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Theorem polsubN 30718
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 2296 . . 3  |-  ( lub `  K )  =  ( lub `  K )
2 eqid 2296 . . 3  |-  ( oc
`  K )  =  ( oc `  K
)
3 polsubsp.a . . 3  |-  A  =  ( Atoms `  K )
4 eqid 2296 . . 3  |-  ( pmap `  K )  =  (
pmap `  K )
5 polsubsp.p . . 3  |-  ._|_  =  ( _|_ P `  K
)
61, 2, 3, 4, 5polval2N 30717 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  X )  =  ( ( pmap `  K ) `  (
( oc `  K
) `  ( ( lub `  K ) `  X ) ) ) )
7 hllat 30175 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
87adantr 451 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  K  e.  Lat )
9 hlop 30174 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
109adantr 451 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  K  e.  OP )
11 hlclat 30170 . . . . 5  |-  ( K  e.  HL  ->  K  e.  CLat )
12 eqid 2296 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1312, 3atssbase 30102 . . . . . 6  |-  A  C_  ( Base `  K )
14 sstr 3200 . . . . . 6  |-  ( ( X  C_  A  /\  A  C_  ( Base `  K
) )  ->  X  C_  ( Base `  K
) )
1513, 14mpan2 652 . . . . 5  |-  ( X 
C_  A  ->  X  C_  ( Base `  K
) )
1612, 1clatlubcl 14233 . . . . 5  |-  ( ( K  e.  CLat  /\  X  C_  ( Base `  K
) )  ->  (
( lub `  K
) `  X )  e.  ( Base `  K
) )
1711, 15, 16syl2an 463 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( lub `  K
) `  X )  e.  ( Base `  K
) )
1812, 2opoccl 30006 . . . 4  |-  ( ( K  e.  OP  /\  ( ( lub `  K
) `  X )  e.  ( Base `  K
) )  ->  (
( oc `  K
) `  ( ( lub `  K ) `  X ) )  e.  ( Base `  K
) )
1910, 17, 18syl2anc 642 . . 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 30579 . . 3  |-  ( ( K  e.  Lat  /\  ( ( oc `  K ) `  (
( lub `  K
) `  X )
)  e.  ( Base `  K ) )  -> 
( ( pmap `  K
) `  ( ( oc `  K ) `  ( ( lub `  K
) `  X )
) )  e.  S
)
228, 19, 21syl2anc 642 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( pmap `  K
) `  ( ( oc `  K ) `  ( ( lub `  K
) `  X )
) )  e.  S
)
236, 22eqeltrd 2370 1  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  X )  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    C_ wss 3165   ` cfv 5271   Basecbs 13164   occoc 13232   lubclub 14092   Latclat 14167   CLatccla 14229   OPcops 29984   Atomscatm 30075   HLchlt 30162   PSubSpcpsubsp 30307   pmapcpmap 30308   _|_
PcpolN 30713
This theorem is referenced by:  polssatN  30719  pclss2polN  30732  psubclsubN  30751  osumcllem1N  30767
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-psubsp 30314  df-pmap 30315  df-polarityN 30714
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