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Theorem 2polvalN 33398
Description: Value of double polarity. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
2polval.u  |-  U  =  ( lub `  K
)
2polval.a  |-  A  =  ( Atoms `  K )
2polval.m  |-  M  =  ( pmap `  K
)
2polval.p  |-  ._|_  =  ( _|_P `  K
)
Assertion
Ref Expression
2polvalN  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  (  ._|_  `  X ) )  =  ( M `  ( U `  X )
) )

Proof of Theorem 2polvalN
StepHypRef Expression
1 2polval.u . . . 4  |-  U  =  ( lub `  K
)
2 eqid 2438 . . . 4  |-  ( oc
`  K )  =  ( oc `  K
)
3 2polval.a . . . 4  |-  A  =  ( Atoms `  K )
4 2polval.m . . . 4  |-  M  =  ( pmap `  K
)
5 2polval.p . . . 4  |-  ._|_  =  ( _|_P `  K
)
61, 2, 3, 4, 5polval2N 33390 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  X )  =  ( M `  ( ( oc `  K ) `  ( U `  X )
) ) )
76fveq2d 5690 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  (  ._|_  `  X ) )  =  (  ._|_  `  ( M `
 ( ( oc
`  K ) `  ( U `  X ) ) ) ) )
8 hlop 32847 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
98adantr 465 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  K  e.  OP )
10 hlclat 32843 . . . . 5  |-  ( K  e.  HL  ->  K  e.  CLat )
11 eqid 2438 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1211, 3atssbase 32775 . . . . . 6  |-  A  C_  ( Base `  K )
13 sstr 3359 . . . . . 6  |-  ( ( X  C_  A  /\  A  C_  ( Base `  K
) )  ->  X  C_  ( Base `  K
) )
1412, 13mpan2 671 . . . . 5  |-  ( X 
C_  A  ->  X  C_  ( Base `  K
) )
1511, 1clatlubcl 15274 . . . . 5  |-  ( ( K  e.  CLat  /\  X  C_  ( Base `  K
) )  ->  ( U `  X )  e.  ( Base `  K
) )
1610, 14, 15syl2an 477 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( U `  X
)  e.  ( Base `  K ) )
1711, 2opoccl 32679 . . . 4  |-  ( ( K  e.  OP  /\  ( U `  X )  e.  ( Base `  K
) )  ->  (
( oc `  K
) `  ( U `  X ) )  e.  ( Base `  K
) )
189, 16, 17syl2anc 661 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( oc `  K ) `  ( U `  X )
)  e.  ( Base `  K ) )
1911, 2, 4, 5polpmapN 33396 . . 3  |-  ( ( K  e.  HL  /\  ( ( oc `  K ) `  ( U `  X )
)  e.  ( Base `  K ) )  -> 
(  ._|_  `  ( M `  ( ( oc `  K ) `  ( U `  X )
) ) )  =  ( M `  (
( oc `  K
) `  ( ( oc `  K ) `  ( U `  X ) ) ) ) )
2018, 19syldan 470 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  ( M `  ( ( oc `  K ) `  ( U `  X )
) ) )  =  ( M `  (
( oc `  K
) `  ( ( oc `  K ) `  ( U `  X ) ) ) ) )
2111, 2opococ 32680 . . . 4  |-  ( ( K  e.  OP  /\  ( U `  X )  e.  ( Base `  K
) )  ->  (
( oc `  K
) `  ( ( oc `  K ) `  ( U `  X ) ) )  =  ( U `  X ) )
229, 16, 21syl2anc 661 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( oc `  K ) `  (
( oc `  K
) `  ( U `  X ) ) )  =  ( U `  X ) )
2322fveq2d 5690 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( M `  (
( oc `  K
) `  ( ( oc `  K ) `  ( U `  X ) ) ) )  =  ( M `  ( U `  X )
) )
247, 20, 233eqtrd 2474 1  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  (  ._|_  `  X ) )  =  ( M `  ( U `  X )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    C_ wss 3323   ` cfv 5413   Basecbs 14166   occoc 14238   lubclub 15104   CLatccla 15269   OPcops 32657   Atomscatm 32748   HLchlt 32835   pmapcpmap 32981   _|_PcpolN 33386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-riotaBAD 32444
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-undef 6784  df-poset 15108  df-plt 15120  df-lub 15136  df-glb 15137  df-join 15138  df-meet 15139  df-p0 15201  df-p1 15202  df-lat 15208  df-clat 15270  df-oposet 32661  df-ol 32663  df-oml 32664  df-covers 32751  df-ats 32752  df-atl 32783  df-cvlat 32807  df-hlat 32836  df-pmap 32988  df-polarityN 33387
This theorem is referenced by:  2polssN  33399  3polN  33400  sspmaplubN  33409  2pmaplubN  33410  paddunN  33411  pnonsingN  33417  pmapidclN  33426  poml4N  33437
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