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Theorem islpir 21218
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 6897 . . . 4 (π‘Ÿ = 𝑅 β†’ (LIdealβ€˜π‘Ÿ) = (LIdealβ€˜π‘…))
2 fveq2 6897 . . . 4 (π‘Ÿ = 𝑅 β†’ (LPIdealβ€˜π‘Ÿ) = (LPIdealβ€˜π‘…))
31, 2eqeq12d 2744 . . 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 2746 . . 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 21213 . 2 LPIR = {π‘Ÿ ∈ Ring ∣ (LIdealβ€˜π‘Ÿ) = (LPIdealβ€˜π‘Ÿ)}
97, 8elrab2 3685 1 (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ π‘ˆ = 𝑃))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  β€˜cfv 6548  Ringcrg 20173  LIdealclidl 21102  LPIdealclpidl 21210  LPIRclpir 21211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-iota 6500  df-fv 6556  df-lpir 21213
This theorem is referenced by:  islpir2  21220  lpirring  21221  mxidlirred  33198  lpirlnr  42541
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