Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lhp2atne Structured version   Unicode version

Theorem lhp2atne 30831
Description: Inequality for joins with 2 different atoms under co-atom  W. (Contributed by NM, 22-Jul-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
lhp2atne  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  U  =/= 
V )  ->  ( P  .\/  U )  =/=  ( Q  .\/  V
) )

Proof of Theorem lhp2atne
StepHypRef Expression
1 simp11 987 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  U  =/= 
V )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp12 988 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  U  =/= 
V )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
3 simp3 959 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  U  =/= 
V )  ->  U  =/=  V )
4 simp2l 983 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  U  =/= 
V )  ->  ( U  e.  A  /\  U  .<_  W ) )
5 simp2r 984 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  U  =/= 
V )  ->  ( V  e.  A  /\  V  .<_  W ) )
6 lhp2atnle.l . . . 4  |-  .<_  =  ( le `  K )
7 lhp2atnle.j . . . 4  |-  .\/  =  ( join `  K )
8 lhp2atnle.a . . . 4  |-  A  =  ( Atoms `  K )
9 lhp2atnle.h . . . 4  |-  H  =  ( LHyp `  K
)
106, 7, 8, 9lhp2atnle 30830 . . 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 ) )
111, 2, 3, 4, 5, 10syl311anc 1198 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  U  =/= 
V )  ->  -.  V  .<_  ( P  .\/  U ) )
12 simp11l 1068 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  U  =/= 
V )  ->  K  e.  HL )
13 simp13 989 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  U  =/= 
V )  ->  Q  e.  A )
14 simp2rl 1026 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  U  =/= 
V )  ->  V  e.  A )
156, 7, 8hlatlej2 30173 . . . . . . 7  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  V  e.  A )  ->  V  .<_  ( Q  .\/  V ) )
1612, 13, 14, 15syl3anc 1184 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  U  =/= 
V )  ->  V  .<_  ( Q  .\/  V
) )
1716adantr 452 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  U  =/= 
V )  /\  ( P  .\/  U )  =  ( Q  .\/  V
) )  ->  V  .<_  ( Q  .\/  V
) )
18 simpr 448 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  U  =/= 
V )  /\  ( P  .\/  U )  =  ( Q  .\/  V
) )  ->  ( P  .\/  U )  =  ( Q  .\/  V
) )
1917, 18breqtrrd 4238 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  U  =/= 
V )  /\  ( P  .\/  U )  =  ( Q  .\/  V
) )  ->  V  .<_  ( P  .\/  U
) )
2019ex 424 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  U  =/= 
V )  ->  (
( P  .\/  U
)  =  ( Q 
.\/  V )  ->  V  .<_  ( P  .\/  U ) ) )
2120necon3bd 2638 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  U  =/= 
V )  ->  ( -.  V  .<_  ( P 
.\/  U )  -> 
( P  .\/  U
)  =/=  ( Q 
.\/  V ) ) )
2211, 21mpd 15 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  U  =/= 
V )  ->  ( P  .\/  U )  =/=  ( Q  .\/  V
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   lecple 13536   joincjn 14401   Atomscatm 30061   HLchlt 30148   LHypclh 30781
This theorem is referenced by:  cdlemg31b0N  31491  cdlemk47  31746
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-psubsp 30300  df-pmap 30301  df-padd 30593  df-lhyp 30785
  Copyright terms: Public domain W3C validator