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Theorem 2polvalN 30396
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 2404 . . . 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 30388 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  X )  =  ( M `  ( ( oc `  K ) `  ( U `  X )
) ) )
76fveq2d 5691 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  (  ._|_  `  X ) )  =  (  ._|_  `  ( M `
 ( ( oc
`  K ) `  ( U `  X ) ) ) ) )
8 hlop 29845 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
98adantr 452 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  K  e.  OP )
10 hlclat 29841 . . . . 5  |-  ( K  e.  HL  ->  K  e.  CLat )
11 eqid 2404 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1211, 3atssbase 29773 . . . . . 6  |-  A  C_  ( Base `  K )
13 sstr 3316 . . . . . 6  |-  ( ( X  C_  A  /\  A  C_  ( Base `  K
) )  ->  X  C_  ( Base `  K
) )
1412, 13mpan2 653 . . . . 5  |-  ( X 
C_  A  ->  X  C_  ( Base `  K
) )
1511, 1clatlubcl 14495 . . . . 5  |-  ( ( K  e.  CLat  /\  X  C_  ( Base `  K
) )  ->  ( U `  X )  e.  ( Base `  K
) )
1610, 14, 15syl2an 464 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( U `  X
)  e.  ( Base `  K ) )
1711, 2opoccl 29677 . . . 4  |-  ( ( K  e.  OP  /\  ( U `  X )  e.  ( Base `  K
) )  ->  (
( oc `  K
) `  ( U `  X ) )  e.  ( Base `  K
) )
189, 16, 17syl2anc 643 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( oc `  K ) `  ( U `  X )
)  e.  ( Base `  K ) )
1911, 2, 4, 5polpmapN 30394 . . 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 457 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  ( M `  ( ( oc `  K ) `  ( U `  X )
) ) )  =  ( M `  (
( oc `  K
) `  ( ( oc `  K ) `  ( U `  X ) ) ) ) )
2111, 2opococ 29678 . . . 4  |-  ( ( K  e.  OP  /\  ( U `  X )  e.  ( Base `  K
) )  ->  (
( oc `  K
) `  ( ( oc `  K ) `  ( U `  X ) ) )  =  ( U `  X ) )
229, 16, 21syl2anc 643 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( oc `  K ) `  (
( oc `  K
) `  ( U `  X ) ) )  =  ( U `  X ) )
2322fveq2d 5691 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( M `  (
( oc `  K
) `  ( ( oc `  K ) `  ( U `  X ) ) ) )  =  ( M `  ( U `  X )
) )
247, 20, 233eqtrd 2440 1  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  (  ._|_  `  X ) )  =  ( M `  ( U `  X )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    C_ wss 3280   ` cfv 5413   Basecbs 13424   occoc 13492   lubclub 14354   CLatccla 14491   OPcops 29655   Atomscatm 29746   HLchlt 29833   pmapcpmap 29979   _|_ PcpolN 30384
This theorem is referenced by:  2polssN  30397  3polN  30398  sspmaplubN  30407  2pmaplubN  30408  paddunN  30409  pnonsingN  30415  pmapidclN  30424  poml4N  30435
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-pmap 29986  df-polarityN 30385
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