Theorem lpival 19066

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 6103	. . . 4 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
2		fveq2 6103	. . . . . 6 ⊢ (𝑟 = 𝑅 → (RSpan‘𝑟) = (RSpan‘𝑅))
3	2	fveq1d 6105	. . . . 5 ⊢ (𝑟 = 𝑅 → ((RSpan‘𝑟)‘{𝑔}) = ((RSpan‘𝑅)‘{𝑔}))
4	3	sneqd 4137	. . . 4 ⊢ (𝑟 = 𝑅 → {((RSpan‘𝑟)‘{𝑔})} = {((RSpan‘𝑅)‘{𝑔})})
5	1, 4	iuneq12d 4482	. . 3 ⊢ (𝑟 = 𝑅 → ∪ 𝑔 ∈ (Base‘𝑟){((RSpan‘𝑟)‘{𝑔})} = ∪ 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})})
6		df-lpidl 19064	. . 3 ⊢ LPIdeal = (𝑟 ∈ Ring ↦ ∪ 𝑔 ∈ (Base‘𝑟){((RSpan‘𝑟)‘{𝑔})})
7		fvex 6113	. . . . . 6 ⊢ (RSpan‘𝑅) ∈ V
8	7	rnex 6992	. . . . 5 ⊢ ran (RSpan‘𝑅) ∈ V
9		p0ex 4779	. . . . 5 ⊢ {∅} ∈ V
10	8, 9	unex 6854	. . . 4 ⊢ (ran (RSpan‘𝑅) ∪ {∅}) ∈ V
11		iunss 4497	. . . . 5 ⊢ (∪ 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})} ⊆ (ran (RSpan‘𝑅) ∪ {∅}) ↔ ∀𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})} ⊆ (ran (RSpan‘𝑅) ∪ {∅}))
12		fvrn0 6126	. . . . . . 7 ⊢ ((RSpan‘𝑅)‘{𝑔}) ∈ (ran (RSpan‘𝑅) ∪ {∅})
13		snssi 4280	. . . . . . 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 2911	. . . 4 ⊢ ∪ 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})} ⊆ (ran (RSpan‘𝑅) ∪ {∅})
17	10, 16	ssexi 4731	. . 3 ⊢ ∪ 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})} ∈ V
18	5, 6, 17	fvmpt 6191	. 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 4470	. . . 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 6104	. . . . . 6 ⊢ (𝐾‘{𝑔}) = ((RSpan‘𝑅)‘{𝑔})
25	24	sneqi 4136	. . . . 5 ⊢ {(𝐾‘{𝑔})} = {((RSpan‘𝑅)‘{𝑔})}
26	25	a1i 11	. . . 4 ⊢ (𝑔 ∈ (Base‘𝑅) → {(𝐾‘{𝑔})} = {((RSpan‘𝑅)‘{𝑔})})
27	26	iuneq2i 4475	. . 3 ⊢ ∪ 𝑔 ∈ (Base‘𝑅){(𝐾‘{𝑔})} = ∪ 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})}
28	22, 27	eqtri 2632	. 2 ⊢ ∪ 𝑔 ∈ 𝐵 {(𝐾‘{𝑔})} = ∪ 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})}
29	18, 19, 28	3eqtr4g 2669	1 ⊢ (𝑅 ∈ Ring → 𝑃 = ∪ 𝑔 ∈ 𝐵 {(𝐾‘{𝑔})})

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   ∪ cun 3538   ⊆ wss 3540  ∅c0 3874  {csn 4125  ∪ ciun 4455  ran crn 5039  ‘cfv 5804  Basecbs 15695  Ringcrg 18370  RSpancrsp 18992  LPIdealclpidl 19062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-lpidl 19064

This theorem is referenced by:  islpidl  19067