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

Theorem islpir 16283
Description: Principal ideal rings are where all ideals are principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
Hypotheses
Ref Expression
lpival.p  |-  P  =  (LPIdeal `  R )
lpiss.u  |-  U  =  (LIdeal `  R )
Assertion
Ref Expression
islpir  |-  ( R  e. LPIR 
<->  ( R  e.  Ring  /\  U  =  P ) )

Proof of Theorem islpir
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 fveq2 5695 . . . 4  |-  ( r  =  R  ->  (LIdeal `  r )  =  (LIdeal `  R ) )
2 fveq2 5695 . . . 4  |-  ( r  =  R  ->  (LPIdeal `  r )  =  (LPIdeal `  R ) )
31, 2eqeq12d 2426 . . 3  |-  ( r  =  R  ->  (
(LIdeal `  r )  =  (LPIdeal `  r )  <->  (LIdeal `  R )  =  (LPIdeal `  R ) ) )
4 lpiss.u . . . 4  |-  U  =  (LIdeal `  R )
5 lpival.p . . . 4  |-  P  =  (LPIdeal `  R )
64, 5eqeq12i 2425 . . 3  |-  ( U  =  P  <->  (LIdeal `  R
)  =  (LPIdeal `  R
) )
73, 6syl6bbr 255 . 2  |-  ( r  =  R  ->  (
(LIdeal `  r )  =  (LPIdeal `  r )  <->  U  =  P ) )
8 df-lpir 16278 . 2  |- LPIR  =  {
r  e.  Ring  |  (LIdeal `  r )  =  (LPIdeal `  r ) }
97, 8elrab2 3062 1  |-  ( R  e. LPIR 
<->  ( R  e.  Ring  /\  U  =  P ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   ` cfv 5421   Ringcrg 15623  LIdealclidl 16205  LPIdealclpidl 16275  LPIRclpir 16276
This theorem is referenced by:  islpir2  16285  lpirrng  16286  lpirlnr  27197
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-iota 5385  df-fv 5429  df-lpir 16278
  Copyright terms: Public domain W3C validator