MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lpirrng Structured version   Unicode version

Theorem lpirrng 17467
Description: Principal ideal rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
Assertion
Ref Expression
lpirrng  |-  ( R  e. LPIR  ->  R  e.  Ring )

Proof of Theorem lpirrng
StepHypRef Expression
1 eqid 2454 . . 3  |-  (LPIdeal `  R
)  =  (LPIdeal `  R
)
2 eqid 2454 . . 3  |-  (LIdeal `  R )  =  (LIdeal `  R )
31, 2islpir 17464 . 2  |-  ( R  e. LPIR 
<->  ( R  e.  Ring  /\  (LIdeal `  R )  =  (LPIdeal `  R )
) )
43simplbi 460 1  |-  ( R  e. LPIR  ->  R  e.  Ring )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   ` cfv 5529   Ringcrg 16778  LIdealclidl 17384  LPIdealclpidl 17456  LPIRclpir 17457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-iota 5492  df-fv 5537  df-lpir 17459
This theorem is referenced by:  lpirlnr  29644
  Copyright terms: Public domain W3C validator