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Theorem islpir 19646
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 6446 . . . 4 (𝑟 = 𝑅 → (LIdeal‘𝑟) = (LIdeal‘𝑅))
2 fveq2 6446 . . . 4 (𝑟 = 𝑅 → (LPIdeal‘𝑟) = (LPIdeal‘𝑅))
31, 2eqeq12d 2793 . . 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 2792 . . 3 (𝑈 = 𝑃 ↔ (LIdeal‘𝑅) = (LPIdeal‘𝑅))
73, 6syl6bbr 281 . 2 (𝑟 = 𝑅 → ((LIdeal‘𝑟) = (LPIdeal‘𝑟) ↔ 𝑈 = 𝑃))
8 df-lpir 19641 . 2 LPIR = {𝑟 ∈ Ring ∣ (LIdeal‘𝑟) = (LPIdeal‘𝑟)}
97, 8elrab2 3576 1 (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ 𝑈 = 𝑃))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 386   = wceq 1601  wcel 2107  cfv 6135  Ringcrg 18934  LIdealclidl 19567  LPIdealclpidl 19638  LPIRclpir 19639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-iota 6099  df-fv 6143  df-lpir 19641
This theorem is referenced by:  islpir2  19648  lpirring  19649  lpirlnr  38646
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