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Theorem lpirring 21228
Description: Principal ideal rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
Assertion
Ref Expression
lpirring (𝑅 ∈ LPIR β†’ 𝑅 ∈ Ring)

Proof of Theorem lpirring
StepHypRef Expression
1 eqid 2728 . . 3 (LPIdealβ€˜π‘…) = (LPIdealβ€˜π‘…)
2 eqid 2728 . . 3 (LIdealβ€˜π‘…) = (LIdealβ€˜π‘…)
31, 2islpir 21225 . 2 (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ (LIdealβ€˜π‘…) = (LPIdealβ€˜π‘…)))
43simplbi 496 1 (𝑅 ∈ LPIR β†’ 𝑅 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  β€˜cfv 6553  Ringcrg 20180  LIdealclidl 21109  LPIdealclpidl 21217  LPIRclpir 21218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-lpir 21220
This theorem is referenced by:  lpirlnr  42572
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