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Theorem islpidl 18089
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 18088 . . 3  |-  ( R  e.  Ring  ->  P  = 
U_ g  e.  B  { ( K `  { g } ) } )
54eleq2d 2524 . 2  |-  ( R  e.  Ring  ->  ( I  e.  P  <->  I  e.  U_ g  e.  B  {
( K `  {
g } ) } ) )
6 eliun 4320 . . 3  |-  ( I  e.  U_ g  e.  B  { ( K `
 { g } ) }  <->  E. g  e.  B  I  e.  { ( K `  {
g } ) } )
7 fvex 5858 . . . . 5  |-  ( K `
 { g } )  e.  _V
87elsnc2 4047 . . . 4  |-  ( I  e.  { ( K `
 { g } ) }  <->  I  =  ( K `  { g } ) )
98rexbii 2956 . . 3  |-  ( E. g  e.  B  I  e.  { ( K `
 { g } ) }  <->  E. g  e.  B  I  =  ( K `  { g } ) )
106, 9bitri 249 . 2  |-  ( I  e.  U_ g  e.  B  { ( K `
 { g } ) }  <->  E. g  e.  B  I  =  ( K `  { g } ) )
115, 10syl6bb 261 1  |-  ( R  e.  Ring  ->  ( I  e.  P  <->  E. g  e.  B  I  =  ( K `  { g } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1398    e. wcel 1823   E.wrex 2805   {csn 4016   U_ciun 4315   ` cfv 5570   Basecbs 14716   Ringcrg 17393  RSpancrsp 18012  LPIdealclpidl 18084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fv 5578  df-lpidl 18086
This theorem is referenced by:  lpi0  18090  lpi1  18091  lpiss  18093  lpigen  18099  ply1lpir  22745  lpirlnr  31307
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