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Theorem pclcmpatN 30163
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 30162 . . . . 5  |-  ( ( K  e.  AtLat  /\  X  C_  A )  ->  ( U `  X )  =  U_ y  e.  ( Fin  i^i  ~P X
) ( U `  y ) )
43eleq2d 2352 . . . 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 3911 . . . 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 3360 . . . . . . 7  |-  ( y  e.  ( Fin  i^i  ~P X )  <->  ( y  e.  Fin  /\  y  e. 
~P X ) )
8 elpwi 3635 . . . . . . . 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 2651 . . 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 1625    e. wcel 1686   E.wrex 2546    i^i cin 3153    C_ wss 3154   ~Pcpw 3627   U_ciun 3907   ` cfv 5257   Fincfn 6865   Atomscatm 29526   AtLatcal 29527   PClcpclN 30149
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-undef 6300  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-oadd 6485  df-er 6662  df-en 6866  df-fin 6869  df-poset 14082  df-plt 14094  df-lub 14110  df-join 14112  df-lat 14154  df-covers 29529  df-ats 29530  df-atl 29561  df-psubsp 29765  df-pclN 30150
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