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Theorem polpmapN 30398
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 2408 . . . 4  |-  ( Atoms `  K )  =  (
Atoms `  K )
3 polpmap.m . . . 4  |-  M  =  ( pmap `  K
)
41, 2, 3pmapssat 30245 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( M `  X
)  C_  ( Atoms `  K ) )
5 eqid 2408 . . . 4  |-  ( lub `  K )  =  ( lub `  K )
6 polpmap.o . . . 4  |-  ._|_  =  ( oc `  K )
7 polpmap.p . . . 4  |-  P  =  ( _|_ P `  K )
85, 6, 2, 3, 7polval2N 30392 . . 3  |-  ( ( K  e.  HL  /\  ( M `  X ) 
C_  ( Atoms `  K
) )  ->  ( P `  ( M `  X ) )  =  ( M `  (  ._|_  `  ( ( lub `  K ) `  ( M `  X )
) ) ) )
94, 8syldan 457 . 2  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( P `  ( M `  X )
)  =  ( M `
 (  ._|_  `  (
( lub `  K
) `  ( M `  X ) ) ) ) )
10 eqid 2408 . . . . . . 7  |-  ( le
`  K )  =  ( le `  K
)
111, 10, 2, 3pmapval 30243 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( M `  X
)  =  { p  e.  ( Atoms `  K )  |  p ( le `  K ) X }
)
1211fveq2d 5695 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( ( lub `  K
) `  ( M `  X ) )  =  ( ( lub `  K
) `  { p  e.  ( Atoms `  K )  |  p ( le `  K ) X }
) )
13 hlomcmat 29851 . . . . . 6  |-  ( K  e.  HL  ->  ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat
) )
141, 10, 5, 2atlatmstc 29806 . . . . . 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 458 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( ( lub `  K
) `  { p  e.  ( Atoms `  K )  |  p ( le `  K ) X }
)  =  X )
1612, 15eqtrd 2440 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( ( lub `  K
) `  ( M `  X ) )  =  X )
1716fveq2d 5695 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  (  ._|_  `  ( ( lub `  K ) `
 ( M `  X ) ) )  =  (  ._|_  `  X
) )
1817fveq2d 5695 . 2  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( M `  (  ._|_  `  ( ( lub `  K ) `  ( M `  X )
) ) )  =  ( M `  (  ._|_  `  X ) ) )
199, 18eqtrd 2440 1  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( P `  ( M `  X )
)  =  ( M `
 (  ._|_  `  X
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   {crab 2674    C_ wss 3284   class class class wbr 4176   ` cfv 5417   Basecbs 13428   lecple 13495   occoc 13496   lubclub 14358   CLatccla 14495   OMLcoml 29662   Atomscatm 29750   AtLatcal 29751   HLchlt 29837   pmapcpmap 29983   _|_
PcpolN 30388
This theorem is referenced by:  2polpmapN  30399  2polvalN  30400  3polN  30402  pmapj2N  30415  pmapocjN  30416  2polatN  30418  poml4N  30439  pmapojoinN  30454
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 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-iin 4060  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-undef 6506  df-riota 6512  df-poset 14362  df-plt 14374  df-lub 14390  df-glb 14391  df-join 14392  df-meet 14393  df-p0 14427  df-p1 14428  df-lat 14434  df-clat 14496  df-oposet 29663  df-ol 29665  df-oml 29666  df-covers 29753  df-ats 29754  df-atl 29785  df-cvlat 29809  df-hlat 29838  df-pmap 29990  df-polarityN 30389
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