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Theorem islpidl 15998
Description: Property of being a principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
Hypotheses
Ref Expression
lpival.p  |-  P  =  (LPIdeal `  R )
lpival.k  |-  K  =  (RSpan `  R )
lpival.b  |-  B  =  ( Base `  R
)
Assertion
Ref Expression
islpidl  |-  ( R  e.  Ring  ->  ( I  e.  P  <->  E. g  e.  B  I  =  ( K `  { g } ) ) )
Distinct variable groups:    R, g    P, g    B, g    g, K   
g, I

Proof of Theorem islpidl
StepHypRef Expression
1 lpival.p . . . 4  |-  P  =  (LPIdeal `  R )
2 lpival.k . . . 4  |-  K  =  (RSpan `  R )
3 lpival.b . . . 4  |-  B  =  ( Base `  R
)
41, 2, 3lpival 15997 . . 3  |-  ( R  e.  Ring  ->  P  = 
U_ g  e.  B  { ( K `  { g } ) } )
54eleq2d 2350 . 2  |-  ( R  e.  Ring  ->  ( I  e.  P  <->  I  e.  U_ g  e.  B  {
( K `  {
g } ) } ) )
6 eliun 3909 . . 3  |-  ( I  e.  U_ g  e.  B  { ( K `
 { g } ) }  <->  E. g  e.  B  I  e.  { ( K `  {
g } ) } )
7 fvex 5539 . . . . 5  |-  ( K `
 { g } )  e.  _V
87elsnc2 3669 . . . 4  |-  ( I  e.  { ( K `
 { g } ) }  <->  I  =  ( K `  { g } ) )
98rexbii 2568 . . 3  |-  ( E. g  e.  B  I  e.  { ( K `
 { g } ) }  <->  E. g  e.  B  I  =  ( K `  { g } ) )
106, 9bitri 240 . 2  |-  ( I  e.  U_ g  e.  B  { ( K `
 { g } ) }  <->  E. g  e.  B  I  =  ( K `  { g } ) )
115, 10syl6bb 252 1  |-  ( R  e.  Ring  ->  ( I  e.  P  <->  E. g  e.  B  I  =  ( K `  { g } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   E.wrex 2544   {csn 3640   U_ciun 3905   ` cfv 5255   Basecbs 13148   Ringcrg 15337  RSpancrsp 15924  LPIdealclpidl 15993
This theorem is referenced by:  lpi0  15999  lpi1  16000  lpiss  16002  lpigen  16008  ply1lpir  19564  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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fv 5263  df-lpidl 15995
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