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Theorem pclvalN 36011
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 5858 . . . 4  |-  ( Atoms `  K )  e.  _V
31, 2eqeltri 2538 . . 3  |-  A  e. 
_V
43elpw2 4601 . 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 36010 . . . . 5  |-  ( K  e.  V  ->  U  =  ( x  e. 
~P A  |->  |^| { y  e.  S  |  x 
C_  y } ) )
87fveq1d 5850 . . . 4  |-  ( K  e.  V  ->  ( U `  X )  =  ( ( x  e.  ~P A  |->  |^|
{ y  e.  S  |  x  C_  y } ) `  X ) )
98adantr 463 . . 3  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  ( U `  X )  =  ( ( x  e.  ~P A  |->  |^| { y  e.  S  |  x  C_  y } ) `  X
) )
10 simpr 459 . . . 4  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  X  e.  ~P A )
11 elpwi 4008 . . . . . . . 8  |-  ( X  e.  ~P A  ->  X  C_  A )
1211adantl 464 . . . . . . 7  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  X  C_  A
)
131, 5atpsubN 35874 . . . . . . . . 9  |-  ( K  e.  V  ->  A  e.  S )
1413adantr 463 . . . . . . . 8  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  A  e.  S )
15 sseq2 3511 . . . . . . . . 9  |-  ( y  =  A  ->  ( X  C_  y  <->  X  C_  A
) )
1615elrab3 3255 . . . . . . . 8  |-  ( A  e.  S  ->  ( A  e.  { y  e.  S  |  X  C_  y }  <->  X  C_  A
) )
1714, 16syl 16 . . . . . . 7  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  ( A  e.  { y  e.  S  |  X  C_  y }  <-> 
X  C_  A )
)
1812, 17mpbird 232 . . . . . 6  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  A  e.  { y  e.  S  |  X  C_  y } )
19 ne0i 3789 . . . . . 6  |-  ( A  e.  { y  e.  S  |  X  C_  y }  ->  { y  e.  S  |  X  C_  y }  =/=  (/) )
2018, 19syl 16 . . . . 5  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  { y  e.  S  |  X  C_  y }  =/=  (/) )
21 intex 4593 . . . . 5  |-  ( { y  e.  S  |  X  C_  y }  =/=  (/)  <->  |^|
{ y  e.  S  |  X  C_  y }  e.  _V )
2220, 21sylib 196 . . . 4  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  |^| { y  e.  S  |  X  C_  y }  e.  _V )
23 sseq1 3510 . . . . . . 7  |-  ( x  =  X  ->  (
x  C_  y  <->  X  C_  y
) )
2423rabbidv 3098 . . . . . 6  |-  ( x  =  X  ->  { y  e.  S  |  x 
C_  y }  =  { y  e.  S  |  X  C_  y } )
2524inteqd 4276 . . . . 5  |-  ( x  =  X  ->  |^| { y  e.  S  |  x 
C_  y }  =  |^| { y  e.  S  |  X  C_  y } )
26 eqid 2454 . . . . 5  |-  ( x  e.  ~P A  |->  |^|
{ y  e.  S  |  x  C_  y } )  =  ( x  e.  ~P A  |->  |^|
{ y  e.  S  |  x  C_  y } )
2725, 26fvmptg 5929 . . . 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 659 . . 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 2495 . 2  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  ( U `  X )  =  |^| { y  e.  S  |  X  C_  y } )
304, 29sylan2br 474 1  |-  ( ( K  e.  V  /\  X  C_  A )  -> 
( U `  X
)  =  |^| { y  e.  S  |  X  C_  y } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   {crab 2808   _Vcvv 3106    C_ wss 3461   (/)c0 3783   ~Pcpw 3999   |^|cint 4271    |-> cmpt 4497   ` cfv 5570   Atomscatm 35385   PSubSpcpsubsp 35617   PClcpclN 36008
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-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
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-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  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-int 4272  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-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-psubsp 35624  df-pclN 36009
This theorem is referenced by:  pclclN  36012  elpclN  36013  elpcliN  36014  pclssN  36015  pclssidN  36016  pclidN  36017
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