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Theorem polsubclN 36092
Description: A polarity is a closed projective subspace. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
polsubcl.a  |-  A  =  ( Atoms `  K )
polsubcl.p  |-  ._|_  =  ( _|_P `  K
)
polsubcl.c  |-  C  =  ( PSubCl `  K )
Assertion
Ref Expression
polsubclN  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  X )  e.  C )

Proof of Theorem polsubclN
StepHypRef Expression
1 eqid 2454 . . 3  |-  ( lub `  K )  =  ( lub `  K )
2 eqid 2454 . . 3  |-  ( oc
`  K )  =  ( oc `  K
)
3 polsubcl.a . . 3  |-  A  =  ( Atoms `  K )
4 eqid 2454 . . 3  |-  ( pmap `  K )  =  (
pmap `  K )
5 polsubcl.p . . 3  |-  ._|_  =  ( _|_P `  K
)
61, 2, 3, 4, 5polval2N 36046 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  X )  =  ( ( pmap `  K ) `  (
( oc `  K
) `  ( ( lub `  K ) `  X ) ) ) )
7 hlop 35503 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
87adantr 463 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  K  e.  OP )
9 hlclat 35499 . . . . 5  |-  ( K  e.  HL  ->  K  e.  CLat )
10 eqid 2454 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1110, 3atssbase 35431 . . . . . 6  |-  A  C_  ( Base `  K )
12 sstr 3497 . . . . . 6  |-  ( ( X  C_  A  /\  A  C_  ( Base `  K
) )  ->  X  C_  ( Base `  K
) )
1311, 12mpan2 669 . . . . 5  |-  ( X 
C_  A  ->  X  C_  ( Base `  K
) )
1410, 1clatlubcl 15944 . . . . 5  |-  ( ( K  e.  CLat  /\  X  C_  ( Base `  K
) )  ->  (
( lub `  K
) `  X )  e.  ( Base `  K
) )
159, 13, 14syl2an 475 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( lub `  K
) `  X )  e.  ( Base `  K
) )
1610, 2opoccl 35335 . . . 4  |-  ( ( K  e.  OP  /\  ( ( lub `  K
) `  X )  e.  ( Base `  K
) )  ->  (
( oc `  K
) `  ( ( lub `  K ) `  X ) )  e.  ( Base `  K
) )
178, 15, 16syl2anc 659 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( oc `  K ) `  (
( lub `  K
) `  X )
)  e.  ( Base `  K ) )
18 polsubcl.c . . . 4  |-  C  =  ( PSubCl `  K )
1910, 4, 18pmapsubclN 36086 . . 3  |-  ( ( K  e.  HL  /\  ( ( oc `  K ) `  (
( lub `  K
) `  X )
)  e.  ( Base `  K ) )  -> 
( ( pmap `  K
) `  ( ( oc `  K ) `  ( ( lub `  K
) `  X )
) )  e.  C
)
2017, 19syldan 468 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( pmap `  K
) `  ( ( oc `  K ) `  ( ( lub `  K
) `  X )
) )  e.  C
)
216, 20eqeltrd 2542 1  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  X )  e.  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823    C_ wss 3461   ` cfv 5570   Basecbs 14719   occoc 14795   lubclub 15773   CLatccla 15939   OPcops 35313   Atomscatm 35404   HLchlt 35491   pmapcpmap 35637   _|_PcpolN 36042   PSubClcpscN 36074
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 35100
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 15759  df-poset 15777  df-plt 15790  df-lub 15806  df-glb 15807  df-join 15808  df-meet 15809  df-p0 15871  df-p1 15872  df-lat 15878  df-clat 15940  df-oposet 35317  df-ol 35319  df-oml 35320  df-covers 35407  df-ats 35408  df-atl 35439  df-cvlat 35463  df-hlat 35492  df-pmap 35644  df-polarityN 36043  df-psubclN 36075
This theorem is referenced by:  osumcllem9N  36104  pexmidN  36109
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