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Theorem lhp2atnle 33510
Description: Inequality for 2 different atoms under co-atom  W. (Contributed by NM, 17-Jun-2013.)
Hypotheses
Ref Expression
lhp2atnle.l  |-  .<_  =  ( le `  K )
lhp2atnle.j  |-  .\/  =  ( join `  K )
lhp2atnle.a  |-  A  =  ( Atoms `  K )
lhp2atnle.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
lhp2atnle  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  ->  -.  V  .<_  ( P 
.\/  U ) )

Proof of Theorem lhp2atnle
StepHypRef Expression
1 simp11l 1116 . . . 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.  HL )
2 hlatl 32838 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
31, 2syl 17 . . 3  |-  ( ( ( ( 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 )
4 simp3l 1033 . . 3  |-  ( ( ( ( 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 )
5 eqid 2428 . . . 4  |-  ( 0.
`  K )  =  ( 0. `  K
)
6 lhp2atnle.a . . . 4  |-  A  =  ( Atoms `  K )
75, 6atn0 32786 . . 3  |-  ( ( K  e.  AtLat  /\  V  e.  A )  ->  V  =/=  ( 0. `  K
) )
83, 4, 7syl2anc 665 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  ->  V  =/=  ( 0. `  K ) )
9 hllat 32841 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
101, 9syl 17 . . . . 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 )
11 eqid 2428 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1211, 6atbase 32767 . . . . . 6  |-  ( V  e.  A  ->  V  e.  ( Base `  K
) )
134, 12syl 17 . . . . 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.  ( Base `  K ) )
14 simp12l 1118 . . . . . 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  e.  A )
15 simp2l 1031 . . . . . 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.  A )
16 lhp2atnle.j . . . . . . 7  |-  .\/  =  ( join `  K )
1711, 16, 6hlatjcl 32844 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  U  e.  A )  ->  ( P  .\/  U
)  e.  ( Base `  K ) )
181, 14, 15, 17syl3anc 1264 . . . . 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  .\/  U
)  e.  ( Base `  K ) )
19 lhp2atnle.l . . . . . 6  |-  .<_  =  ( le `  K )
20 eqid 2428 . . . . . 6  |-  ( meet `  K )  =  (
meet `  K )
2111, 19, 20latleeqm2 16269 . . . . 5  |-  ( ( K  e.  Lat  /\  V  e.  ( Base `  K )  /\  ( P  .\/  U )  e.  ( Base `  K
) )  ->  ( V  .<_  ( P  .\/  U )  <->  ( ( P 
.\/  U ) (
meet `  K ) V )  =  V ) )
2210, 13, 18, 21syl3anc 1264 . . . 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  .<_  ( P 
.\/  U )  <->  ( ( P  .\/  U ) (
meet `  K ) V )  =  V ) )
23 lhp2atnle.h . . . . . 6  |-  H  =  ( LHyp `  K
)
2419, 16, 20, 5, 6, 23lhp2at0 33509 . . . . 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  .\/  U ) ( meet `  K
) V )  =  ( 0. `  K
) )
25 eqeq1 2432 . . . . 5  |-  ( ( ( P  .\/  U
) ( meet `  K
) V )  =  V  ->  ( (
( P  .\/  U
) ( meet `  K
) V )  =  ( 0. `  K
)  <->  V  =  ( 0. `  K ) ) )
2624, 25syl5ibcom 223 . . . 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 ) (
meet `  K ) V )  =  V  ->  V  =  ( 0. `  K ) ) )
2722, 26sylbid 218 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  -> 
( V  .<_  ( P 
.\/  U )  ->  V  =  ( 0. `  K ) ) )
2827necon3ad 2614 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  -> 
( V  =/=  ( 0. `  K )  ->  -.  V  .<_  ( P 
.\/  U ) ) )
298, 28mpd 15 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  ->  -.  V  .<_  ( P 
.\/  U ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2599   class class class wbr 4366   ` cfv 5544  (class class class)co 6249   Basecbs 15064   lecple 15140   joincjn 16132   meetcmee 16133   0.cp0 16226   Latclat 16234   Atomscatm 32741   AtLatcal 32742   HLchlt 32828   LHypclh 33461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-iun 4244  df-iin 4245  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-1st 6751  df-2nd 6752  df-preset 16116  df-poset 16134  df-plt 16147  df-lub 16163  df-glb 16164  df-join 16165  df-meet 16166  df-p0 16228  df-lat 16235  df-clat 16297  df-oposet 32654  df-ol 32656  df-oml 32657  df-covers 32744  df-ats 32745  df-atl 32776  df-cvlat 32800  df-hlat 32829  df-psubsp 32980  df-pmap 32981  df-padd 33273  df-lhyp 33465
This theorem is referenced by:  lhp2atne  33511  lhp2at0nle  33512  cdlemg27a  34171  cdlemg31c  34178  cdlemh  34296  cdlemk12  34329  cdlemk12u  34351
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