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Theorem 2polvalN 33152
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 2433 . . . 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 33144 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  X )  =  ( M `  ( ( oc `  K ) `  ( U `  X )
) ) )
76fveq2d 5683 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  (  ._|_  `  X ) )  =  (  ._|_  `  ( M `
 ( ( oc
`  K ) `  ( U `  X ) ) ) ) )
8 hlop 32601 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
98adantr 462 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  K  e.  OP )
10 hlclat 32597 . . . . 5  |-  ( K  e.  HL  ->  K  e.  CLat )
11 eqid 2433 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1211, 3atssbase 32529 . . . . . 6  |-  A  C_  ( Base `  K )
13 sstr 3352 . . . . . 6  |-  ( ( X  C_  A  /\  A  C_  ( Base `  K
) )  ->  X  C_  ( Base `  K
) )
1412, 13mpan2 664 . . . . 5  |-  ( X 
C_  A  ->  X  C_  ( Base `  K
) )
1511, 1clatlubcl 15265 . . . . 5  |-  ( ( K  e.  CLat  /\  X  C_  ( Base `  K
) )  ->  ( U `  X )  e.  ( Base `  K
) )
1610, 14, 15syl2an 474 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( U `  X
)  e.  ( Base `  K ) )
1711, 2opoccl 32433 . . . 4  |-  ( ( K  e.  OP  /\  ( U `  X )  e.  ( Base `  K
) )  ->  (
( oc `  K
) `  ( U `  X ) )  e.  ( Base `  K
) )
189, 16, 17syl2anc 654 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( oc `  K ) `  ( U `  X )
)  e.  ( Base `  K ) )
1911, 2, 4, 5polpmapN 33150 . . 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 467 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  ( M `  ( ( oc `  K ) `  ( U `  X )
) ) )  =  ( M `  (
( oc `  K
) `  ( ( oc `  K ) `  ( U `  X ) ) ) ) )
2111, 2opococ 32434 . . . 4  |-  ( ( K  e.  OP  /\  ( U `  X )  e.  ( Base `  K
) )  ->  (
( oc `  K
) `  ( ( oc `  K ) `  ( U `  X ) ) )  =  ( U `  X ) )
229, 16, 21syl2anc 654 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( oc `  K ) `  (
( oc `  K
) `  ( U `  X ) ) )  =  ( U `  X ) )
2322fveq2d 5683 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( M `  (
( oc `  K
) `  ( ( oc `  K ) `  ( U `  X ) ) ) )  =  ( M `  ( U `  X )
) )
247, 20, 233eqtrd 2469 1  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  (  ._|_  `  X ) )  =  ( M `  ( U `  X )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1362    e. wcel 1755    C_ wss 3316   ` cfv 5406   Basecbs 14157   occoc 14229   lubclub 15095   CLatccla 15260   OPcops 32411   Atomscatm 32502   HLchlt 32589   pmapcpmap 32735   _|_PcpolN 33140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-riotaBAD 32198
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-undef 6778  df-poset 15099  df-plt 15111  df-lub 15127  df-glb 15128  df-join 15129  df-meet 15130  df-p0 15192  df-p1 15193  df-lat 15199  df-clat 15261  df-oposet 32415  df-ol 32417  df-oml 32418  df-covers 32505  df-ats 32506  df-atl 32537  df-cvlat 32561  df-hlat 32590  df-pmap 32742  df-polarityN 33141
This theorem is referenced by:  2polssN  33153  3polN  33154  sspmaplubN  33163  2pmaplubN  33164  paddunN  33165  pnonsingN  33171  pmapidclN  33180  poml4N  33191
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