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Theorem islpir 16325
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 5731 . . . 4  |-  ( r  =  R  ->  (LIdeal `  r )  =  (LIdeal `  R ) )
2 fveq2 5731 . . . 4  |-  ( r  =  R  ->  (LPIdeal `  r )  =  (LPIdeal `  R ) )
31, 2eqeq12d 2452 . . 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 2451 . . 3  |-  ( U  =  P  <->  (LIdeal `  R
)  =  (LPIdeal `  R
) )
73, 6syl6bbr 256 . 2  |-  ( r  =  R  ->  (
(LIdeal `  r )  =  (LPIdeal `  r )  <->  U  =  P ) )
8 df-lpir 16320 . 2  |- LPIR  =  {
r  e.  Ring  |  (LIdeal `  r )  =  (LPIdeal `  r ) }
97, 8elrab2 3096 1  |-  ( R  e. LPIR 
<->  ( R  e.  Ring  /\  U  =  P ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   ` cfv 5457   Ringcrg 15665  LIdealclidl 16247  LPIdealclpidl 16317  LPIRclpir 16318
This theorem is referenced by:  islpir2  16327  lpirrng  16328  lpirlnr  27312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-iota 5421  df-fv 5465  df-lpir 16320
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