Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dalem8 Structured version   Unicode version

Theorem dalem8 34341
Description: Lemma for dath 34407. Plane  Z belongs to the 3-dimensional space. (Contributed by NM, 21-Jul-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 )
dalem6.o  |-  O  =  ( LPlanes `  K )
dalem6.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem6.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
dalem6.w  |-  W  =  ( Y  .\/  C
)
Assertion
Ref Expression
dalem8  |-  ( ph  ->  Z  .<_  W )

Proof of Theorem dalem8
StepHypRef Expression
1 dalem6.z . 2  |-  Z  =  ( ( S  .\/  T )  .\/  U )
2 dalema.ph . . . . 5  |-  ( 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 ) ) ) ) )
3 dalemc.l . . . . 5  |-  .<_  =  ( le `  K )
4 dalemc.j . . . . 5  |-  .\/  =  ( join `  K )
5 dalemc.a . . . . 5  |-  A  =  ( Atoms `  K )
6 dalem6.o . . . . 5  |-  O  =  ( LPlanes `  K )
7 dalem6.y . . . . 5  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
8 dalem6.w . . . . 5  |-  W  =  ( Y  .\/  C
)
92, 3, 4, 5, 6, 7, 1, 8dalem6 34339 . . . 4  |-  ( ph  ->  S  .<_  W )
102, 3, 4, 5, 6, 7, 1, 8dalem7 34340 . . . 4  |-  ( ph  ->  T  .<_  W )
112dalemkelat 34295 . . . . 5  |-  ( ph  ->  K  e.  Lat )
122, 5dalemseb 34313 . . . . 5  |-  ( ph  ->  S  e.  ( Base `  K ) )
132, 5dalemteb 34314 . . . . 5  |-  ( ph  ->  T  e.  ( Base `  K ) )
142, 6dalemyeb 34320 . . . . . . 7  |-  ( ph  ->  Y  e.  ( Base `  K ) )
152, 5dalemceb 34309 . . . . . . 7  |-  ( ph  ->  C  e.  ( Base `  K ) )
16 eqid 2460 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
1716, 4latjcl 15527 . . . . . . 7  |-  ( ( K  e.  Lat  /\  Y  e.  ( Base `  K )  /\  C  e.  ( Base `  K
) )  ->  ( Y  .\/  C )  e.  ( Base `  K
) )
1811, 14, 15, 17syl3anc 1223 . . . . . 6  |-  ( ph  ->  ( Y  .\/  C
)  e.  ( Base `  K ) )
198, 18syl5eqel 2552 . . . . 5  |-  ( ph  ->  W  e.  ( Base `  K ) )
2016, 3, 4latjle12 15538 . . . . 5  |-  ( ( K  e.  Lat  /\  ( S  e.  ( Base `  K )  /\  T  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) ) )  -> 
( ( S  .<_  W  /\  T  .<_  W )  <-> 
( S  .\/  T
)  .<_  W ) )
2111, 12, 13, 19, 20syl13anc 1225 . . . 4  |-  ( ph  ->  ( ( S  .<_  W  /\  T  .<_  W )  <-> 
( S  .\/  T
)  .<_  W ) )
229, 10, 21mpbi2and 914 . . 3  |-  ( ph  ->  ( S  .\/  T
)  .<_  W )
232, 3, 4, 5, 6, 7, 8dalem5 34338 . . 3  |-  ( ph  ->  U  .<_  W )
242, 4, 5dalemsjteb 34317 . . . 4  |-  ( ph  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
252, 5dalemueb 34315 . . . 4  |-  ( ph  ->  U  e.  ( Base `  K ) )
2616, 3, 4latjle12 15538 . . . 4  |-  ( ( K  e.  Lat  /\  ( ( S  .\/  T )  e.  ( Base `  K )  /\  U  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
) )  ->  (
( ( S  .\/  T )  .<_  W  /\  U  .<_  W )  <->  ( ( S  .\/  T )  .\/  U )  .<_  W )
)
2711, 24, 25, 19, 26syl13anc 1225 . . 3  |-  ( ph  ->  ( ( ( S 
.\/  T )  .<_  W  /\  U  .<_  W )  <-> 
( ( S  .\/  T )  .\/  U ) 
.<_  W ) )
2822, 23, 27mpbi2and 914 . 2  |-  ( ph  ->  ( ( S  .\/  T )  .\/  U ) 
.<_  W )
291, 28syl5eqbr 4473 1  |-  ( ph  ->  Z  .<_  W )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   Basecbs 14479   lecple 14551   joincjn 15420   Latclat 15521   Atomscatm 33935   HLchlt 34022   LPlanesclpl 34163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-poset 15422  df-plt 15434  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-p0 15515  df-lat 15522  df-clat 15584  df-oposet 33848  df-ol 33850  df-oml 33851  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023  df-llines 34169  df-lplanes 34170
This theorem is referenced by:  dalem13  34347
  Copyright terms: Public domain W3C validator