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Theorem lpival 19939
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 6666 . . . 4 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
2 fveq2 6666 . . . . . 6 (𝑟 = 𝑅 → (RSpan‘𝑟) = (RSpan‘𝑅))
32fveq1d 6668 . . . . 5 (𝑟 = 𝑅 → ((RSpan‘𝑟)‘{𝑔}) = ((RSpan‘𝑅)‘{𝑔}))
43sneqd 4575 . . . 4 (𝑟 = 𝑅 → {((RSpan‘𝑟)‘{𝑔})} = {((RSpan‘𝑅)‘{𝑔})})
51, 4iuneq12d 4943 . . 3 (𝑟 = 𝑅 𝑔 ∈ (Base‘𝑟){((RSpan‘𝑟)‘{𝑔})} = 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})})
6 df-lpidl 19937 . . 3 LPIdeal = (𝑟 ∈ Ring ↦ 𝑔 ∈ (Base‘𝑟){((RSpan‘𝑟)‘{𝑔})})
7 fvex 6679 . . . . . 6 (RSpan‘𝑅) ∈ V
87rnex 7608 . . . . 5 ran (RSpan‘𝑅) ∈ V
9 p0ex 5280 . . . . 5 {∅} ∈ V
108, 9unex 7461 . . . 4 (ran (RSpan‘𝑅) ∪ {∅}) ∈ V
11 iunss 4965 . . . . 5 ( 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})} ⊆ (ran (RSpan‘𝑅) ∪ {∅}) ↔ ∀𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})} ⊆ (ran (RSpan‘𝑅) ∪ {∅}))
12 fvrn0 6694 . . . . . . 7 ((RSpan‘𝑅)‘{𝑔}) ∈ (ran (RSpan‘𝑅) ∪ {∅})
13 snssi 4739 . . . . . . 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 3157 . . . 4 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})} ⊆ (ran (RSpan‘𝑅) ∪ {∅})
1710, 16ssexi 5222 . . 3 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})} ∈ V
185, 6, 17fvmpt 6764 . 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 4931 . . . 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 6667 . . . . . 6 (𝐾‘{𝑔}) = ((RSpan‘𝑅)‘{𝑔})
2524sneqi 4574 . . . . 5 {(𝐾‘{𝑔})} = {((RSpan‘𝑅)‘{𝑔})}
2625a1i 11 . . . 4 (𝑔 ∈ (Base‘𝑅) → {(𝐾‘{𝑔})} = {((RSpan‘𝑅)‘{𝑔})})
2726iuneq2i 4936 . . 3 𝑔 ∈ (Base‘𝑅){(𝐾‘{𝑔})} = 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})}
2822, 27eqtri 2848 . 2 𝑔𝐵 {(𝐾‘{𝑔})} = 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})}
2918, 19, 283eqtr4g 2885 1 (𝑅 ∈ Ring → 𝑃 = 𝑔𝐵 {(𝐾‘{𝑔})})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  wcel 2107  cun 3937  wss 3939  c0 4294  {csn 4563   ciun 4916  ran crn 5554  cfv 6351  Basecbs 16475  Ringcrg 19219  RSpancrsp 19865  LPIdealclpidl 19935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-iota 6311  df-fun 6353  df-fv 6359  df-lpidl 19937
This theorem is referenced by:  islpidl  19940
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