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Theorem polsubN 36028
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 2454 . . 3  |-  ( lub `  K )  =  ( lub `  K )
2 eqid 2454 . . 3  |-  ( oc
`  K )  =  ( oc `  K
)
3 polsubsp.a . . 3  |-  A  =  ( Atoms `  K )
4 eqid 2454 . . 3  |-  ( pmap `  K )  =  (
pmap `  K )
5 polsubsp.p . . 3  |-  ._|_  =  ( _|_P `  K
)
61, 2, 3, 4, 5polval2N 36027 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  X )  =  ( ( pmap `  K ) `  (
( oc `  K
) `  ( ( lub `  K ) `  X ) ) ) )
7 hllat 35485 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
87adantr 463 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  K  e.  Lat )
9 hlop 35484 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
109adantr 463 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  K  e.  OP )
11 hlclat 35480 . . . . 5  |-  ( K  e.  HL  ->  K  e.  CLat )
12 eqid 2454 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1312, 3atssbase 35412 . . . . . 6  |-  A  C_  ( Base `  K )
14 sstr 3497 . . . . . 6  |-  ( ( X  C_  A  /\  A  C_  ( Base `  K
) )  ->  X  C_  ( Base `  K
) )
1513, 14mpan2 669 . . . . 5  |-  ( X 
C_  A  ->  X  C_  ( Base `  K
) )
1612, 1clatlubcl 15941 . . . . 5  |-  ( ( K  e.  CLat  /\  X  C_  ( Base `  K
) )  ->  (
( lub `  K
) `  X )  e.  ( Base `  K
) )
1711, 15, 16syl2an 475 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( lub `  K
) `  X )  e.  ( Base `  K
) )
1812, 2opoccl 35316 . . . 4  |-  ( ( K  e.  OP  /\  ( ( lub `  K
) `  X )  e.  ( Base `  K
) )  ->  (
( oc `  K
) `  ( ( lub `  K ) `  X ) )  e.  ( Base `  K
) )
1910, 17, 18syl2anc 659 . . 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 35889 . . 3  |-  ( ( K  e.  Lat  /\  ( ( oc `  K ) `  (
( lub `  K
) `  X )
)  e.  ( Base `  K ) )  -> 
( ( pmap `  K
) `  ( ( oc `  K ) `  ( ( lub `  K
) `  X )
) )  e.  S
)
228, 19, 21syl2anc 659 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( pmap `  K
) `  ( ( oc `  K ) `  ( ( lub `  K
) `  X )
) )  e.  S
)
236, 22eqeltrd 2542 1  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  X )  e.  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823    C_ wss 3461   ` cfv 5570   Basecbs 14716   occoc 14792   lubclub 15770   Latclat 15874   CLatccla 15936   OPcops 35294   Atomscatm 35385   HLchlt 35472   PSubSpcpsubsp 35617   pmapcpmap 35618   _|_PcpolN 36023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-riotaBAD 35081
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-undef 6994  df-preset 15756  df-poset 15774  df-lub 15803  df-glb 15804  df-join 15805  df-meet 15806  df-p1 15869  df-lat 15875  df-clat 15937  df-oposet 35298  df-ol 35300  df-oml 35301  df-ats 35389  df-atl 35420  df-cvlat 35444  df-hlat 35473  df-psubsp 35624  df-pmap 35625  df-polarityN 36024
This theorem is referenced by:  polssatN  36029  pclss2polN  36042  psubclsubN  36061  osumcllem1N  36077
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