Theorem lpival 20071

Description: Value of the set of principal ideals. (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
lpival	⊢ (𝑅 ∈ Ring → 𝑃 = ∪ 𝑔 ∈ 𝐵 {(𝐾‘{𝑔})})

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

Proof of Theorem lpival

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6651	. . . 4 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
2		fveq2 6651	. . . . . 6 ⊢ (𝑟 = 𝑅 → (RSpan‘𝑟) = (RSpan‘𝑅))
3	2	fveq1d 6653	. . . . 5 ⊢ (𝑟 = 𝑅 → ((RSpan‘𝑟)‘{𝑔}) = ((RSpan‘𝑅)‘{𝑔}))
4	3	sneqd 4527	. . . 4 ⊢ (𝑟 = 𝑅 → {((RSpan‘𝑟)‘{𝑔})} = {((RSpan‘𝑅)‘{𝑔})})
5	1, 4	iuneq12d 4904	. . 3 ⊢ (𝑟 = 𝑅 → ∪ 𝑔 ∈ (Base‘𝑟){((RSpan‘𝑟)‘{𝑔})} = ∪ 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})})
6		df-lpidl 20069	. . 3 ⊢ LPIdeal = (𝑟 ∈ Ring ↦ ∪ 𝑔 ∈ (Base‘𝑟){((RSpan‘𝑟)‘{𝑔})})
7		fvex 6664	. . . . . 6 ⊢ (RSpan‘𝑅) ∈ V
8	7	rnex 7615	. . . . 5 ⊢ ran (RSpan‘𝑅) ∈ V
9		p0ex 5246	. . . . 5 ⊢ {∅} ∈ V
10	8, 9	unex 7460	. . . 4 ⊢ (ran (RSpan‘𝑅) ∪ {∅}) ∈ V
11		iunss 4927	. . . . 5 ⊢ (∪ 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})} ⊆ (ran (RSpan‘𝑅) ∪ {∅}) ↔ ∀𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})} ⊆ (ran (RSpan‘𝑅) ∪ {∅}))
12		fvrn0 6679	. . . . . . 7 ⊢ ((RSpan‘𝑅)‘{𝑔}) ∈ (ran (RSpan‘𝑅) ∪ {∅})
13		snssi 4691	. . . . . . 7 ⊢ (((RSpan‘𝑅)‘{𝑔}) ∈ (ran (RSpan‘𝑅) ∪ {∅}) → {((RSpan‘𝑅)‘{𝑔})} ⊆ (ran (RSpan‘𝑅) ∪ {∅}))
14	12, 13	ax-mp 5	. . . . . 6 ⊢ {((RSpan‘𝑅)‘{𝑔})} ⊆ (ran (RSpan‘𝑅) ∪ {∅})
15	14	a1i 11	. . . . 5 ⊢ (𝑔 ∈ (Base‘𝑅) → {((RSpan‘𝑅)‘{𝑔})} ⊆ (ran (RSpan‘𝑅) ∪ {∅}))
16	11, 15	mprgbir 3083	. . . 4 ⊢ ∪ 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})} ⊆ (ran (RSpan‘𝑅) ∪ {∅})
17	10, 16	ssexi 5185	. . 3 ⊢ ∪ 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})} ∈ V
18	5, 6, 17	fvmpt 6752	. 2 ⊢ (𝑅 ∈ Ring → (LPIdeal‘𝑅) = ∪ 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})})
19		lpival.p	. 2 ⊢ 𝑃 = (LPIdeal‘𝑅)
20		lpival.b	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
21		iuneq1 4892	. . . 4 ⊢ (𝐵 = (Base‘𝑅) → ∪ 𝑔 ∈ 𝐵 {(𝐾‘{𝑔})} = ∪ 𝑔 ∈ (Base‘𝑅){(𝐾‘{𝑔})})
22	20, 21	ax-mp 5	. . 3 ⊢ ∪ 𝑔 ∈ 𝐵 {(𝐾‘{𝑔})} = ∪ 𝑔 ∈ (Base‘𝑅){(𝐾‘{𝑔})}
23		lpival.k	. . . . . . 7 ⊢ 𝐾 = (RSpan‘𝑅)
24	23	fveq1i 6652	. . . . . 6 ⊢ (𝐾‘{𝑔}) = ((RSpan‘𝑅)‘{𝑔})
25	24	sneqi 4526	. . . . 5 ⊢ {(𝐾‘{𝑔})} = {((RSpan‘𝑅)‘{𝑔})}
26	25	a1i 11	. . . 4 ⊢ (𝑔 ∈ (Base‘𝑅) → {(𝐾‘{𝑔})} = {((RSpan‘𝑅)‘{𝑔})})
27	26	iuneq2i 4897	. . 3 ⊢ ∪ 𝑔 ∈ (Base‘𝑅){(𝐾‘{𝑔})} = ∪ 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})}
28	22, 27	eqtri 2782	. 2 ⊢ ∪ 𝑔 ∈ 𝐵 {(𝐾‘{𝑔})} = ∪ 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})}
29	18, 19, 28	3eqtr4g 2819	1 ⊢ (𝑅 ∈ Ring → 𝑃 = ∪ 𝑔 ∈ 𝐵 {(𝐾‘{𝑔})})

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2112   ∪ cun 3852   ⊆ wss 3854  ∅c0 4221  {csn 4515  ∪ ciun 4876  ran crn 5518  ‘cfv 6328  Basecbs 16526  Ringcrg 19350  RSpancrsp 19996  LPIdealclpidl 20067

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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5162  ax-nul 5169  ax-pow 5227  ax-pr 5291  ax-un 7452

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2899  df-ne 2950  df-ral 3073  df-rex 3074  df-rab 3077  df-v 3409  df-sbc 3694  df-dif 3857  df-un 3859  df-in 3861  df-ss 3871  df-nul 4222  df-if 4414  df-pw 4489  df-sn 4516  df-pr 4518  df-op 4522  df-uni 4792  df-iun 4878  df-br 5026  df-opab 5088  df-mpt 5106  df-id 5423  df-xp 5523  df-rel 5524  df-cnv 5525  df-co 5526  df-dm 5527  df-rn 5528  df-iota 6287  df-fun 6330  df-fv 6336  df-lpidl 20069

This theorem is referenced by:  islpidl  20072