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Theorem pmapocjN 33572
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 33571 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( F `  ( X  .\/  Y ) )  =  ( N `  ( N `  ( ( F `  X ) 
.+  ( F `  Y ) ) ) ) )
76fveq2d 5694 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  ( F `  ( X  .\/  Y ) ) )  =  ( N `  ( N `  ( N `
 ( ( F `
 X )  .+  ( F `  Y ) ) ) ) ) )
8 simp1 988 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  HL )
9 hllat 33006 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
101, 2latjcl 15220 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
119, 10syl3an1 1251 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
12 pmapocj.o . . . 4  |-  ._|_  =  ( oc `  K )
131, 12, 3, 5polpmapN 33554 . . 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 2442 . . . . . 6  |-  ( Atoms `  K )  =  (
Atoms `  K )
161, 15, 3pmapssat 33401 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( F `  X
)  C_  ( Atoms `  K ) )
17163adant3 1008 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( F `  X
)  C_  ( Atoms `  K ) )
181, 15, 3pmapssat 33401 . . . . 5  |-  ( ( K  e.  HL  /\  Y  e.  B )  ->  ( F `  Y
)  C_  ( Atoms `  K ) )
19183adant2 1007 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( F `  Y
)  C_  ( Atoms `  K ) )
2015, 4paddssat 33456 . . . 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 1218 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( F `  X )  .+  ( F `  Y )
)  C_  ( Atoms `  K ) )
2215, 53polN 33558 . . 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 2482 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 965    = wceq 1369    e. wcel 1756    C_ wss 3327   ` cfv 5417  (class class class)co 6090   Basecbs 14173   occoc 14245   joincjn 15113   meetcmee 15114   Latclat 15214   Atomscatm 32906   HLchlt 32993   pmapcpmap 33139   +Pcpadd 33437   _|_PcpolN 33544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-riotaBAD 32602
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-iin 4173  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-undef 6791  df-poset 15115  df-plt 15127  df-lub 15143  df-glb 15144  df-join 15145  df-meet 15146  df-p0 15208  df-p1 15209  df-lat 15215  df-clat 15277  df-oposet 32819  df-ol 32821  df-oml 32822  df-covers 32909  df-ats 32910  df-atl 32941  df-cvlat 32965  df-hlat 32994  df-psubsp 33145  df-pmap 33146  df-padd 33438  df-polarityN 33545
This theorem is referenced by: (None)
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