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

Theorem polsubN 30704
Description: The polarity of a set of atoms is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
polsubsp.a  |-  A  =  ( Atoms `  K )
polsubsp.s  |-  S  =  ( PSubSp `  K )
polsubsp.p  |-  ._|_  =  ( _|_ P `  K
)
Assertion
Ref Expression
polsubN  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  X )  e.  S )

Proof of Theorem polsubN
StepHypRef Expression
1 eqid 2436 . . 3  |-  ( lub `  K )  =  ( lub `  K )
2 eqid 2436 . . 3  |-  ( oc
`  K )  =  ( oc `  K
)
3 polsubsp.a . . 3  |-  A  =  ( Atoms `  K )
4 eqid 2436 . . 3  |-  ( pmap `  K )  =  (
pmap `  K )
5 polsubsp.p . . 3  |-  ._|_  =  ( _|_ P `  K
)
61, 2, 3, 4, 5polval2N 30703 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  X )  =  ( ( pmap `  K ) `  (
( oc `  K
) `  ( ( lub `  K ) `  X ) ) ) )
7 hllat 30161 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
87adantr 452 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  K  e.  Lat )
9 hlop 30160 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
109adantr 452 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  K  e.  OP )
11 hlclat 30156 . . . . 5  |-  ( K  e.  HL  ->  K  e.  CLat )
12 eqid 2436 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1312, 3atssbase 30088 . . . . . 6  |-  A  C_  ( Base `  K )
14 sstr 3356 . . . . . 6  |-  ( ( X  C_  A  /\  A  C_  ( Base `  K
) )  ->  X  C_  ( Base `  K
) )
1513, 14mpan2 653 . . . . 5  |-  ( X 
C_  A  ->  X  C_  ( Base `  K
) )
1612, 1clatlubcl 14540 . . . . 5  |-  ( ( K  e.  CLat  /\  X  C_  ( Base `  K
) )  ->  (
( lub `  K
) `  X )  e.  ( Base `  K
) )
1711, 15, 16syl2an 464 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( lub `  K
) `  X )  e.  ( Base `  K
) )
1812, 2opoccl 29992 . . . 4  |-  ( ( K  e.  OP  /\  ( ( lub `  K
) `  X )  e.  ( Base `  K
) )  ->  (
( oc `  K
) `  ( ( lub `  K ) `  X ) )  e.  ( Base `  K
) )
1910, 17, 18syl2anc 643 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( oc `  K ) `  (
( lub `  K
) `  X )
)  e.  ( Base `  K ) )
20 polsubsp.s . . . 4  |-  S  =  ( PSubSp `  K )
2112, 20, 4pmapsub 30565 . . 3  |-  ( ( K  e.  Lat  /\  ( ( oc `  K ) `  (
( lub `  K
) `  X )
)  e.  ( Base `  K ) )  -> 
( ( pmap `  K
) `  ( ( oc `  K ) `  ( ( lub `  K
) `  X )
) )  e.  S
)
228, 19, 21syl2anc 643 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( pmap `  K
) `  ( ( oc `  K ) `  ( ( lub `  K
) `  X )
) )  e.  S
)
236, 22eqeltrd 2510 1  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  X )  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    C_ wss 3320   ` cfv 5454   Basecbs 13469   occoc 13537   lubclub 14399   Latclat 14474   CLatccla 14536   OPcops 29970   Atomscatm 30061   HLchlt 30148   PSubSpcpsubsp 30293   pmapcpmap 30294   _|_
PcpolN 30699
This theorem is referenced by:  polssatN  30705  pclss2polN  30718  psubclsubN  30737  osumcllem1N  30753
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p1 14469  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-psubsp 30300  df-pmap 30301  df-polarityN 30700
  Copyright terms: Public domain W3C validator