbii 3247	. . 3 ⊢ (∃𝑔 ∈ 𝐵 𝐼 ∈ {(𝐾‘{𝑔})} ↔ ∃𝑔 ∈ 𝐵 𝐼 = (𝐾‘{𝑔}))
10	6, 9	bitri 277	. 2 ⊢ (𝐼 ∈ ∪ 𝑔 ∈ 𝐵 {(𝐾‘{𝑔})} ↔ ∃𝑔 ∈ 𝐵 𝐼 = (𝐾‘{𝑔}))
11	5, 10	syl6bb 289	1 ⊢ (𝑅 ∈ Ring → (𝐼 ∈ 𝑃 ↔ ∃𝑔 ∈ 𝐵 𝐼 = (𝐾‘{𝑔})))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1533   ∈ wcel 2110  ∃wrex 3139  {csn 4560  ∪ ciun 4911  ‘cfv 6349  Basecbs 16477  Ringcrg 19291  RSpancrsp 19937  LPIdealclpidl 20008

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-iota 6308  df-fun 6351  df-fv 6357  df-lpidl 20010

This theorem is referenced by:  lpi0  20014  lpi1  20015  lpiss  20017  lpigen  20023  ply1lpir  24766  lsmsnidl  30944  lpirlnr  39710