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Theorem lpival 16306
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 5720 . . . 4  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
2 fveq2 5720 . . . . . 6  |-  ( r  =  R  ->  (RSpan `  r )  =  (RSpan `  R ) )
32fveq1d 5722 . . . . 5  |-  ( r  =  R  ->  (
(RSpan `  r ) `  { g } )  =  ( (RSpan `  R ) `  {
g } ) )
43sneqd 3819 . . . 4  |-  ( r  =  R  ->  { ( (RSpan `  r ) `  { g } ) }  =  { ( (RSpan `  R ) `  { g } ) } )
51, 4iuneq12d 4109 . . 3  |-  ( r  =  R  ->  U_ g  e.  ( Base `  r
) { ( (RSpan `  r ) `  {
g } ) }  =  U_ g  e.  ( Base `  R
) { ( (RSpan `  R ) `  {
g } ) } )
6 df-lpidl 16304 . . 3  |- LPIdeal  =  ( r  e.  Ring  |->  U_ g  e.  ( Base `  r
) { ( (RSpan `  r ) `  {
g } ) } )
7 fvex 5734 . . . . . 6  |-  (RSpan `  R )  e.  _V
87rnex 5125 . . . . 5  |-  ran  (RSpan `  R )  e.  _V
9 p0ex 4378 . . . . 5  |-  { (/) }  e.  _V
108, 9unex 4699 . . . 4  |-  ( ran  (RSpan `  R )  u.  { (/) } )  e. 
_V
11 iunss 4124 . . . . 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 5745 . . . . . . 7  |-  ( (RSpan `  R ) `  {
g } )  e.  ( ran  (RSpan `  R )  u.  { (/)
} )
13 snssi 3934 . . . . . . 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 2768 . . . 4  |-  U_ g  e.  ( Base `  R
) { ( (RSpan `  R ) `  {
g } ) } 
C_  ( ran  (RSpan `  R )  u.  { (/)
} )
1710, 16ssexi 4340 . . 3  |-  U_ g  e.  ( Base `  R
) { ( (RSpan `  R ) `  {
g } ) }  e.  _V
185, 6, 17fvmpt 5798 . 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 4098 . . . 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 5721 . . . . . 6  |-  ( K `
 { g } )  =  ( (RSpan `  R ) `  {
g } )
2524sneqi 3818 . . . . 5  |-  { ( K `  { g } ) }  =  { ( (RSpan `  R ) `  {
g } ) }
2625a1i 11 . . . 4  |-  ( g  e.  ( Base `  R
)  ->  { ( K `  { g } ) }  =  { ( (RSpan `  R ) `  {
g } ) } )
2726iuneq2i 4103 . . 3  |-  U_ g  e.  ( Base `  R
) { ( K `
 { g } ) }  =  U_ g  e.  ( Base `  R ) { ( (RSpan `  R ) `  { g } ) }
2822, 27eqtri 2455 . 2  |-  U_ g  e.  B  { ( K `  { g } ) }  =  U_ g  e.  ( Base `  R ) { ( (RSpan `  R ) `  { g } ) }
2918, 19, 283eqtr4g 2492 1  |-  ( R  e.  Ring  ->  P  = 
U_ g  e.  B  { ( K `  { g } ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725    u. cun 3310    C_ wss 3312   (/)c0 3620   {csn 3806   U_ciun 4085   ran crn 4871   ` cfv 5446   Basecbs 13459   Ringcrg 15650  RSpancrsp 16233  LPIdealclpidl 16302
This theorem is referenced by:  islpidl  16307
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fv 5454  df-lpidl 16304
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