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Theorem pclvalN 30701
Description: Value of the projective subspace closure function. (Contributed by NM, 7-Sep-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
pclfval.a  |-  A  =  ( Atoms `  K )
pclfval.s  |-  S  =  ( PSubSp `  K )
pclfval.c  |-  U  =  ( PCl `  K
)
Assertion
Ref Expression
pclvalN  |-  ( ( K  e.  V  /\  X  C_  A )  -> 
( U `  X
)  =  |^| { y  e.  S  |  X  C_  y } )
Distinct variable groups:    y, A    y, K    y, S    y, X
Allowed substitution hints:    U( y)    V( y)

Proof of Theorem pclvalN
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 pclfval.a . . . 4  |-  A  =  ( Atoms `  K )
2 fvex 5555 . . . 4  |-  ( Atoms `  K )  e.  _V
31, 2eqeltri 2366 . . 3  |-  A  e. 
_V
43elpw2 4191 . 2  |-  ( X  e.  ~P A  <->  X  C_  A
)
5 pclfval.s . . . . . 6  |-  S  =  ( PSubSp `  K )
6 pclfval.c . . . . . 6  |-  U  =  ( PCl `  K
)
71, 5, 6pclfvalN 30700 . . . . 5  |-  ( K  e.  V  ->  U  =  ( x  e. 
~P A  |->  |^| { y  e.  S  |  x 
C_  y } ) )
87fveq1d 5543 . . . 4  |-  ( K  e.  V  ->  ( U `  X )  =  ( ( x  e.  ~P A  |->  |^|
{ y  e.  S  |  x  C_  y } ) `  X ) )
98adantr 451 . . 3  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  ( U `  X )  =  ( ( x  e.  ~P A  |->  |^| { y  e.  S  |  x  C_  y } ) `  X
) )
10 simpr 447 . . . 4  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  X  e.  ~P A )
11 elpwi 3646 . . . . . . . 8  |-  ( X  e.  ~P A  ->  X  C_  A )
1211adantl 452 . . . . . . 7  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  X  C_  A
)
131, 5atpsubN 30564 . . . . . . . . 9  |-  ( K  e.  V  ->  A  e.  S )
1413adantr 451 . . . . . . . 8  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  A  e.  S )
15 sseq2 3213 . . . . . . . . 9  |-  ( y  =  A  ->  ( X  C_  y  <->  X  C_  A
) )
1615elrab3 2937 . . . . . . . 8  |-  ( A  e.  S  ->  ( A  e.  { y  e.  S  |  X  C_  y }  <->  X  C_  A
) )
1714, 16syl 15 . . . . . . 7  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  ( A  e.  { y  e.  S  |  X  C_  y }  <-> 
X  C_  A )
)
1812, 17mpbird 223 . . . . . 6  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  A  e.  { y  e.  S  |  X  C_  y } )
19 ne0i 3474 . . . . . 6  |-  ( A  e.  { y  e.  S  |  X  C_  y }  ->  { y  e.  S  |  X  C_  y }  =/=  (/) )
2018, 19syl 15 . . . . 5  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  { y  e.  S  |  X  C_  y }  =/=  (/) )
21 intex 4183 . . . . 5  |-  ( { y  e.  S  |  X  C_  y }  =/=  (/)  <->  |^|
{ y  e.  S  |  X  C_  y }  e.  _V )
2220, 21sylib 188 . . . 4  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  |^| { y  e.  S  |  X  C_  y }  e.  _V )
23 sseq1 3212 . . . . . . 7  |-  ( x  =  X  ->  (
x  C_  y  <->  X  C_  y
) )
2423rabbidv 2793 . . . . . 6  |-  ( x  =  X  ->  { y  e.  S  |  x 
C_  y }  =  { y  e.  S  |  X  C_  y } )
2524inteqd 3883 . . . . 5  |-  ( x  =  X  ->  |^| { y  e.  S  |  x 
C_  y }  =  |^| { y  e.  S  |  X  C_  y } )
26 eqid 2296 . . . . 5  |-  ( x  e.  ~P A  |->  |^|
{ y  e.  S  |  x  C_  y } )  =  ( x  e.  ~P A  |->  |^|
{ y  e.  S  |  x  C_  y } )
2725, 26fvmptg 5616 . . . 4  |-  ( ( X  e.  ~P A  /\  |^| { y  e.  S  |  X  C_  y }  e.  _V )  ->  ( ( x  e.  ~P A  |->  |^|
{ y  e.  S  |  x  C_  y } ) `  X )  =  |^| { y  e.  S  |  X  C_  y } )
2810, 22, 27syl2anc 642 . . 3  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  ( (
x  e.  ~P A  |-> 
|^| { y  e.  S  |  x  C_  y } ) `  X )  =  |^| { y  e.  S  |  X  C_  y } )
299, 28eqtrd 2328 . 2  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  ( U `  X )  =  |^| { y  e.  S  |  X  C_  y } )
304, 29sylan2br 462 1  |-  ( ( K  e.  V  /\  X  C_  A )  -> 
( U `  X
)  =  |^| { y  e.  S  |  X  C_  y } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   {crab 2560   _Vcvv 2801    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   |^|cint 3878    e. cmpt 4093   ` cfv 5271   Atomscatm 30075   PSubSpcpsubsp 30307   PClcpclN 30698
This theorem is referenced by:  pclclN  30702  elpclN  30703  elpcliN  30704  pclssN  30705  pclssidN  30706  pclidN  30707
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
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-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  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-int 3879  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-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-psubsp 30314  df-pclN 30699
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