Theorem lpival 21241

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 6861	. . . 4 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
2		fveq2 6861	. . . . . 6 ⊢ (𝑟 = 𝑅 → (RSpan‘𝑟) = (RSpan‘𝑅))
3	2	fveq1d 6863	. . . . 5 ⊢ (𝑟 = 𝑅 → ((RSpan‘𝑟)‘{𝑔}) = ((RSpan‘𝑅)‘{𝑔}))
4	3	sneqd 4604	. . . 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 21239	. . 3 ⊢ LPIdeal = (𝑟 ∈ Ring ↦ ∪ 𝑔 ∈ (Base‘𝑟){((RSpan‘𝑟)‘{𝑔})})
7		fvex 6874	. . . . . 6 ⊢ (RSpan‘𝑅) ∈ V
8	7	rnex 7889	. . . . 5 ⊢ ran (RSpan‘𝑅) ∈ V
9		p0ex 5342	. . . . 5 ⊢ {∅} ∈ V
10	8, 9	unex 7723	. . . 4 ⊢ (ran (RSpan‘𝑅) ∪ {∅}) ∈ V
11		iunss 5012	. . . . 5 ⊢ (∪ 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})} ⊆ (ran (RSpan‘𝑅) ∪ {∅}) ↔ ∀𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})} ⊆ (ran (RSpan‘𝑅) ∪ {∅}))
12		fvrn0 6891	. . . . . . 7 ⊢ ((RSpan‘𝑅)‘{𝑔}) ∈ (ran (RSpan‘𝑅) ∪ {∅})
13		snssi 4775	. . . . . . 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 3052	. . . 4 ⊢ ∪ 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})} ⊆ (ran (RSpan‘𝑅) ∪ {∅})
17	10, 16	ssexi 5280	. . 3 ⊢ ∪ 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})} ∈ V
18	5, 6, 17	fvmpt 6971	. 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 6862	. . . . . 6 ⊢ (𝐾‘{𝑔}) = ((RSpan‘𝑅)‘{𝑔})
25	24	sneqi 4603	. . . . 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 2753	. 2 ⊢ ∪ 𝑔 ∈ 𝐵 {(𝐾‘{𝑔})} = ∪ 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})}
29	18, 19, 28	3eqtr4g 2790	1 ⊢ (𝑅 ∈ Ring → 𝑃 = ∪ 𝑔 ∈ 𝐵 {(𝐾‘{𝑔})})

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ∪ cun 3915   ⊆ wss 3917  ∅c0 4299  {csn 4592  ∪ ciun 4958  ran crn 5642  ‘cfv 6514  Basecbs 17186  Ringcrg 20149  RSpancrsp 21124  LPIdealclpidl 21237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-iota 6467  df-fun 6516  df-fv 6522  df-lpidl 21239

This theorem is referenced by:  islpidl  21242