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Theorem islpir 18033
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 5791 . . . 4  |-  ( r  =  R  ->  (LIdeal `  r )  =  (LIdeal `  R ) )
2 fveq2 5791 . . . 4  |-  ( r  =  R  ->  (LPIdeal `  r )  =  (LPIdeal `  R ) )
31, 2eqeq12d 2418 . . 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 2416 . . 3  |-  ( U  =  P  <->  (LIdeal `  R
)  =  (LPIdeal `  R
) )
73, 6syl6bbr 263 . 2  |-  ( r  =  R  ->  (
(LIdeal `  r )  =  (LPIdeal `  r )  <->  U  =  P ) )
8 df-lpir 18028 . 2  |- LPIR  =  {
r  e.  Ring  |  (LIdeal `  r )  =  (LPIdeal `  r ) }
97, 8elrab2 3201 1  |-  ( R  e. LPIR 
<->  ( R  e.  Ring  /\  U  =  P ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1836   ` cfv 5513   Ringcrg 17334  LIdealclidl 17952  LPIdealclpidl 18025  LPIRclpir 18026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-rex 2752  df-rab 2755  df-v 3053  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-br 4385  df-iota 5477  df-fv 5521  df-lpir 18028
This theorem is referenced by:  islpir2  18035  lpirring  18036  lpirlnr  31274
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