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Theorem pmapocjN 35755
Description: The projective map of the orthocomplement of the join of two lattice elements. (Contributed by NM, 14-Mar-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
pmapocj.b  |-  B  =  ( Base `  K
)
pmapocj.j  |-  .\/  =  ( join `  K )
pmapocj.m  |-  ./\  =  ( meet `  K )
pmapocj.o  |-  ._|_  =  ( oc `  K )
pmapocj.f  |-  F  =  ( pmap `  K
)
pmapocj.p  |-  .+  =  ( +P `  K
)
pmapocj.r  |-  N  =  ( _|_P `  K )
Assertion
Ref Expression
pmapocjN  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( F `  (  ._|_  `  ( X  .\/  Y ) ) )  =  ( N `  (
( F `  X
)  .+  ( F `  Y ) ) ) )

Proof of Theorem pmapocjN
StepHypRef Expression
1 pmapocj.b . . . 4  |-  B  =  ( Base `  K
)
2 pmapocj.j . . . 4  |-  .\/  =  ( join `  K )
3 pmapocj.f . . . 4  |-  F  =  ( pmap `  K
)
4 pmapocj.p . . . 4  |-  .+  =  ( +P `  K
)
5 pmapocj.r . . . 4  |-  N  =  ( _|_P `  K )
61, 2, 3, 4, 5pmapj2N 35754 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( F `  ( X  .\/  Y ) )  =  ( N `  ( N `  ( ( F `  X ) 
.+  ( F `  Y ) ) ) ) )
76fveq2d 5876 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  ( F `  ( X  .\/  Y ) ) )  =  ( N `  ( N `  ( N `
 ( ( F `
 X )  .+  ( F `  Y ) ) ) ) ) )
8 simp1 996 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  HL )
9 hllat 35189 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
101, 2latjcl 15807 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
119, 10syl3an1 1261 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
12 pmapocj.o . . . 4  |-  ._|_  =  ( oc `  K )
131, 12, 3, 5polpmapN 35737 . . 3  |-  ( ( K  e.  HL  /\  ( X  .\/  Y )  e.  B )  -> 
( N `  ( F `  ( X  .\/  Y ) ) )  =  ( F `  (  ._|_  `  ( X  .\/  Y ) ) ) )
148, 11, 13syl2anc 661 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  ( F `  ( X  .\/  Y ) ) )  =  ( F `  (  ._|_  `  ( X  .\/  Y ) ) ) )
15 eqid 2457 . . . . . 6  |-  ( Atoms `  K )  =  (
Atoms `  K )
161, 15, 3pmapssat 35584 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( F `  X
)  C_  ( Atoms `  K ) )
17163adant3 1016 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( F `  X
)  C_  ( Atoms `  K ) )
181, 15, 3pmapssat 35584 . . . . 5  |-  ( ( K  e.  HL  /\  Y  e.  B )  ->  ( F `  Y
)  C_  ( Atoms `  K ) )
19183adant2 1015 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( F `  Y
)  C_  ( Atoms `  K ) )
2015, 4paddssat 35639 . . . 4  |-  ( ( K  e.  HL  /\  ( F `  X ) 
C_  ( Atoms `  K
)  /\  ( F `  Y )  C_  ( Atoms `  K ) )  ->  ( ( F `
 X )  .+  ( F `  Y ) )  C_  ( Atoms `  K ) )
218, 17, 19, 20syl3anc 1228 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( F `  X )  .+  ( F `  Y )
)  C_  ( Atoms `  K ) )
2215, 53polN 35741 . . 3  |-  ( ( K  e.  HL  /\  ( ( F `  X )  .+  ( F `  Y )
)  C_  ( Atoms `  K ) )  -> 
( N `  ( N `  ( N `  ( ( F `  X )  .+  ( F `  Y )
) ) ) )  =  ( N `  ( ( F `  X )  .+  ( F `  Y )
) ) )
238, 21, 22syl2anc 661 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  ( N `  ( N `  ( ( F `  X )  .+  ( F `  Y )
) ) ) )  =  ( N `  ( ( F `  X )  .+  ( F `  Y )
) ) )
247, 14, 233eqtr3d 2506 1  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( F `  (  ._|_  `  ( X  .\/  Y ) ) )  =  ( N `  (
( F `  X
)  .+  ( F `  Y ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1395    e. wcel 1819    C_ wss 3471   ` cfv 5594  (class class class)co 6296   Basecbs 14643   occoc 14719   joincjn 15699   meetcmee 15700   Latclat 15801   Atomscatm 35089   HLchlt 35176   pmapcpmap 35322   +Pcpadd 35620   _|_PcpolN 35727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-riotaBAD 34785
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-undef 7020  df-preset 15683  df-poset 15701  df-plt 15714  df-lub 15730  df-glb 15731  df-join 15732  df-meet 15733  df-p0 15795  df-p1 15796  df-lat 15802  df-clat 15864  df-oposet 35002  df-ol 35004  df-oml 35005  df-covers 35092  df-ats 35093  df-atl 35124  df-cvlat 35148  df-hlat 35177  df-psubsp 35328  df-pmap 35329  df-padd 35621  df-polarityN 35728
This theorem is referenced by: (None)
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