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

Theorem polpmapN 30783
Description: The polarity of a projective map. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
polpmap.b  |-  B  =  ( Base `  K
)
polpmap.o  |-  ._|_  =  ( oc `  K )
polpmap.m  |-  M  =  ( pmap `  K
)
polpmap.p  |-  P  =  ( _|_ P `  K )
Assertion
Ref Expression
polpmapN  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( P `  ( M `  X )
)  =  ( M `
 (  ._|_  `  X
) ) )

Proof of Theorem polpmapN
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 polpmap.b . . . 4  |-  B  =  ( Base `  K
)
2 eqid 2438 . . . 4  |-  ( Atoms `  K )  =  (
Atoms `  K )
3 polpmap.m . . . 4  |-  M  =  ( pmap `  K
)
41, 2, 3pmapssat 30630 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( M `  X
)  C_  ( Atoms `  K ) )
5 eqid 2438 . . . 4  |-  ( lub `  K )  =  ( lub `  K )
6 polpmap.o . . . 4  |-  ._|_  =  ( oc `  K )
7 polpmap.p . . . 4  |-  P  =  ( _|_ P `  K )
85, 6, 2, 3, 7polval2N 30777 . . 3  |-  ( ( K  e.  HL  /\  ( M `  X ) 
C_  ( Atoms `  K
) )  ->  ( P `  ( M `  X ) )  =  ( M `  (  ._|_  `  ( ( lub `  K ) `  ( M `  X )
) ) ) )
94, 8syldan 458 . 2  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( P `  ( M `  X )
)  =  ( M `
 (  ._|_  `  (
( lub `  K
) `  ( M `  X ) ) ) ) )
10 eqid 2438 . . . . . . 7  |-  ( le
`  K )  =  ( le `  K
)
111, 10, 2, 3pmapval 30628 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( M `  X
)  =  { p  e.  ( Atoms `  K )  |  p ( le `  K ) X }
)
1211fveq2d 5735 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( ( lub `  K
) `  ( M `  X ) )  =  ( ( lub `  K
) `  { p  e.  ( Atoms `  K )  |  p ( le `  K ) X }
) )
13 hlomcmat 30236 . . . . . 6  |-  ( K  e.  HL  ->  ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat
) )
141, 10, 5, 2atlatmstc 30191 . . . . . 6  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat )  /\  X  e.  B )  ->  (
( lub `  K
) `  { p  e.  ( Atoms `  K )  |  p ( le `  K ) X }
)  =  X )
1513, 14sylan 459 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( ( lub `  K
) `  { p  e.  ( Atoms `  K )  |  p ( le `  K ) X }
)  =  X )
1612, 15eqtrd 2470 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( ( lub `  K
) `  ( M `  X ) )  =  X )
1716fveq2d 5735 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  (  ._|_  `  ( ( lub `  K ) `
 ( M `  X ) ) )  =  (  ._|_  `  X
) )
1817fveq2d 5735 . 2  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( M `  (  ._|_  `  ( ( lub `  K ) `  ( M `  X )
) ) )  =  ( M `  (  ._|_  `  X ) ) )
199, 18eqtrd 2470 1  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( P `  ( M `  X )
)  =  ( M `
 (  ._|_  `  X
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   {crab 2711    C_ wss 3322   class class class wbr 4215   ` cfv 5457   Basecbs 13474   lecple 13541   occoc 13542   lubclub 14404   CLatccla 14541   OMLcoml 30047   Atomscatm 30135   AtLatcal 30136   HLchlt 30222   pmapcpmap 30368   _|_
PcpolN 30773
This theorem is referenced by:  2polpmapN  30784  2polvalN  30785  3polN  30787  pmapj2N  30800  pmapocjN  30801  2polatN  30803  poml4N  30824  pmapojoinN  30839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-poset 14408  df-plt 14420  df-lub 14436  df-glb 14437  df-join 14438  df-meet 14439  df-p0 14473  df-p1 14474  df-lat 14480  df-clat 14542  df-oposet 30048  df-ol 30050  df-oml 30051  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223  df-pmap 30375  df-polarityN 30774
  Copyright terms: Public domain W3C validator