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Theorem lpival 17445
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 5794 . . . 4  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
2 fveq2 5794 . . . . . 6  |-  ( r  =  R  ->  (RSpan `  r )  =  (RSpan `  R ) )
32fveq1d 5796 . . . . 5  |-  ( r  =  R  ->  (
(RSpan `  r ) `  { g } )  =  ( (RSpan `  R ) `  {
g } ) )
43sneqd 3992 . . . 4  |-  ( r  =  R  ->  { ( (RSpan `  r ) `  { g } ) }  =  { ( (RSpan `  R ) `  { g } ) } )
51, 4iuneq12d 4299 . . 3  |-  ( r  =  R  ->  U_ g  e.  ( Base `  r
) { ( (RSpan `  r ) `  {
g } ) }  =  U_ g  e.  ( Base `  R
) { ( (RSpan `  R ) `  {
g } ) } )
6 df-lpidl 17443 . . 3  |- LPIdeal  =  ( r  e.  Ring  |->  U_ g  e.  ( Base `  r
) { ( (RSpan `  r ) `  {
g } ) } )
7 fvex 5804 . . . . . 6  |-  (RSpan `  R )  e.  _V
87rnex 6617 . . . . 5  |-  ran  (RSpan `  R )  e.  _V
9 p0ex 4582 . . . . 5  |-  { (/) }  e.  _V
108, 9unex 6483 . . . 4  |-  ( ran  (RSpan `  R )  u.  { (/) } )  e. 
_V
11 iunss 4314 . . . . 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 5816 . . . . . . 7  |-  ( (RSpan `  R ) `  {
g } )  e.  ( ran  (RSpan `  R )  u.  { (/)
} )
13 snssi 4120 . . . . . . 7  |-  ( ( (RSpan `  R ) `  { g } )  e.  ( ran  (RSpan `  R )  u.  { (/)
} )  ->  { ( (RSpan `  R ) `  { g } ) }  C_  ( ran  (RSpan `  R )  u. 
{ (/) } ) )
1412, 13ax-mp 5 . . . . . 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 2898 . . . 4  |-  U_ g  e.  ( Base `  R
) { ( (RSpan `  R ) `  {
g } ) } 
C_  ( ran  (RSpan `  R )  u.  { (/)
} )
1710, 16ssexi 4540 . . 3  |-  U_ g  e.  ( Base `  R
) { ( (RSpan `  R ) `  {
g } ) }  e.  _V
185, 6, 17fvmpt 5878 . 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 4287 . . . 4  |-  ( B  =  ( Base `  R
)  ->  U_ g  e.  B  { ( K `
 { g } ) }  =  U_ g  e.  ( Base `  R ) { ( K `  { g } ) } )
2220, 21ax-mp 5 . . 3  |-  U_ g  e.  B  { ( K `  { g } ) }  =  U_ g  e.  ( Base `  R ) { ( K `  { g } ) }
23 lpival.k . . . . . . 7  |-  K  =  (RSpan `  R )
2423fveq1i 5795 . . . . . 6  |-  ( K `
 { g } )  =  ( (RSpan `  R ) `  {
g } )
2524sneqi 3991 . . . . 5  |-  { ( K `  { g } ) }  =  { ( (RSpan `  R ) `  {
g } ) }
2625a1i 11 . . . 4  |-  ( g  e.  ( Base `  R
)  ->  { ( K `  { g } ) }  =  { ( (RSpan `  R ) `  {
g } ) } )
2726iuneq2i 4292 . . 3  |-  U_ g  e.  ( Base `  R
) { ( K `
 { g } ) }  =  U_ g  e.  ( Base `  R ) { ( (RSpan `  R ) `  { g } ) }
2822, 27eqtri 2481 . 2  |-  U_ g  e.  B  { ( K `  { g } ) }  =  U_ g  e.  ( Base `  R ) { ( (RSpan `  R ) `  { g } ) }
2918, 19, 283eqtr4g 2518 1  |-  ( R  e.  Ring  ->  P  = 
U_ g  e.  B  { ( K `  { g } ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758    u. cun 3429    C_ wss 3431   (/)c0 3740   {csn 3980   U_ciun 4274   ran crn 4944   ` cfv 5521   Basecbs 14287   Ringcrg 16763  RSpancrsp 17370  LPIdealclpidl 17441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-iota 5484  df-fun 5523  df-fv 5529  df-lpidl 17443
This theorem is referenced by:  islpidl  17446
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