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

Theorem dalemqnet 29767
Description: Lemma for dath 29851. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
Hypotheses
Ref Expression
dalema.ph  |-  ( ph  <->  ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  /\  ( Y  e.  O  /\  Z  e.  O )  /\  ( ( -.  C  .<_  ( P  .\/  Q
)  /\  -.  C  .<_  ( Q  .\/  R
)  /\  -.  C  .<_  ( R  .\/  P
) )  /\  ( -.  C  .<_  ( S 
.\/  T )  /\  -.  C  .<_  ( T 
.\/  U )  /\  -.  C  .<_  ( U 
.\/  S ) )  /\  ( C  .<_  ( P  .\/  S )  /\  C  .<_  ( Q 
.\/  T )  /\  C  .<_  ( R  .\/  U ) ) ) ) )
dalemc.l  |-  .<_  =  ( le `  K )
dalemc.j  |-  .\/  =  ( join `  K )
dalemc.a  |-  A  =  ( Atoms `  K )
dalempnes.o  |-  O  =  ( LPlanes `  K )
dalempnes.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
Assertion
Ref Expression
dalemqnet  |-  ( ph  ->  Q  =/=  T )

Proof of Theorem dalemqnet
StepHypRef Expression
1 dalema.ph . . . 4  |-  ( ph  <->  ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  /\  ( Y  e.  O  /\  Z  e.  O )  /\  ( ( -.  C  .<_  ( P  .\/  Q
)  /\  -.  C  .<_  ( Q  .\/  R
)  /\  -.  C  .<_  ( R  .\/  P
) )  /\  ( -.  C  .<_  ( S 
.\/  T )  /\  -.  C  .<_  ( T 
.\/  U )  /\  -.  C  .<_  ( U 
.\/  S ) )  /\  ( C  .<_  ( P  .\/  S )  /\  C  .<_  ( Q 
.\/  T )  /\  C  .<_  ( R  .\/  U ) ) ) ) )
21dalemkelat 29739 . . 3  |-  ( ph  ->  K  e.  Lat )
3 dalemc.a . . . 4  |-  A  =  ( Atoms `  K )
41, 3dalemceb 29753 . . 3  |-  ( ph  ->  C  e.  ( Base `  K ) )
51, 3dalemteb 29758 . . 3  |-  ( ph  ->  T  e.  ( Base `  K ) )
61, 3dalemueb 29759 . . 3  |-  ( ph  ->  U  e.  ( Base `  K ) )
7 simp322 1108 . . . 4  |-  ( ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  /\  ( Y  e.  O  /\  Z  e.  O )  /\  ( ( -.  C  .<_  ( P  .\/  Q
)  /\  -.  C  .<_  ( Q  .\/  R
)  /\  -.  C  .<_  ( R  .\/  P
) )  /\  ( -.  C  .<_  ( S 
.\/  T )  /\  -.  C  .<_  ( T 
.\/  U )  /\  -.  C  .<_  ( U 
.\/  S ) )  /\  ( C  .<_  ( P  .\/  S )  /\  C  .<_  ( Q 
.\/  T )  /\  C  .<_  ( R  .\/  U ) ) ) )  ->  -.  C  .<_  ( T  .\/  U ) )
81, 7sylbi 188 . . 3  |-  ( ph  ->  -.  C  .<_  ( T 
.\/  U ) )
9 eqid 2388 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
10 dalemc.l . . . 4  |-  .<_  =  ( le `  K )
11 dalemc.j . . . 4  |-  .\/  =  ( join `  K )
129, 10, 11latnlej2l 14429 . . 3  |-  ( ( K  e.  Lat  /\  ( C  e.  ( Base `  K )  /\  T  e.  ( Base `  K )  /\  U  e.  ( Base `  K
) )  /\  -.  C  .<_  ( T  .\/  U ) )  ->  -.  C  .<_  T )
132, 4, 5, 6, 8, 12syl131anc 1197 . 2  |-  ( ph  ->  -.  C  .<_  T )
141dalemclqjt 29750 . . . . 5  |-  ( ph  ->  C  .<_  ( Q  .\/  T ) )
15 oveq1 6028 . . . . . 6  |-  ( Q  =  T  ->  ( Q  .\/  T )  =  ( T  .\/  T
) )
1615breq2d 4166 . . . . 5  |-  ( Q  =  T  ->  ( C  .<_  ( Q  .\/  T )  <->  C  .<_  ( T 
.\/  T ) ) )
1714, 16syl5ibcom 212 . . . 4  |-  ( ph  ->  ( Q  =  T  ->  C  .<_  ( T 
.\/  T ) ) )
181dalemkehl 29738 . . . . . 6  |-  ( ph  ->  K  e.  HL )
191dalemtea 29745 . . . . . 6  |-  ( ph  ->  T  e.  A )
2011, 3hlatjidm 29484 . . . . . 6  |-  ( ( K  e.  HL  /\  T  e.  A )  ->  ( T  .\/  T
)  =  T )
2118, 19, 20syl2anc 643 . . . . 5  |-  ( ph  ->  ( T  .\/  T
)  =  T )
2221breq2d 4166 . . . 4  |-  ( ph  ->  ( C  .<_  ( T 
.\/  T )  <->  C  .<_  T ) )
2317, 22sylibd 206 . . 3  |-  ( ph  ->  ( Q  =  T  ->  C  .<_  T ) )
2423necon3bd 2588 . 2  |-  ( ph  ->  ( -.  C  .<_  T  ->  Q  =/=  T
) )
2513, 24mpd 15 1  |-  ( ph  ->  Q  =/=  T )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2551   class class class wbr 4154   ` cfv 5395  (class class class)co 6021   Basecbs 13397   lecple 13464   joincjn 14329   Latclat 14402   Atomscatm 29379   HLchlt 29466   LPlanesclpl 29607
This theorem is referenced by:  dalemcea  29775  dalem2  29776  dalemdnee  29781
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-undef 6480  df-riota 6486  df-poset 14331  df-lub 14359  df-join 14361  df-lat 14403  df-ats 29383  df-atl 29414  df-cvlat 29438  df-hlat 29467
  Copyright terms: Public domain W3C validator