Theorem islpidl 20515

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 20514	. . 3 ⊢ (𝑅 ∈ Ring → 𝑃 = ∪ 𝑔 ∈ 𝐵 {(𝐾‘{𝑔})})
5	4	eleq2d 2824	. 2 ⊢ (𝑅 ∈ Ring → (𝐼 ∈ 𝑃 ↔ 𝐼 ∈ ∪ 𝑔 ∈ 𝐵 {(𝐾‘{𝑔})}))
6		eliun 4930	. . 3 ⊢ (𝐼 ∈ ∪ 𝑔 ∈ 𝐵 {(𝐾‘{𝑔})} ↔ ∃𝑔 ∈ 𝐵 𝐼 ∈ {(𝐾‘{𝑔})})
7		fvex 6789	. . . . 5 ⊢ (𝐾‘{𝑔}) ∈ V
8	7	elsn2 4602	. . . 4 ⊢ (𝐼 ∈ {(𝐾‘{𝑔})} ↔ 𝐼 = (𝐾‘{𝑔}))
9	8	rexbii 3180	. . 3 ⊢ (∃𝑔 ∈ 𝐵 𝐼 ∈ {(𝐾‘{𝑔})} ↔ ∃𝑔 ∈ 𝐵 𝐼 = (𝐾‘{𝑔}))
10	6, 9	bitri 274	. 2 ⊢ (𝐼 ∈ ∪ 𝑔 ∈ 𝐵 {(𝐾‘{𝑔})} ↔ ∃𝑔 ∈ 𝐵 𝐼 = (𝐾‘{𝑔}))
11	5, 10	bitrdi 287	1 ⊢ (𝑅 ∈ Ring → (𝐼 ∈ 𝑃 ↔ ∃𝑔 ∈ 𝐵 𝐼 = (𝐾‘{𝑔})))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2106  ∃wrex 3065  {csn 4563  ∪ ciun 4926  ‘cfv 6435  Basecbs 16910  Ringcrg 19781  RSpancrsp 20431  LPIdealclpidl 20510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5225  ax-nul 5232  ax-pow 5290  ax-pr 5354  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3433  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-pw 4537  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-iun 4928  df-br 5077  df-opab 5139  df-mpt 5160  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-iota 6393  df-fun 6437  df-fv 6443  df-lpidl 20512

This theorem is referenced by:  lpi0  20516  lpi1  20517  lpiss  20519  lpigen  20525  ply1lpir  25341  lsmsnidl  31584  lpirlnr  40939