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Theorem lhple 30853
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 979 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  K  e.  HL )
2 hllat 30175 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
31, 2syl 15 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  K  e.  Lat )
4 simp2l 981 . . . . 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 30101 . . . . 5  |-  ( P  e.  A  ->  P  e.  B )
84, 7syl 15 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  P  e.  B
)
9 simp3l 983 . . . 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 14181 . . . 4  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  X  e.  B )  ->  ( P  .\/  X
)  =  ( X 
.\/  P ) )
123, 8, 9, 11syl3anc 1182 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( P  .\/  X )  =  ( X 
.\/  P ) )
1312oveq1d 5889 . 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 955 . . 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 984 . . 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 30850 . . 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 1187 . 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 2296 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
2216, 17, 21, 6, 18lhpmat 30841 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( P  ./\  W
)  =  ( 0.
`  K ) )
23223adant3 975 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( P  ./\  W )  =  ( 0.
`  K ) )
2423oveq2d 5890 . . 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 30173 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OL )
261, 25syl 15 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  K  e.  OL )
275, 10, 21olj01 30037 . . . 4  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( X  .\/  ( 0. `  K ) )  =  X )
2826, 9, 27syl2anc 642 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( X  .\/  ( 0. `  K ) )  =  X )
2924, 28eqtrd 2328 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( X  .\/  ( P  ./\  W ) )  =  X )
3013, 20, 293eqtr2d 2334 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   meetcmee 14095   0.cp0 14159   Latclat 14167   OLcol 29986   Atomscatm 30075   HLchlt 30162   LHypclh 30795
This theorem is referenced by:  lhpat4N  30855  cdlemn2  32007  dihord5apre  32074
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-psubsp 30314  df-pmap 30315  df-padd 30607  df-lhyp 30799
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