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Theorem lhp2at0nle 34831
Description: Inequality for 2 different atoms (or an atom and zero) under co-atom  W. (Contributed by NM, 28-Jul-2013.)
Hypotheses
Ref Expression
lhp2at0nle.l  |-  .<_  =  ( le `  K )
lhp2at0nle.j  |-  .\/  =  ( join `  K )
lhp2at0nle.z  |-  .0.  =  ( 0. `  K )
lhp2at0nle.a  |-  A  =  ( Atoms `  K )
lhp2at0nle.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
lhp2at0nle  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( ( U  e.  A  \/  U  =  .0.  )  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  ->  -.  V  .<_  ( P  .\/  U ) )

Proof of Theorem lhp2at0nle
StepHypRef Expression
1 simpl1 999 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( ( U  e.  A  \/  U  =  .0.  )  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  U  e.  A )  ->  (
( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V ) )
2 simpr 461 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( ( U  e.  A  \/  U  =  .0.  )  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  U  e.  A )  ->  U  e.  A )
3 simpl2r 1050 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( ( U  e.  A  \/  U  =  .0.  )  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  U  e.  A )  ->  U  .<_  W )
4 simpl3 1001 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( ( U  e.  A  \/  U  =  .0.  )  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  U  e.  A )  ->  ( V  e.  A  /\  V  .<_  W ) )
5 lhp2at0nle.l . . . 4  |-  .<_  =  ( le `  K )
6 lhp2at0nle.j . . . 4  |-  .\/  =  ( join `  K )
7 lhp2at0nle.a . . . 4  |-  A  =  ( Atoms `  K )
8 lhp2at0nle.h . . . 4  |-  H  =  ( LHyp `  K
)
95, 6, 7, 8lhp2atnle 34829 . . 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 ) )
101, 2, 3, 4, 9syl121anc 1233 . 2  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( ( U  e.  A  \/  U  =  .0.  )  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  U  e.  A )  ->  -.  V  .<_  ( P  .\/  U ) )
11 simp3r 1025 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( ( U  e.  A  \/  U  =  .0.  )  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  ->  V  .<_  W )
12 simp12r 1110 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( ( U  e.  A  \/  U  =  .0.  )  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  ->  -.  P  .<_  W )
13 nbrne2 4465 . . . . . . 7  |-  ( ( V  .<_  W  /\  -.  P  .<_  W )  ->  V  =/=  P
)
1411, 12, 13syl2anc 661 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( ( U  e.  A  \/  U  =  .0.  )  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  ->  V  =/=  P )
1514neneqd 2669 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( ( U  e.  A  \/  U  =  .0.  )  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  ->  -.  V  =  P )
16 simp11l 1107 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( ( U  e.  A  \/  U  =  .0.  )  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  ->  K  e.  HL )
17 hlatl 34157 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  AtLat )
1816, 17syl 16 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( ( U  e.  A  \/  U  =  .0.  )  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  ->  K  e.  AtLat )
19 simp3l 1024 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( ( U  e.  A  \/  U  =  .0.  )  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  ->  V  e.  A )
20 simp12l 1109 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( ( U  e.  A  \/  U  =  .0.  )  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  ->  P  e.  A )
215, 7atcmp 34108 . . . . . 6  |-  ( ( K  e.  AtLat  /\  V  e.  A  /\  P  e.  A )  ->  ( V  .<_  P  <->  V  =  P ) )
2218, 19, 20, 21syl3anc 1228 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( ( U  e.  A  \/  U  =  .0.  )  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  ->  ( V  .<_  P  <->  V  =  P ) )
2315, 22mtbird 301 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( ( U  e.  A  \/  U  =  .0.  )  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  ->  -.  V  .<_  P )
2423adantr 465 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( ( U  e.  A  \/  U  =  .0.  )  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  U  =  .0.  )  ->  -.  V  .<_  P )
25 oveq2 6290 . . . . 5  |-  ( U  =  .0.  ->  ( P  .\/  U )  =  ( P  .\/  .0.  ) )
26 hlol 34158 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OL )
2716, 26syl 16 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( ( U  e.  A  \/  U  =  .0.  )  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  ->  K  e.  OL )
28 eqid 2467 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2928, 7atbase 34086 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
3020, 29syl 16 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( ( U  e.  A  \/  U  =  .0.  )  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  ->  P  e.  ( Base `  K
) )
31 lhp2at0nle.z . . . . . . 7  |-  .0.  =  ( 0. `  K )
3228, 6, 31olj01 34022 . . . . . 6  |-  ( ( K  e.  OL  /\  P  e.  ( Base `  K ) )  -> 
( P  .\/  .0.  )  =  P )
3327, 30, 32syl2anc 661 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( ( U  e.  A  \/  U  =  .0.  )  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  ->  ( P  .\/  .0.  )  =  P )
3425, 33sylan9eqr 2530 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( ( U  e.  A  \/  U  =  .0.  )  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  U  =  .0.  )  ->  ( P  .\/  U )  =  P )
3534breq2d 4459 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( ( U  e.  A  \/  U  =  .0.  )  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  U  =  .0.  )  ->  ( V  .<_  ( P  .\/  U )  <->  V  .<_  P ) )
3624, 35mtbird 301 . 2  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( ( U  e.  A  \/  U  =  .0.  )  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  U  =  .0.  )  ->  -.  V  .<_  ( P  .\/  U ) )
37 simp2l 1022 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( ( U  e.  A  \/  U  =  .0.  )  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  ->  ( U  e.  A  \/  U  =  .0.  )
)
3810, 36, 37mpjaodan 784 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V
)  /\  ( ( U  e.  A  \/  U  =  .0.  )  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  ->  -.  V  .<_  ( P  .\/  U ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   Basecbs 14486   lecple 14558   joincjn 15427   0.cp0 15520   OLcol 33971   Atomscatm 34060   AtLatcal 34061   HLchlt 34147   LHypclh 34780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-poset 15429  df-plt 15441  df-lub 15457  df-glb 15458  df-join 15459  df-meet 15460  df-p0 15522  df-lat 15529  df-clat 15591  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-psubsp 34299  df-pmap 34300  df-padd 34592  df-lhyp 34784
This theorem is referenced by:  lhp2at0ne  34832  cdlemkfid1N  35717
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