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

Theorem pclcmpatN 30159
Description: The set of projective subspaces is compactly atomistic: if an atom is in the projective subspace closure of a set of atoms, it also belongs to the projective subspace closure of a finite subset of that set. Analogous to Lemma 3.3.10 of [PtakPulmannova] p. 74. (Contributed by NM, 10-Sep-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
pclfin.a  |-  A  =  ( Atoms `  K )
pclfin.c  |-  U  =  ( PCl `  K
)
Assertion
Ref Expression
pclcmpatN  |-  ( ( K  e.  AtLat  /\  X  C_  A  /\  P  e.  ( U `  X
) )  ->  E. y  e.  Fin  ( y  C_  X  /\  P  e.  ( U `  y ) ) )
Distinct variable groups:    y, A    y, U    y, K    y, X    y, P

Proof of Theorem pclcmpatN
StepHypRef Expression
1 pclfin.a . . . . . 6  |-  A  =  ( Atoms `  K )
2 pclfin.c . . . . . 6  |-  U  =  ( PCl `  K
)
31, 2pclfinN 30158 . . . . 5  |-  ( ( K  e.  AtLat  /\  X  C_  A )  ->  ( U `  X )  =  U_ y  e.  ( Fin  i^i  ~P X
) ( U `  y ) )
43eleq2d 2425 . . . 4  |-  ( ( K  e.  AtLat  /\  X  C_  A )  ->  ( P  e.  ( U `  X )  <->  P  e.  U_ y  e.  ( Fin 
i^i  ~P X ) ( U `  y ) ) )
5 eliun 3990 . . . 4  |-  ( P  e.  U_ y  e.  ( Fin  i^i  ~P X ) ( U `
 y )  <->  E. y  e.  ( Fin  i^i  ~P X ) P  e.  ( U `  y
) )
64, 5syl6bb 252 . . 3  |-  ( ( K  e.  AtLat  /\  X  C_  A )  ->  ( P  e.  ( U `  X )  <->  E. y  e.  ( Fin  i^i  ~P X ) P  e.  ( U `  y
) ) )
7 elin 3434 . . . . . . 7  |-  ( y  e.  ( Fin  i^i  ~P X )  <->  ( y  e.  Fin  /\  y  e. 
~P X ) )
8 elpwi 3709 . . . . . . . 8  |-  ( y  e.  ~P X  -> 
y  C_  X )
98anim2i 552 . . . . . . 7  |-  ( ( y  e.  Fin  /\  y  e.  ~P X
)  ->  ( y  e.  Fin  /\  y  C_  X ) )
107, 9sylbi 187 . . . . . 6  |-  ( y  e.  ( Fin  i^i  ~P X )  ->  (
y  e.  Fin  /\  y  C_  X ) )
1110anim1i 551 . . . . 5  |-  ( ( y  e.  ( Fin 
i^i  ~P X )  /\  P  e.  ( U `  y ) )  -> 
( ( y  e. 
Fin  /\  y  C_  X )  /\  P  e.  ( U `  y
) ) )
12 anass 630 . . . . 5  |-  ( ( ( y  e.  Fin  /\  y  C_  X )  /\  P  e.  ( U `  y )
)  <->  ( y  e. 
Fin  /\  ( y  C_  X  /\  P  e.  ( U `  y
) ) ) )
1311, 12sylib 188 . . . 4  |-  ( ( y  e.  ( Fin 
i^i  ~P X )  /\  P  e.  ( U `  y ) )  -> 
( y  e.  Fin  /\  ( y  C_  X  /\  P  e.  ( U `  y )
) ) )
1413reximi2 2725 . . 3  |-  ( E. y  e.  ( Fin 
i^i  ~P X ) P  e.  ( U `  y )  ->  E. y  e.  Fin  ( y  C_  X  /\  P  e.  ( U `  y ) ) )
156, 14syl6bi 219 . 2  |-  ( ( K  e.  AtLat  /\  X  C_  A )  ->  ( P  e.  ( U `  X )  ->  E. y  e.  Fin  ( y  C_  X  /\  P  e.  ( U `  y ) ) ) )
16153impia 1148 1  |-  ( ( K  e.  AtLat  /\  X  C_  A  /\  P  e.  ( U `  X
) )  ->  E. y  e.  Fin  ( y  C_  X  /\  P  e.  ( U `  y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   E.wrex 2620    i^i cin 3227    C_ wss 3228   ~Pcpw 3701   U_ciun 3986   ` cfv 5337   Fincfn 6951   Atomscatm 29522   AtLatcal 29523   PClcpclN 30145
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-undef 6385  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-oadd 6570  df-er 6747  df-en 6952  df-fin 6955  df-poset 14179  df-plt 14191  df-lub 14207  df-join 14209  df-lat 14251  df-covers 29525  df-ats 29526  df-atl 29557  df-psubsp 29761  df-pclN 30146
  Copyright terms: Public domain W3C validator