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Theorem islpir 16211
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 5632 . . . 4  |-  ( r  =  R  ->  (LIdeal `  r )  =  (LIdeal `  R ) )
2 fveq2 5632 . . . 4  |-  ( r  =  R  ->  (LPIdeal `  r )  =  (LPIdeal `  R ) )
31, 2eqeq12d 2380 . . 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 2379 . . 3  |-  ( U  =  P  <->  (LIdeal `  R
)  =  (LPIdeal `  R
) )
73, 6syl6bbr 254 . 2  |-  ( r  =  R  ->  (
(LIdeal `  r )  =  (LPIdeal `  r )  <->  U  =  P ) )
8 df-lpir 16206 . 2  |- LPIR  =  {
r  e.  Ring  |  (LIdeal `  r )  =  (LPIdeal `  r ) }
97, 8elrab2 3011 1  |-  ( R  e. LPIR 
<->  ( R  e.  Ring  /\  U  =  P ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1647    e. wcel 1715   ` cfv 5358   Ringcrg 15547  LIdealclidl 16133  LPIdealclpidl 16203  LPIRclpir 16204
This theorem is referenced by:  islpir2  16213  lpirrng  16214  lpirlnr  26912
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-iota 5322  df-fv 5366  df-lpir 16206
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