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Theorem lpival 21334
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 = 𝑔𝐵 {(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.
StepHypRef Expression
1 fveq2 6906 . . . 4 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
2 fveq2 6906 . . . . . 6 (𝑟 = 𝑅 → (RSpan‘𝑟) = (RSpan‘𝑅))
32fveq1d 6908 . . . . 5 (𝑟 = 𝑅 → ((RSpan‘𝑟)‘{𝑔}) = ((RSpan‘𝑅)‘{𝑔}))
43sneqd 4638 . . . 4 (𝑟 = 𝑅 → {((RSpan‘𝑟)‘{𝑔})} = {((RSpan‘𝑅)‘{𝑔})})
51, 4iuneq12d 5021 . . 3 (𝑟 = 𝑅 𝑔 ∈ (Base‘𝑟){((RSpan‘𝑟)‘{𝑔})} = 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})})
6 df-lpidl 21332 . . 3 LPIdeal = (𝑟 ∈ Ring ↦ 𝑔 ∈ (Base‘𝑟){((RSpan‘𝑟)‘{𝑔})})
7 fvex 6919 . . . . . 6 (RSpan‘𝑅) ∈ V
87rnex 7932 . . . . 5 ran (RSpan‘𝑅) ∈ V
9 p0ex 5384 . . . . 5 {∅} ∈ V
108, 9unex 7764 . . . 4 (ran (RSpan‘𝑅) ∪ {∅}) ∈ V
11 iunss 5045 . . . . 5 ( 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})} ⊆ (ran (RSpan‘𝑅) ∪ {∅}) ↔ ∀𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})} ⊆ (ran (RSpan‘𝑅) ∪ {∅}))
12 fvrn0 6936 . . . . . . 7 ((RSpan‘𝑅)‘{𝑔}) ∈ (ran (RSpan‘𝑅) ∪ {∅})
13 snssi 4808 . . . . . . 7 (((RSpan‘𝑅)‘{𝑔}) ∈ (ran (RSpan‘𝑅) ∪ {∅}) → {((RSpan‘𝑅)‘{𝑔})} ⊆ (ran (RSpan‘𝑅) ∪ {∅}))
1412, 13ax-mp 5 . . . . . 6 {((RSpan‘𝑅)‘{𝑔})} ⊆ (ran (RSpan‘𝑅) ∪ {∅})
1514a1i 11 . . . . 5 (𝑔 ∈ (Base‘𝑅) → {((RSpan‘𝑅)‘{𝑔})} ⊆ (ran (RSpan‘𝑅) ∪ {∅}))
1611, 15mprgbir 3068 . . . 4 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})} ⊆ (ran (RSpan‘𝑅) ∪ {∅})
1710, 16ssexi 5322 . . 3 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})} ∈ V
185, 6, 17fvmpt 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 5008 . . . 4 (𝐵 = (Base‘𝑅) → 𝑔𝐵 {(𝐾‘{𝑔})} = 𝑔 ∈ (Base‘𝑅){(𝐾‘{𝑔})})
2220, 21ax-mp 5 . . 3 𝑔𝐵 {(𝐾‘{𝑔})} = 𝑔 ∈ (Base‘𝑅){(𝐾‘{𝑔})}
23 lpival.k . . . . . . 7 𝐾 = (RSpan‘𝑅)
2423fveq1i 6907 . . . . . 6 (𝐾‘{𝑔}) = ((RSpan‘𝑅)‘{𝑔})
2524sneqi 4637 . . . . 5 {(𝐾‘{𝑔})} = {((RSpan‘𝑅)‘{𝑔})}
2625a1i 11 . . . 4 (𝑔 ∈ (Base‘𝑅) → {(𝐾‘{𝑔})} = {((RSpan‘𝑅)‘{𝑔})})
2726iuneq2i 5013 . . 3 𝑔 ∈ (Base‘𝑅){(𝐾‘{𝑔})} = 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})}
2822, 27eqtri 2765 . 2 𝑔𝐵 {(𝐾‘{𝑔})} = 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})}
2918, 19, 283eqtr4g 2802 1 (𝑅 ∈ Ring → 𝑃 = 𝑔𝐵 {(𝐾‘{𝑔})})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  cun 3949  wss 3951  c0 4333  {csn 4626   ciun 4991  ran crn 5686  cfv 6561  Basecbs 17247  Ringcrg 20230  RSpancrsp 21217  LPIdealclpidl 21330
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-lpidl 21332
This theorem is referenced by:  islpidl  21335
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