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Theorem islpir 20735
Description: Principal ideal rings are where all ideals are principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
Hypotheses
Ref Expression
lpival.p 𝑃 = (LPIdealβ€˜π‘…)
lpiss.u π‘ˆ = (LIdealβ€˜π‘…)
Assertion
Ref Expression
islpir (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ π‘ˆ = 𝑃))

Proof of Theorem islpir
Dummy variable π‘Ÿ is distinct from all other variables.
StepHypRef Expression
1 fveq2 6843 . . . 4 (π‘Ÿ = 𝑅 β†’ (LIdealβ€˜π‘Ÿ) = (LIdealβ€˜π‘…))
2 fveq2 6843 . . . 4 (π‘Ÿ = 𝑅 β†’ (LPIdealβ€˜π‘Ÿ) = (LPIdealβ€˜π‘…))
31, 2eqeq12d 2749 . . 3 (π‘Ÿ = 𝑅 β†’ ((LIdealβ€˜π‘Ÿ) = (LPIdealβ€˜π‘Ÿ) ↔ (LIdealβ€˜π‘…) = (LPIdealβ€˜π‘…)))
4 lpiss.u . . . 4 π‘ˆ = (LIdealβ€˜π‘…)
5 lpival.p . . . 4 𝑃 = (LPIdealβ€˜π‘…)
64, 5eqeq12i 2751 . . 3 (π‘ˆ = 𝑃 ↔ (LIdealβ€˜π‘…) = (LPIdealβ€˜π‘…))
73, 6bitr4di 289 . 2 (π‘Ÿ = 𝑅 β†’ ((LIdealβ€˜π‘Ÿ) = (LPIdealβ€˜π‘Ÿ) ↔ π‘ˆ = 𝑃))
8 df-lpir 20730 . 2 LPIR = {π‘Ÿ ∈ Ring ∣ (LIdealβ€˜π‘Ÿ) = (LPIdealβ€˜π‘Ÿ)}
97, 8elrab2 3649 1 (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ π‘ˆ = 𝑃))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  β€˜cfv 6497  Ringcrg 19969  LIdealclidl 20647  LPIdealclpidl 20727  LPIRclpir 20728
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-lpir 20730
This theorem is referenced by:  islpir2  20737  lpirring  20738  lpirlnr  41487
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