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Theorem islpir 19171
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 6150 . . . 4 (𝑟 = 𝑅 → (LIdeal‘𝑟) = (LIdeal‘𝑅))
2 fveq2 6150 . . . 4 (𝑟 = 𝑅 → (LPIdeal‘𝑟) = (LPIdeal‘𝑅))
31, 2eqeq12d 2636 . . 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 2635 . . 3 (𝑈 = 𝑃 ↔ (LIdeal‘𝑅) = (LPIdeal‘𝑅))
73, 6syl6bbr 278 . 2 (𝑟 = 𝑅 → ((LIdeal‘𝑟) = (LPIdeal‘𝑟) ↔ 𝑈 = 𝑃))
8 df-lpir 19166 . 2 LPIR = {𝑟 ∈ Ring ∣ (LIdeal‘𝑟) = (LPIdeal‘𝑟)}
97, 8elrab2 3349 1 (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ 𝑈 = 𝑃))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1987  cfv 5849  Ringcrg 18471  LIdealclidl 19092  LPIdealclpidl 19163  LPIRclpir 19164
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-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
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-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-br 4616  df-iota 5812  df-fv 5857  df-lpir 19166
This theorem is referenced by:  islpir2  19173  lpirring  19174  lpirlnr  37189
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