Theorem lpival 21352

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 6907	. . . 4 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
2		fveq2 6907	. . . . . 6 ⊢ (𝑟 = 𝑅 → (RSpan‘𝑟) = (RSpan‘𝑅))
3	2	fveq1d 6909	. . . . 5 ⊢ (𝑟 = 𝑅 → ((RSpan‘𝑟)‘{𝑔}) = ((RSpan‘𝑅)‘{𝑔}))
4	3	sneqd 4643	. . . 4 ⊢ (𝑟 = 𝑅 → {((RSpan‘𝑟)‘{𝑔})} = {((RSpan‘𝑅)‘{𝑔})})
5	1, 4	iuneq12d 5026	. . 3 ⊢ (𝑟 = 𝑅 → ∪ 𝑔 ∈ (Base‘𝑟){((RSpan‘𝑟)‘{𝑔})} = ∪ 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})})
6		df-lpidl 21350	. . 3 ⊢ LPIdeal = (𝑟 ∈ Ring ↦ ∪ 𝑔 ∈ (Base‘𝑟){((RSpan‘𝑟)‘{𝑔})})
7		fvex 6920	. . . . . 6 ⊢ (RSpan‘𝑅) ∈ V
8	7	rnex 7933	. . . . 5 ⊢ ran (RSpan‘𝑅) ∈ V
9		p0ex 5390	. . . . 5 ⊢ {∅} ∈ V
10	8, 9	unex 7763	. . . 4 ⊢ (ran (RSpan‘𝑅) ∪ {∅}) ∈ V
11		iunss 5050	. . . . 5 ⊢ (∪ 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})} ⊆ (ran (RSpan‘𝑅) ∪ {∅}) ↔ ∀𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})} ⊆ (ran (RSpan‘𝑅) ∪ {∅}))
12		fvrn0 6937	. . . . . . 7 ⊢ ((RSpan‘𝑅)‘{𝑔}) ∈ (ran (RSpan‘𝑅) ∪ {∅})
13		snssi 4813	. . . . . . 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 3066	. . . 4 ⊢ ∪ 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})} ⊆ (ran (RSpan‘𝑅) ∪ {∅})
17	10, 16	ssexi 5328	. . 3 ⊢ ∪ 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})} ∈ V
18	5, 6, 17	fvmpt 7016	. 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 5013	. . . 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 6908	. . . . . 6 ⊢ (𝐾‘{𝑔}) = ((RSpan‘𝑅)‘{𝑔})
25	24	sneqi 4642	. . . . 5 ⊢ {(𝐾‘{𝑔})} = {((RSpan‘𝑅)‘{𝑔})}
26	25	a1i 11	. . . 4 ⊢ (𝑔 ∈ (Base‘𝑅) → {(𝐾‘{𝑔})} = {((RSpan‘𝑅)‘{𝑔})})
27	26	iuneq2i 5018	. . 3 ⊢ ∪ 𝑔 ∈ (Base‘𝑅){(𝐾‘{𝑔})} = ∪ 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})}
28	22, 27	eqtri 2763	. 2 ⊢ ∪ 𝑔 ∈ 𝐵 {(𝐾‘{𝑔})} = ∪ 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})}
29	18, 19, 28	3eqtr4g 2800	1 ⊢ (𝑅 ∈ Ring → 𝑃 = ∪ 𝑔 ∈ 𝐵 {(𝐾‘{𝑔})})

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106   ∪ cun 3961   ⊆ wss 3963  ∅c0 4339  {csn 4631  ∪ ciun 4996  ran crn 5690  ‘cfv 6563  Basecbs 17245  Ringcrg 20251  RSpancrsp 21235  LPIdealclpidl 21348

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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fv 6571  df-lpidl 21350

This theorem is referenced by:  islpidl  21353