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Theorem lpival 19167
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 6150 . . . 4 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
2 fveq2 6150 . . . . . 6 (𝑟 = 𝑅 → (RSpan‘𝑟) = (RSpan‘𝑅))
32fveq1d 6152 . . . . 5 (𝑟 = 𝑅 → ((RSpan‘𝑟)‘{𝑔}) = ((RSpan‘𝑅)‘{𝑔}))
43sneqd 4162 . . . 4 (𝑟 = 𝑅 → {((RSpan‘𝑟)‘{𝑔})} = {((RSpan‘𝑅)‘{𝑔})})
51, 4iuneq12d 4514 . . 3 (𝑟 = 𝑅 𝑔 ∈ (Base‘𝑟){((RSpan‘𝑟)‘{𝑔})} = 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})})
6 df-lpidl 19165 . . 3 LPIdeal = (𝑟 ∈ Ring ↦ 𝑔 ∈ (Base‘𝑟){((RSpan‘𝑟)‘{𝑔})})
7 fvex 6160 . . . . . 6 (RSpan‘𝑅) ∈ V
87rnex 7050 . . . . 5 ran (RSpan‘𝑅) ∈ V
9 p0ex 4815 . . . . 5 {∅} ∈ V
108, 9unex 6912 . . . 4 (ran (RSpan‘𝑅) ∪ {∅}) ∈ V
11 iunss 4529 . . . . 5 ( 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})} ⊆ (ran (RSpan‘𝑅) ∪ {∅}) ↔ ∀𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})} ⊆ (ran (RSpan‘𝑅) ∪ {∅}))
12 fvrn0 6175 . . . . . . 7 ((RSpan‘𝑅)‘{𝑔}) ∈ (ran (RSpan‘𝑅) ∪ {∅})
13 snssi 4310 . . . . . . 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 2922 . . . 4 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})} ⊆ (ran (RSpan‘𝑅) ∪ {∅})
1710, 16ssexi 4765 . . 3 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})} ∈ V
185, 6, 17fvmpt 6241 . 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 4502 . . . 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 6151 . . . . . 6 (𝐾‘{𝑔}) = ((RSpan‘𝑅)‘{𝑔})
2524sneqi 4161 . . . . 5 {(𝐾‘{𝑔})} = {((RSpan‘𝑅)‘{𝑔})}
2625a1i 11 . . . 4 (𝑔 ∈ (Base‘𝑅) → {(𝐾‘{𝑔})} = {((RSpan‘𝑅)‘{𝑔})})
2726iuneq2i 4507 . . 3 𝑔 ∈ (Base‘𝑅){(𝐾‘{𝑔})} = 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})}
2822, 27eqtri 2643 . 2 𝑔𝐵 {(𝐾‘{𝑔})} = 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})}
2918, 19, 283eqtr4g 2680 1 (𝑅 ∈ Ring → 𝑃 = 𝑔𝐵 {(𝐾‘{𝑔})})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  cun 3554  wss 3556  c0 3893  {csn 4150   ciun 4487  ran crn 5077  cfv 5849  Basecbs 15784  Ringcrg 18471  RSpancrsp 19093  LPIdealclpidl 19163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-iota 5812  df-fun 5851  df-fv 5857  df-lpidl 19165
This theorem is referenced by:  islpidl  19168
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