Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pclvalN Structured version   Unicode version

Theorem pclvalN 30624
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 5734 . . . 4  |-  ( Atoms `  K )  e.  _V
31, 2eqeltri 2505 . . 3  |-  A  e. 
_V
43elpw2 4356 . 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 30623 . . . . 5  |-  ( K  e.  V  ->  U  =  ( x  e. 
~P A  |->  |^| { y  e.  S  |  x 
C_  y } ) )
87fveq1d 5722 . . . 4  |-  ( K  e.  V  ->  ( U `  X )  =  ( ( x  e.  ~P A  |->  |^|
{ y  e.  S  |  x  C_  y } ) `  X ) )
98adantr 452 . . 3  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  ( U `  X )  =  ( ( x  e.  ~P A  |->  |^| { y  e.  S  |  x  C_  y } ) `  X
) )
10 simpr 448 . . . 4  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  X  e.  ~P A )
11 elpwi 3799 . . . . . . . 8  |-  ( X  e.  ~P A  ->  X  C_  A )
1211adantl 453 . . . . . . 7  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  X  C_  A
)
131, 5atpsubN 30487 . . . . . . . . 9  |-  ( K  e.  V  ->  A  e.  S )
1413adantr 452 . . . . . . . 8  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  A  e.  S )
15 sseq2 3362 . . . . . . . . 9  |-  ( y  =  A  ->  ( X  C_  y  <->  X  C_  A
) )
1615elrab3 3085 . . . . . . . 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 224 . . . . . 6  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  A  e.  { y  e.  S  |  X  C_  y } )
19 ne0i 3626 . . . . . 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 4348 . . . . 5  |-  ( { y  e.  S  |  X  C_  y }  =/=  (/)  <->  |^|
{ y  e.  S  |  X  C_  y }  e.  _V )
2220, 21sylib 189 . . . 4  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  |^| { y  e.  S  |  X  C_  y }  e.  _V )
23 sseq1 3361 . . . . . . 7  |-  ( x  =  X  ->  (
x  C_  y  <->  X  C_  y
) )
2423rabbidv 2940 . . . . . 6  |-  ( x  =  X  ->  { y  e.  S  |  x 
C_  y }  =  { y  e.  S  |  X  C_  y } )
2524inteqd 4047 . . . . 5  |-  ( x  =  X  ->  |^| { y  e.  S  |  x 
C_  y }  =  |^| { y  e.  S  |  X  C_  y } )
26 eqid 2435 . . . . 5  |-  ( x  e.  ~P A  |->  |^|
{ y  e.  S  |  x  C_  y } )  =  ( x  e.  ~P A  |->  |^|
{ y  e.  S  |  x  C_  y } )
2725, 26fvmptg 5796 . . . 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 643 . . 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 2467 . 2  |-  ( ( K  e.  V  /\  X  e.  ~P A
)  ->  ( U `  X )  =  |^| { y  e.  S  |  X  C_  y } )
304, 29sylan2br 463 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 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   {crab 2701   _Vcvv 2948    C_ wss 3312   (/)c0 3620   ~Pcpw 3791   |^|cint 4042    e. cmpt 4258   ` cfv 5446   Atomscatm 29998   PSubSpcpsubsp 30230   PClcpclN 30621
This theorem is referenced by:  pclclN  30625  elpclN  30626  elpcliN  30627  pclssN  30628  pclssidN  30629  pclidN  30630
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-psubsp 30237  df-pclN 30622
  Copyright terms: Public domain W3C validator