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Theorem lpival 19535
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 6379 . . . 4 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
2 fveq2 6379 . . . . . 6 (𝑟 = 𝑅 → (RSpan‘𝑟) = (RSpan‘𝑅))
32fveq1d 6381 . . . . 5 (𝑟 = 𝑅 → ((RSpan‘𝑟)‘{𝑔}) = ((RSpan‘𝑅)‘{𝑔}))
43sneqd 4348 . . . 4 (𝑟 = 𝑅 → {((RSpan‘𝑟)‘{𝑔})} = {((RSpan‘𝑅)‘{𝑔})})
51, 4iuneq12d 4704 . . 3 (𝑟 = 𝑅 𝑔 ∈ (Base‘𝑟){((RSpan‘𝑟)‘{𝑔})} = 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})})
6 df-lpidl 19533 . . 3 LPIdeal = (𝑟 ∈ Ring ↦ 𝑔 ∈ (Base‘𝑟){((RSpan‘𝑟)‘{𝑔})})
7 fvex 6392 . . . . . 6 (RSpan‘𝑅) ∈ V
87rnex 7302 . . . . 5 ran (RSpan‘𝑅) ∈ V
9 p0ex 5021 . . . . 5 {∅} ∈ V
108, 9unex 7158 . . . 4 (ran (RSpan‘𝑅) ∪ {∅}) ∈ V
11 iunss 4719 . . . . 5 ( 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})} ⊆ (ran (RSpan‘𝑅) ∪ {∅}) ↔ ∀𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})} ⊆ (ran (RSpan‘𝑅) ∪ {∅}))
12 fvrn0 6407 . . . . . . 7 ((RSpan‘𝑅)‘{𝑔}) ∈ (ran (RSpan‘𝑅) ∪ {∅})
13 snssi 4495 . . . . . . 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 3074 . . . 4 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})} ⊆ (ran (RSpan‘𝑅) ∪ {∅})
1710, 16ssexi 4966 . . 3 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})} ∈ V
185, 6, 17fvmpt 6475 . 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 4692 . . . 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 6380 . . . . . 6 (𝐾‘{𝑔}) = ((RSpan‘𝑅)‘{𝑔})
2524sneqi 4347 . . . . 5 {(𝐾‘{𝑔})} = {((RSpan‘𝑅)‘{𝑔})}
2625a1i 11 . . . 4 (𝑔 ∈ (Base‘𝑅) → {(𝐾‘{𝑔})} = {((RSpan‘𝑅)‘{𝑔})})
2726iuneq2i 4697 . . 3 𝑔 ∈ (Base‘𝑅){(𝐾‘{𝑔})} = 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})}
2822, 27eqtri 2787 . 2 𝑔𝐵 {(𝐾‘{𝑔})} = 𝑔 ∈ (Base‘𝑅){((RSpan‘𝑅)‘{𝑔})}
2918, 19, 283eqtr4g 2824 1 (𝑅 ∈ Ring → 𝑃 = 𝑔𝐵 {(𝐾‘{𝑔})})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1652  wcel 2155  cun 3732  wss 3734  c0 4081  {csn 4336   ciun 4678  ran crn 5280  cfv 6070  Basecbs 16146  Ringcrg 18830  RSpancrsp 19461  LPIdealclpidl 19531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-iota 6033  df-fun 6072  df-fv 6078  df-lpidl 19533
This theorem is referenced by:  islpidl  19536
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