Theorem lpival 21544

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 6885	. . . 4 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
2		fveq2 6885	. . . . . 6 ⊢ (𝑟 = 𝑅 → (RSpan‘𝑟) = (RSpan‘𝑅))
3	2	fveq1d 6887	. . . . 5 ⊢ (𝑟 = 𝑅 → ((RSpan‘𝑟)‘{𝑔}) = ((RSpan‘𝑅)‘{𝑔}))
4	3	sneqd 4603	. . . 4 ⊢ (𝑟 = 𝑅 → {((RSpan‘𝑟)‘{𝑔})} = {((RSpan‘𝑅)‘{𝑔})})
5	1, 4	iuneq12d 4988	. . 3 ⊢ (𝑟 = 𝑅 → ∪ 𝑔 ∈ (Base‘𝑟){((RSpan‘𝑟)‘{𝑔})} = ∪ 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})})
6		df-lpidl 21542	. . 3 ⊢ LPIdeal = (𝑟 ∈ Ring ↦ ∪ 𝑔 ∈ (Base‘𝑟){((RSpan‘𝑟)‘{𝑔})})
7		fvex 6898	. . . . . 6 ⊢ (RSpan‘𝑅) ∈ V
8	7	rnex 7913	. . . . 5 ⊢ ran (RSpan‘𝑅) ∈ V
9		snex 5412	. . . . 5 ⊢ {∅} ∈ V
10	8, 9	unex 7752	. . . 4 ⊢ (ran (RSpan‘𝑅) ∪ {∅}) ∈ V
11		iunss 5011	. . . . 5 ⊢ (∪ 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})} ⊆ (ran (RSpan‘𝑅) ∪ {∅}) ↔ ∀𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})} ⊆ (ran (RSpan‘𝑅) ∪ {∅}))
12		fvrn0 6913	. . . . . . 7 ⊢ ((RSpan‘𝑅)‘{𝑔}) ∈ (ran (RSpan‘𝑅) ∪ {∅})
13		snssi 4753	. . . . . . 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 3088	. . . 4 ⊢ ∪ 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})} ⊆ (ran (RSpan‘𝑅) ∪ {∅})
17	10, 16	ssexi 5295	. . 3 ⊢ ∪ 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})} ∈ V
18	5, 6, 17	fvmpt 6993	. 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 4975	. . . 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 6886	. . . . . 6 ⊢ (𝐾‘{𝑔}) = ((RSpan‘𝑅)‘{𝑔})
25	24	sneqi 4602	. . . . 5 ⊢ {(𝐾‘{𝑔})} = {((RSpan‘𝑅)‘{𝑔})}
26	25	a1i 11	. . . 4 ⊢ (𝑔 ∈ (Base‘𝑅) → {(𝐾‘{𝑔})} = {((RSpan‘𝑅)‘{𝑔})})
27	26	iuneq2i 4980	. . 3 ⊢ ∪ 𝑔 ∈ (Base‘𝑅){(𝐾‘{𝑔})} = ∪ 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})}
28	22, 27	eqtri 2788	. 2 ⊢ ∪ 𝑔 ∈ 𝐵 {(𝐾‘{𝑔})} = ∪ 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})}
29	18, 19, 28	3eqtr4g 2825	1 ⊢ (𝑅 ∈ Ring → 𝑃 = ∪ 𝑔 ∈ 𝐵 {(𝐾‘{𝑔})})

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146   ∪ cun 3904   ⊆ wss 3906  ∅c0 4286  {csn 4591  ∪ ciun 4958  ran crn 5664  ‘cfv 6540  Basecbs 17293  Ringcrg 20361  RSpancrsp 21383  LPIdealclpidl 21540

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6496  df-fun 6542  df-fv 6548  df-lpidl 21542

This theorem is used by:  islpidl  21545