Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lhp2atnle Unicode version

Theorem lhp2atnle 30293
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 1067 . . . 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 29621 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
31, 2syl 15 . . 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 984 . . 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 2366 . . . 4  |-  ( 0.
`  K )  =  ( 0. `  K
)
6 lhp2atnle.a . . . 4  |-  A  =  ( Atoms `  K )
75, 6atn0 29569 . . 3  |-  ( ( K  e.  AtLat  /\  V  e.  A )  ->  V  =/=  ( 0. `  K
) )
83, 4, 7syl2anc 642 . 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 29624 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
101, 9syl 15 . . . . 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 2366 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1211, 6atbase 29550 . . . . . 6  |-  ( V  e.  A  ->  V  e.  ( Base `  K
) )
134, 12syl 15 . . . . 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 1069 . . . . . 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 982 . . . . . 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 29627 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  U  e.  A )  ->  ( P  .\/  U
)  e.  ( Base `  K ) )
181, 14, 15, 17syl3anc 1183 . . . . 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 2366 . . . . . 6  |-  ( meet `  K )  =  (
meet `  K )
2111, 19, 20latleeqm2 14396 . . . . 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 1183 . . . 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 30292 . . . . 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 2372 . . . . 5  |-  ( ( ( P  .\/  U
) ( meet `  K
) V )  =  V  ->  ( (
( P  .\/  U
) ( meet `  K
) V )  =  ( 0. `  K
)  <->  V  =  ( 0. `  K ) ) )
2624, 25syl5ibcom 211 . . . 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 206 . . 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 2565 . 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 14 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715    =/= wne 2529   class class class wbr 4125   ` cfv 5358  (class class class)co 5981   Basecbs 13356   lecple 13423   joincjn 14288   meetcmee 14289   0.cp0 14353   Latclat 14361   Atomscatm 29524   AtLatcal 29525   HLchlt 29611   LHypclh 30244
This theorem is referenced by:  lhp2atne  30294  lhp2at0nle  30295  cdlemg27a  30952  cdlemg31c  30959  cdlemh  31077  cdlemk12  31110  cdlemk12u  31132
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-iin 4010  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-undef 6440  df-riota 6446  df-poset 14290  df-plt 14302  df-lub 14318  df-glb 14319  df-join 14320  df-meet 14321  df-p0 14355  df-lat 14362  df-clat 14424  df-oposet 29437  df-ol 29439  df-oml 29440  df-covers 29527  df-ats 29528  df-atl 29559  df-cvlat 29583  df-hlat 29612  df-psubsp 29763  df-pmap 29764  df-padd 30056  df-lhyp 30248
  Copyright terms: Public domain W3C validator