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Theorem islpidl 16317
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 16316 . . 3  |-  ( R  e.  Ring  ->  P  = 
U_ g  e.  B  { ( K `  { g } ) } )
54eleq2d 2503 . 2  |-  ( R  e.  Ring  ->  ( I  e.  P  <->  I  e.  U_ g  e.  B  {
( K `  {
g } ) } ) )
6 eliun 4097 . . 3  |-  ( I  e.  U_ g  e.  B  { ( K `
 { g } ) }  <->  E. g  e.  B  I  e.  { ( K `  {
g } ) } )
7 fvex 5742 . . . . 5  |-  ( K `
 { g } )  e.  _V
87elsnc2 3843 . . . 4  |-  ( I  e.  { ( K `
 { g } ) }  <->  I  =  ( K `  { g } ) )
98rexbii 2730 . . 3  |-  ( E. g  e.  B  I  e.  { ( K `
 { g } ) }  <->  E. g  e.  B  I  =  ( K `  { g } ) )
106, 9bitri 241 . 2  |-  ( I  e.  U_ g  e.  B  { ( K `
 { g } ) }  <->  E. g  e.  B  I  =  ( K `  { g } ) )
115, 10syl6bb 253 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 177    = wceq 1652    e. wcel 1725   E.wrex 2706   {csn 3814   U_ciun 4093   ` cfv 5454   Basecbs 13469   Ringcrg 15660  RSpancrsp 16243  LPIdealclpidl 16312
This theorem is referenced by:  lpi0  16318  lpi1  16319  lpiss  16321  lpigen  16327  ply1lpir  20101  lpirlnr  27298
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fv 5462  df-lpidl 16314
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