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Theorem lpival 16013
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 5541 . . . 4  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
2 fveq2 5541 . . . . . 6  |-  ( r  =  R  ->  (RSpan `  r )  =  (RSpan `  R ) )
32fveq1d 5543 . . . . 5  |-  ( r  =  R  ->  (
(RSpan `  r ) `  { g } )  =  ( (RSpan `  R ) `  {
g } ) )
43sneqd 3666 . . . 4  |-  ( r  =  R  ->  { ( (RSpan `  r ) `  { g } ) }  =  { ( (RSpan `  R ) `  { g } ) } )
51, 4iuneq12d 3945 . . 3  |-  ( r  =  R  ->  U_ g  e.  ( Base `  r
) { ( (RSpan `  r ) `  {
g } ) }  =  U_ g  e.  ( Base `  R
) { ( (RSpan `  R ) `  {
g } ) } )
6 df-lpidl 16011 . . 3  |- LPIdeal  =  ( r  e.  Ring  |->  U_ g  e.  ( Base `  r
) { ( (RSpan `  r ) `  {
g } ) } )
7 fvex 5555 . . . . . 6  |-  (RSpan `  R )  e.  _V
87rnex 4958 . . . . 5  |-  ran  (RSpan `  R )  e.  _V
9 p0ex 4213 . . . . 5  |-  { (/) }  e.  _V
108, 9unex 4534 . . . 4  |-  ( ran  (RSpan `  R )  u.  { (/) } )  e. 
_V
11 iunss 3959 . . . . 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 5566 . . . . . . 7  |-  ( (RSpan `  R ) `  {
g } )  e.  ( ran  (RSpan `  R )  u.  { (/)
} )
13 snssi 3775 . . . . . . 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 10 . . . . 5  |-  ( g  e.  ( Base `  R
)  ->  { (
(RSpan `  R ) `  { g } ) }  C_  ( ran  (RSpan `  R )  u. 
{ (/) } ) )
1611, 15mprgbir 2626 . . . 4  |-  U_ g  e.  ( Base `  R
) { ( (RSpan `  R ) `  {
g } ) } 
C_  ( ran  (RSpan `  R )  u.  { (/)
} )
1710, 16ssexi 4175 . . 3  |-  U_ g  e.  ( Base `  R
) { ( (RSpan `  R ) `  {
g } ) }  e.  _V
185, 6, 17fvmpt 5618 . 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 3934 . . . 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 5542 . . . . . 6  |-  ( K `
 { g } )  =  ( (RSpan `  R ) `  {
g } )
2524sneqi 3665 . . . . 5  |-  { ( K `  { g } ) }  =  { ( (RSpan `  R ) `  {
g } ) }
2625a1i 10 . . . 4  |-  ( g  e.  ( Base `  R
)  ->  { ( K `  { g } ) }  =  { ( (RSpan `  R ) `  {
g } ) } )
2726iuneq2i 3939 . . 3  |-  U_ g  e.  ( Base `  R
) { ( K `
 { g } ) }  =  U_ g  e.  ( Base `  R ) { ( (RSpan `  R ) `  { g } ) }
2822, 27eqtri 2316 . 2  |-  U_ g  e.  B  { ( K `  { g } ) }  =  U_ g  e.  ( Base `  R ) { ( (RSpan `  R ) `  { g } ) }
2918, 19, 283eqtr4g 2353 1  |-  ( R  e.  Ring  ->  P  = 
U_ g  e.  B  { ( K `  { g } ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696    u. cun 3163    C_ wss 3165   (/)c0 3468   {csn 3653   U_ciun 3921   ran crn 4706   ` cfv 5271   Basecbs 13164   Ringcrg 15353  RSpancrsp 15940  LPIdealclpidl 16009
This theorem is referenced by:  islpidl  16014
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fv 5279  df-lpidl 16011
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