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Theorem lhple 33526
Description: Property of a lattice element under a co-atom. (Contributed by NM, 28-Feb-2014.)
Hypotheses
Ref Expression
lhple.b  |-  B  =  ( Base `  K
)
lhple.l  |-  .<_  =  ( le `  K )
lhple.j  |-  .\/  =  ( join `  K )
lhple.m  |-  ./\  =  ( meet `  K )
lhple.a  |-  A  =  ( Atoms `  K )
lhple.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
lhple  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( ( P 
.\/  X )  ./\  W )  =  X )

Proof of Theorem lhple
StepHypRef Expression
1 simp1l 1012 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  K  e.  HL )
2 hllat 32848 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
31, 2syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  K  e.  Lat )
4 simp2l 1014 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  P  e.  A
)
5 lhple.b . . . . . 6  |-  B  =  ( Base `  K
)
6 lhple.a . . . . . 6  |-  A  =  ( Atoms `  K )
75, 6atbase 32774 . . . . 5  |-  ( P  e.  A  ->  P  e.  B )
84, 7syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  P  e.  B
)
9 simp3l 1016 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  X  e.  B
)
10 lhple.j . . . . 5  |-  .\/  =  ( join `  K )
115, 10latjcom 15221 . . . 4  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  X  e.  B )  ->  ( P  .\/  X
)  =  ( X 
.\/  P ) )
123, 8, 9, 11syl3anc 1218 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( P  .\/  X )  =  ( X 
.\/  P ) )
1312oveq1d 6101 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( ( P 
.\/  X )  ./\  W )  =  ( ( X  .\/  P ) 
./\  W ) )
14 simp1 988 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
15 simp3r 1017 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  X  .<_  W )
16 lhple.l . . . 4  |-  .<_  =  ( le `  K )
17 lhple.m . . . 4  |-  ./\  =  ( meet `  K )
18 lhple.h . . . 4  |-  H  =  ( LHyp `  K
)
195, 16, 10, 17, 18lhpmod6i1 33523 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  P  e.  B )  /\  X  .<_  W )  ->  ( X  .\/  ( P  ./\  W ) )  =  ( ( X  .\/  P
)  ./\  W )
)
2014, 9, 8, 15, 19syl121anc 1223 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( X  .\/  ( P  ./\  W ) )  =  ( ( X  .\/  P ) 
./\  W ) )
21 eqid 2438 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
2216, 17, 21, 6, 18lhpmat 33514 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( P  ./\  W
)  =  ( 0.
`  K ) )
23223adant3 1008 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( P  ./\  W )  =  ( 0.
`  K ) )
2423oveq2d 6102 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( X  .\/  ( P  ./\  W ) )  =  ( X 
.\/  ( 0. `  K ) ) )
25 hlol 32846 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OL )
261, 25syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  K  e.  OL )
275, 10, 21olj01 32710 . . . 4  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( X  .\/  ( 0. `  K ) )  =  X )
2826, 9, 27syl2anc 661 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( X  .\/  ( 0. `  K ) )  =  X )
2924, 28eqtrd 2470 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( X  .\/  ( P  ./\  W ) )  =  X )
3013, 20, 293eqtr2d 2476 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( ( P 
.\/  X )  ./\  W )  =  X )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   Basecbs 14166   lecple 14237   joincjn 15106   meetcmee 15107   0.cp0 15199   Latclat 15207   OLcol 32659   Atomscatm 32748   HLchlt 32835   LHypclh 33468
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-1st 6572  df-2nd 6573  df-poset 15108  df-plt 15120  df-lub 15136  df-glb 15137  df-join 15138  df-meet 15139  df-p0 15201  df-p1 15202  df-lat 15208  df-clat 15270  df-oposet 32661  df-ol 32663  df-oml 32664  df-covers 32751  df-ats 32752  df-atl 32783  df-cvlat 32807  df-hlat 32836  df-psubsp 32987  df-pmap 32988  df-padd 33280  df-lhyp 33472
This theorem is referenced by:  lhpat4N  33528  cdlemn2  34680  dihord5apre  34747
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