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Theorem islpir 19070
 Description: Principal ideal rings are where all ideals are principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
Hypotheses
Ref Expression
lpival.p 𝑃 = (LPIdeal‘𝑅)
lpiss.u 𝑈 = (LIdeal‘𝑅)
Assertion
Ref Expression
islpir (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ 𝑈 = 𝑃))

Proof of Theorem islpir
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6103 . . . 4 (𝑟 = 𝑅 → (LIdeal‘𝑟) = (LIdeal‘𝑅))
2 fveq2 6103 . . . 4 (𝑟 = 𝑅 → (LPIdeal‘𝑟) = (LPIdeal‘𝑅))
31, 2eqeq12d 2625 . . 3 (𝑟 = 𝑅 → ((LIdeal‘𝑟) = (LPIdeal‘𝑟) ↔ (LIdeal‘𝑅) = (LPIdeal‘𝑅)))
4 lpiss.u . . . 4 𝑈 = (LIdeal‘𝑅)
5 lpival.p . . . 4 𝑃 = (LPIdeal‘𝑅)
64, 5eqeq12i 2624 . . 3 (𝑈 = 𝑃 ↔ (LIdeal‘𝑅) = (LPIdeal‘𝑅))
73, 6syl6bbr 277 . 2 (𝑟 = 𝑅 → ((LIdeal‘𝑟) = (LPIdeal‘𝑟) ↔ 𝑈 = 𝑃))
8 df-lpir 19065 . 2 LPIR = {𝑟 ∈ Ring ∣ (LIdeal‘𝑟) = (LPIdeal‘𝑟)}
97, 8elrab2 3333 1 (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ 𝑈 = 𝑃))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  Ringcrg 18370  LIdealclidl 18991  LPIdealclpidl 19062  LPIRclpir 19063 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-lpir 19065 This theorem is referenced by:  islpir2  19072  lpirring  19073  lpirlnr  36706
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