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Theorem lpival 18088
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 5848 . . . 4  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
2 fveq2 5848 . . . . . 6  |-  ( r  =  R  ->  (RSpan `  r )  =  (RSpan `  R ) )
32fveq1d 5850 . . . . 5  |-  ( r  =  R  ->  (
(RSpan `  r ) `  { g } )  =  ( (RSpan `  R ) `  {
g } ) )
43sneqd 4028 . . . 4  |-  ( r  =  R  ->  { ( (RSpan `  r ) `  { g } ) }  =  { ( (RSpan `  R ) `  { g } ) } )
51, 4iuneq12d 4341 . . 3  |-  ( r  =  R  ->  U_ g  e.  ( Base `  r
) { ( (RSpan `  r ) `  {
g } ) }  =  U_ g  e.  ( Base `  R
) { ( (RSpan `  R ) `  {
g } ) } )
6 df-lpidl 18086 . . 3  |- LPIdeal  =  ( r  e.  Ring  |->  U_ g  e.  ( Base `  r
) { ( (RSpan `  r ) `  {
g } ) } )
7 fvex 5858 . . . . . 6  |-  (RSpan `  R )  e.  _V
87rnex 6707 . . . . 5  |-  ran  (RSpan `  R )  e.  _V
9 p0ex 4624 . . . . 5  |-  { (/) }  e.  _V
108, 9unex 6571 . . . 4  |-  ( ran  (RSpan `  R )  u.  { (/) } )  e. 
_V
11 iunss 4356 . . . . 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 5870 . . . . . . 7  |-  ( (RSpan `  R ) `  {
g } )  e.  ( ran  (RSpan `  R )  u.  { (/)
} )
13 snssi 4160 . . . . . . 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 2818 . . . 4  |-  U_ g  e.  ( Base `  R
) { ( (RSpan `  R ) `  {
g } ) } 
C_  ( ran  (RSpan `  R )  u.  { (/)
} )
1710, 16ssexi 4582 . . 3  |-  U_ g  e.  ( Base `  R
) { ( (RSpan `  R ) `  {
g } ) }  e.  _V
185, 6, 17fvmpt 5931 . 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 4329 . . . 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 5849 . . . . . 6  |-  ( K `
 { g } )  =  ( (RSpan `  R ) `  {
g } )
2524sneqi 4027 . . . . 5  |-  { ( K `  { g } ) }  =  { ( (RSpan `  R ) `  {
g } ) }
2625a1i 11 . . . 4  |-  ( g  e.  ( Base `  R
)  ->  { ( K `  { g } ) }  =  { ( (RSpan `  R ) `  {
g } ) } )
2726iuneq2i 4334 . . 3  |-  U_ g  e.  ( Base `  R
) { ( K `
 { g } ) }  =  U_ g  e.  ( Base `  R ) { ( (RSpan `  R ) `  { g } ) }
2822, 27eqtri 2483 . 2  |-  U_ g  e.  B  { ( K `  { g } ) }  =  U_ g  e.  ( Base `  R ) { ( (RSpan `  R ) `  { g } ) }
2918, 19, 283eqtr4g 2520 1  |-  ( R  e.  Ring  ->  P  = 
U_ g  e.  B  { ( K `  { g } ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823    u. cun 3459    C_ wss 3461   (/)c0 3783   {csn 4016   U_ciun 4315   ran crn 4989   ` 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:  islpidl  18089
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