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Theorem islpir 16001
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 5525 . . . 4  |-  ( r  =  R  ->  (LIdeal `  r )  =  (LIdeal `  R ) )
2 fveq2 5525 . . . 4  |-  ( r  =  R  ->  (LPIdeal `  r )  =  (LPIdeal `  R ) )
31, 2eqeq12d 2297 . . 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 2296 . . 3  |-  ( U  =  P  <->  (LIdeal `  R
)  =  (LPIdeal `  R
) )
73, 6syl6bbr 254 . 2  |-  ( r  =  R  ->  (
(LIdeal `  r )  =  (LPIdeal `  r )  <->  U  =  P ) )
8 df-lpir 15996 . 2  |- LPIR  =  {
r  e.  Ring  |  (LIdeal `  r )  =  (LPIdeal `  r ) }
97, 8elrab2 2925 1  |-  ( R  e. LPIR 
<->  ( R  e.  Ring  /\  U  =  P ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   ` cfv 5255   Ringcrg 15337  LIdealclidl 15923  LPIdealclpidl 15993  LPIRclpir 15994
This theorem is referenced by:  islpir2  16003  lpirrng  16004  lpirlnr  27321
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-lpir 15996
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