Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  2polvalN Structured version   Unicode version

Theorem 2polvalN 33897
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 2454 . . . 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 33889 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  X )  =  ( M `  ( ( oc `  K ) `  ( U `  X )
) ) )
76fveq2d 5804 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  (  ._|_  `  X ) )  =  (  ._|_  `  ( M `
 ( ( oc
`  K ) `  ( U `  X ) ) ) ) )
8 hlop 33346 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
98adantr 465 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  K  e.  OP )
10 hlclat 33342 . . . . 5  |-  ( K  e.  HL  ->  K  e.  CLat )
11 eqid 2454 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1211, 3atssbase 33274 . . . . . 6  |-  A  C_  ( Base `  K )
13 sstr 3473 . . . . . 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 15402 . . . . 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 33178 . . . 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 33895 . . 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 33179 . . . 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 5804 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( M `  (
( oc `  K
) `  ( ( oc `  K ) `  ( U `  X ) ) ) )  =  ( M `  ( U `  X )
) )
247, 20, 233eqtrd 2499 1  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  (  ._|_  `  X ) )  =  ( M `  ( U `  X )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    C_ wss 3437   ` cfv 5527   Basecbs 14293   occoc 14366   lubclub 15232   CLatccla 15397   OPcops 33156   Atomscatm 33247   HLchlt 33334   pmapcpmap 33480   _|_PcpolN 33885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-riotaBAD 32943
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-iin 4283  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-undef 6903  df-poset 15236  df-plt 15248  df-lub 15264  df-glb 15265  df-join 15266  df-meet 15267  df-p0 15329  df-p1 15330  df-lat 15336  df-clat 15398  df-oposet 33160  df-ol 33162  df-oml 33163  df-covers 33250  df-ats 33251  df-atl 33282  df-cvlat 33306  df-hlat 33335  df-pmap 33487  df-polarityN 33886
This theorem is referenced by:  2polssN  33898  3polN  33899  sspmaplubN  33908  2pmaplubN  33909  paddunN  33910  pnonsingN  33916  pmapidclN  33925  poml4N  33936
  Copyright terms: Public domain W3C validator