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Theorem pclvalN 33455
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 5875 . . . 4  |-  ( Atoms `  K )  e.  _V
31, 2eqeltri 2525 . . 3  |-  A  e. 
_V
43elpw2 4567 . 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 33454 . . . . 5  |-  ( K  e.  V  ->  U  =  ( x  e. 
~P A  |->  |^| { y  e.  S  |  x 
C_  y } ) )
87fveq1d 5867 . . . 4  |-  ( K  e.  V  ->  ( U `  X )  =  ( ( x  e.  ~P A  |->  |^|
{ y  e.  S  |  x  C_  y } ) `  X ) )
98adantr 467 . . 3  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  ( U `  X )  =  ( ( x  e.  ~P A  |->  |^| { y  e.  S  |  x  C_  y } ) `  X
) )
10 simpr 463 . . . 4  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  X  e.  ~P A )
11 elpwi 3960 . . . . . . . 8  |-  ( X  e.  ~P A  ->  X  C_  A )
1211adantl 468 . . . . . . 7  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  X  C_  A
)
131, 5atpsubN 33318 . . . . . . . . 9  |-  ( K  e.  V  ->  A  e.  S )
1413adantr 467 . . . . . . . 8  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  A  e.  S )
15 sseq2 3454 . . . . . . . . 9  |-  ( y  =  A  ->  ( X  C_  y  <->  X  C_  A
) )
1615elrab3 3197 . . . . . . . 8  |-  ( A  e.  S  ->  ( A  e.  { y  e.  S  |  X  C_  y }  <->  X  C_  A
) )
1714, 16syl 17 . . . . . . 7  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  ( A  e.  { y  e.  S  |  X  C_  y }  <-> 
X  C_  A )
)
1812, 17mpbird 236 . . . . . 6  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  A  e.  { y  e.  S  |  X  C_  y } )
19 ne0i 3737 . . . . . 6  |-  ( A  e.  { y  e.  S  |  X  C_  y }  ->  { y  e.  S  |  X  C_  y }  =/=  (/) )
2018, 19syl 17 . . . . 5  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  { y  e.  S  |  X  C_  y }  =/=  (/) )
21 intex 4559 . . . . 5  |-  ( { y  e.  S  |  X  C_  y }  =/=  (/)  <->  |^|
{ y  e.  S  |  X  C_  y }  e.  _V )
2220, 21sylib 200 . . . 4  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  |^| { y  e.  S  |  X  C_  y }  e.  _V )
23 sseq1 3453 . . . . . . 7  |-  ( x  =  X  ->  (
x  C_  y  <->  X  C_  y
) )
2423rabbidv 3036 . . . . . 6  |-  ( x  =  X  ->  { y  e.  S  |  x 
C_  y }  =  { y  e.  S  |  X  C_  y } )
2524inteqd 4239 . . . . 5  |-  ( x  =  X  ->  |^| { y  e.  S  |  x 
C_  y }  =  |^| { y  e.  S  |  X  C_  y } )
26 eqid 2451 . . . . 5  |-  ( x  e.  ~P A  |->  |^|
{ y  e.  S  |  x  C_  y } )  =  ( x  e.  ~P A  |->  |^|
{ y  e.  S  |  x  C_  y } )
2725, 26fvmptg 5946 . . . 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 667 . . 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 2485 . 2  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  ( U `  X )  =  |^| { y  e.  S  |  X  C_  y } )
304, 29sylan2br 479 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 188    /\ wa 371    = wceq 1444    e. wcel 1887    =/= wne 2622   {crab 2741   _Vcvv 3045    C_ wss 3404   (/)c0 3731   ~Pcpw 3951   |^|cint 4234    |-> cmpt 4461   ` cfv 5582   Atomscatm 32829   PSubSpcpsubsp 33061   PClcpclN 33452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-psubsp 33068  df-pclN 33453
This theorem is referenced by:  pclclN  33456  elpclN  33457  elpcliN  33458  pclssN  33459  pclssidN  33460  pclidN  33461
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