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Theorem lpival 16245
Description: Value of the set of principal ideals. (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
lpival  |-  ( R  e.  Ring  ->  P  = 
U_ g  e.  B  { ( K `  { g } ) } )
Distinct variable groups:    R, g    P, g    B, g    g, K

Proof of Theorem lpival
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 fveq2 5670 . . . 4  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
2 fveq2 5670 . . . . . 6  |-  ( r  =  R  ->  (RSpan `  r )  =  (RSpan `  R ) )
32fveq1d 5672 . . . . 5  |-  ( r  =  R  ->  (
(RSpan `  r ) `  { g } )  =  ( (RSpan `  R ) `  {
g } ) )
43sneqd 3772 . . . 4  |-  ( r  =  R  ->  { ( (RSpan `  r ) `  { g } ) }  =  { ( (RSpan `  R ) `  { g } ) } )
51, 4iuneq12d 4061 . . 3  |-  ( r  =  R  ->  U_ g  e.  ( Base `  r
) { ( (RSpan `  r ) `  {
g } ) }  =  U_ g  e.  ( Base `  R
) { ( (RSpan `  R ) `  {
g } ) } )
6 df-lpidl 16243 . . 3  |- LPIdeal  =  ( r  e.  Ring  |->  U_ g  e.  ( Base `  r
) { ( (RSpan `  r ) `  {
g } ) } )
7 fvex 5684 . . . . . 6  |-  (RSpan `  R )  e.  _V
87rnex 5075 . . . . 5  |-  ran  (RSpan `  R )  e.  _V
9 p0ex 4329 . . . . 5  |-  { (/) }  e.  _V
108, 9unex 4649 . . . 4  |-  ( ran  (RSpan `  R )  u.  { (/) } )  e. 
_V
11 iunss 4075 . . . . 5  |-  ( U_ g  e.  ( Base `  R ) { ( (RSpan `  R ) `  { g } ) }  C_  ( ran  (RSpan `  R )  u. 
{ (/) } )  <->  A. g  e.  ( Base `  R
) { ( (RSpan `  R ) `  {
g } ) } 
C_  ( ran  (RSpan `  R )  u.  { (/)
} ) )
12 fvrn0 5695 . . . . . . 7  |-  ( (RSpan `  R ) `  {
g } )  e.  ( ran  (RSpan `  R )  u.  { (/)
} )
13 snssi 3887 . . . . . . 7  |-  ( ( (RSpan `  R ) `  { g } )  e.  ( ran  (RSpan `  R )  u.  { (/)
} )  ->  { ( (RSpan `  R ) `  { g } ) }  C_  ( ran  (RSpan `  R )  u. 
{ (/) } ) )
1412, 13ax-mp 8 . . . . . 6  |-  { ( (RSpan `  R ) `  { g } ) }  C_  ( ran  (RSpan `  R )  u. 
{ (/) } )
1514a1i 11 . . . . 5  |-  ( g  e.  ( Base `  R
)  ->  { (
(RSpan `  R ) `  { g } ) }  C_  ( ran  (RSpan `  R )  u. 
{ (/) } ) )
1611, 15mprgbir 2721 . . . 4  |-  U_ g  e.  ( Base `  R
) { ( (RSpan `  R ) `  {
g } ) } 
C_  ( ran  (RSpan `  R )  u.  { (/)
} )
1710, 16ssexi 4291 . . 3  |-  U_ g  e.  ( Base `  R
) { ( (RSpan `  R ) `  {
g } ) }  e.  _V
185, 6, 17fvmpt 5747 . 2  |-  ( R  e.  Ring  ->  (LPIdeal `  R
)  =  U_ g  e.  ( Base `  R
) { ( (RSpan `  R ) `  {
g } ) } )
19 lpival.p . 2  |-  P  =  (LPIdeal `  R )
20 lpival.b . . . 4  |-  B  =  ( Base `  R
)
21 iuneq1 4050 . . . 4  |-  ( B  =  ( Base `  R
)  ->  U_ g  e.  B  { ( K `
 { g } ) }  =  U_ g  e.  ( Base `  R ) { ( K `  { g } ) } )
2220, 21ax-mp 8 . . 3  |-  U_ g  e.  B  { ( K `  { g } ) }  =  U_ g  e.  ( Base `  R ) { ( K `  { g } ) }
23 lpival.k . . . . . . 7  |-  K  =  (RSpan `  R )
2423fveq1i 5671 . . . . . 6  |-  ( K `
 { g } )  =  ( (RSpan `  R ) `  {
g } )
2524sneqi 3771 . . . . 5  |-  { ( K `  { g } ) }  =  { ( (RSpan `  R ) `  {
g } ) }
2625a1i 11 . . . 4  |-  ( g  e.  ( Base `  R
)  ->  { ( K `  { g } ) }  =  { ( (RSpan `  R ) `  {
g } ) } )
2726iuneq2i 4055 . . 3  |-  U_ g  e.  ( Base `  R
) { ( K `
 { g } ) }  =  U_ g  e.  ( Base `  R ) { ( (RSpan `  R ) `  { g } ) }
2822, 27eqtri 2409 . 2  |-  U_ g  e.  B  { ( K `  { g } ) }  =  U_ g  e.  ( Base `  R ) { ( (RSpan `  R ) `  { g } ) }
2918, 19, 283eqtr4g 2446 1  |-  ( R  e.  Ring  ->  P  = 
U_ g  e.  B  { ( K `  { g } ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717    u. cun 3263    C_ wss 3265   (/)c0 3573   {csn 3759   U_ciun 4037   ran crn 4821   ` cfv 5396   Basecbs 13398   Ringcrg 15589  RSpancrsp 16172  LPIdealclpidl 16241
This theorem is referenced by:  islpidl  16246
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-iota 5360  df-fun 5398  df-fv 5404  df-lpidl 16243
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