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Theorem lpirrng 16315
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 2435 . . 3  |-  (LPIdeal `  R
)  =  (LPIdeal `  R
)
2 eqid 2435 . . 3  |-  (LIdeal `  R )  =  (LIdeal `  R )
31, 2islpir 16312 . 2  |-  ( R  e. LPIR 
<->  ( R  e.  Ring  /\  (LIdeal `  R )  =  (LPIdeal `  R )
) )
43simplbi 447 1  |-  ( R  e. LPIR  ->  R  e.  Ring )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   ` cfv 5446   Ringcrg 15652  LIdealclidl 16234  LPIdealclpidl 16304  LPIRclpir 16305
This theorem is referenced by:  lpirlnr  27289
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-lpir 16307
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