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Theorem lhp2at0 30903
Description: Join and meet with different atoms under co-atom  W. (Contributed by NM, 15-Jun-2013.)
Hypotheses
Ref Expression
lhp2at0.l  |-  .<_  =  ( le `  K )
lhp2at0.j  |-  .\/  =  ( join `  K )
lhp2at0.m  |-  ./\  =  ( meet `  K )
lhp2at0.z  |-  .0.  =  ( 0. `  K )
lhp2at0.a  |-  A  =  ( Atoms `  K )
lhp2at0.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
lhp2at0  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  -> 
( ( P  .\/  U )  ./\  V )  =  .0.  )

Proof of Theorem lhp2at0
StepHypRef Expression
1 simp11l 1069 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  ->  K  e.  HL )
2 hlol 30233 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OL )
31, 2syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  ->  K  e.  OL )
4 simp12l 1071 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  ->  P  e.  A )
5 simp2l 984 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  ->  U  e.  A )
6 eqid 2438 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
7 lhp2at0.j . . . . . 6  |-  .\/  =  ( join `  K )
8 lhp2at0.a . . . . . 6  |-  A  =  ( Atoms `  K )
96, 7, 8hlatjcl 30238 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  U  e.  A )  ->  ( P  .\/  U
)  e.  ( Base `  K ) )
101, 4, 5, 9syl3anc 1185 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  -> 
( P  .\/  U
)  e.  ( Base `  K ) )
11 simp11r 1070 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  ->  W  e.  H )
12 lhp2at0.h . . . . . 6  |-  H  =  ( LHyp `  K
)
136, 12lhpbase 30869 . . . . 5  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
1411, 13syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  ->  W  e.  ( Base `  K ) )
15 simp3l 986 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  ->  V  e.  A )
166, 8atbase 30161 . . . . 5  |-  ( V  e.  A  ->  V  e.  ( Base `  K
) )
1715, 16syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  ->  V  e.  ( Base `  K ) )
18 lhp2at0.m . . . . 5  |-  ./\  =  ( meet `  K )
196, 18latmassOLD 30101 . . . 4  |-  ( ( K  e.  OL  /\  ( ( P  .\/  U )  e.  ( Base `  K )  /\  W  e.  ( Base `  K
)  /\  V  e.  ( Base `  K )
) )  ->  (
( ( P  .\/  U )  ./\  W )  ./\  V )  =  ( ( P  .\/  U
)  ./\  ( W  ./\ 
V ) ) )
203, 10, 14, 17, 19syl13anc 1187 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  -> 
( ( ( P 
.\/  U )  ./\  W )  ./\  V )  =  ( ( P 
.\/  U )  ./\  ( W  ./\  V ) ) )
21 lhp2at0.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
22 lhp2at0.z . . . . . . . . 9  |-  .0.  =  ( 0. `  K )
2321, 18, 22, 8, 12lhpmat 30901 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( P  ./\  W
)  =  .0.  )
24233adant3 978 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V )  ->  ( P  ./\  W )  =  .0.  )
25243ad2ant1 979 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  -> 
( P  ./\  W
)  =  .0.  )
2625oveq1d 6099 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  -> 
( ( P  ./\  W )  .\/  U )  =  (  .0.  .\/  U ) )
276, 8atbase 30161 . . . . . . 7  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
285, 27syl 16 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  ->  U  e.  ( Base `  K ) )
29 simp2r 985 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  ->  U  .<_  W )
306, 21, 7, 18, 8atmod4i2 30738 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  U  e.  ( Base `  K )  /\  W  e.  ( Base `  K ) )  /\  U  .<_  W )  -> 
( ( P  ./\  W )  .\/  U )  =  ( ( P 
.\/  U )  ./\  W ) )
311, 4, 28, 14, 29, 30syl131anc 1198 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  -> 
( ( P  ./\  W )  .\/  U )  =  ( ( P 
.\/  U )  ./\  W ) )
326, 7, 22olj02 30098 . . . . . 6  |-  ( ( K  e.  OL  /\  U  e.  ( Base `  K ) )  -> 
(  .0.  .\/  U
)  =  U )
333, 28, 32syl2anc 644 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  -> 
(  .0.  .\/  U
)  =  U )
3426, 31, 333eqtr3d 2478 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  -> 
( ( P  .\/  U )  ./\  W )  =  U )
3534oveq1d 6099 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  -> 
( ( ( P 
.\/  U )  ./\  W )  ./\  V )  =  ( U  ./\  V ) )
3620, 35eqtr3d 2472 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  -> 
( ( P  .\/  U )  ./\  ( W  ./\ 
V ) )  =  ( U  ./\  V
) )
37 simp3r 987 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  ->  V  .<_  W )
38 hllat 30235 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
391, 38syl 16 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  ->  K  e.  Lat )
406, 21, 18latleeqm2 14514 . . . . 5  |-  ( ( K  e.  Lat  /\  V  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) )  ->  ( V  .<_  W  <->  ( W  ./\ 
V )  =  V ) )
4139, 17, 14, 40syl3anc 1185 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  -> 
( V  .<_  W  <->  ( W  ./\ 
V )  =  V ) )
4237, 41mpbid 203 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  -> 
( W  ./\  V
)  =  V )
4342oveq2d 6100 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  -> 
( ( P  .\/  U )  ./\  ( W  ./\ 
V ) )  =  ( ( P  .\/  U )  ./\  V )
)
44 simp13 990 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  ->  U  =/=  V )
45 hlatl 30232 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
461, 45syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  ->  K  e.  AtLat )
4718, 22, 8atnem0 30190 . . . 4  |-  ( ( K  e.  AtLat  /\  U  e.  A  /\  V  e.  A )  ->  ( U  =/=  V  <->  ( U  ./\ 
V )  =  .0.  ) )
4846, 5, 15, 47syl3anc 1185 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  -> 
( U  =/=  V  <->  ( U  ./\  V )  =  .0.  ) )
4944, 48mpbid 203 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  -> 
( U  ./\  V
)  =  .0.  )
5036, 43, 493eqtr3d 2478 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  -> 
( ( P  .\/  U )  ./\  V )  =  .0.  )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   Basecbs 13474   lecple 13541   joincjn 14406   meetcmee 14407   0.cp0 14471   Latclat 14479   OLcol 30046   Atomscatm 30135   AtLatcal 30136   HLchlt 30222   LHypclh 30855
This theorem is referenced by:  lhp2atnle  30904  cdlemh2  31687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-poset 14408  df-plt 14420  df-lub 14436  df-glb 14437  df-join 14438  df-meet 14439  df-p0 14473  df-lat 14480  df-clat 14542  df-oposet 30048  df-ol 30050  df-oml 30051  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223  df-psubsp 30374  df-pmap 30375  df-padd 30667  df-lhyp 30859
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