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Theorem dalem2 34858
Description: Lemma for dath 34933. Show the lines  P Q and  S T form a plane. (Contributed by NM, 11-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 )
dalem1.o  |-  O  =  ( LPlanes `  K )
dalem1.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
Assertion
Ref Expression
dalem2  |-  ( ph  ->  ( ( P  .\/  Q )  .\/  ( S 
.\/  T ) )  e.  O )

Proof of Theorem dalem2
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 ) ) ) ) )
21dalemkehl 34820 . . 3  |-  ( ph  ->  K  e.  HL )
31dalempea 34823 . . 3  |-  ( ph  ->  P  e.  A )
41dalemqea 34824 . . 3  |-  ( ph  ->  Q  e.  A )
51dalemsea 34826 . . 3  |-  ( ph  ->  S  e.  A )
61dalemtea 34827 . . 3  |-  ( ph  ->  T  e.  A )
7 dalemc.j . . . 4  |-  .\/  =  ( join `  K )
8 dalemc.a . . . 4  |-  A  =  ( Atoms `  K )
97, 8hlatj4 34571 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  -> 
( ( P  .\/  Q )  .\/  ( S 
.\/  T ) )  =  ( ( P 
.\/  S )  .\/  ( Q  .\/  T ) ) )
102, 3, 4, 5, 6, 9syl122anc 1237 . 2  |-  ( ph  ->  ( ( P  .\/  Q )  .\/  ( S 
.\/  T ) )  =  ( ( P 
.\/  S )  .\/  ( Q  .\/  T ) ) )
11 dalemc.l . . . . 5  |-  .<_  =  ( le `  K )
12 dalem1.o . . . . 5  |-  O  =  ( LPlanes `  K )
13 dalem1.y . . . . 5  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
141, 11, 7, 8, 12, 13dalempjsen 34850 . . . 4  |-  ( ph  ->  ( P  .\/  S
)  e.  ( LLines `  K ) )
151, 11, 7, 8, 12, 13dalemqnet 34849 . . . . 5  |-  ( ph  ->  Q  =/=  T )
16 eqid 2467 . . . . . 6  |-  ( LLines `  K )  =  (
LLines `  K )
177, 8, 16llni2 34709 . . . . 5  |-  ( ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  /\  Q  =/=  T
)  ->  ( Q  .\/  T )  e.  (
LLines `  K ) )
182, 4, 6, 15, 17syl31anc 1231 . . . 4  |-  ( ph  ->  ( Q  .\/  T
)  e.  ( LLines `  K ) )
191, 11, 7, 8, 12, 13dalem1 34856 . . . 4  |-  ( ph  ->  ( P  .\/  S
)  =/=  ( Q 
.\/  T ) )
201, 11, 7, 8, 12, 13dalemcea 34857 . . . . 5  |-  ( ph  ->  C  e.  A )
211dalemclpjs 34831 . . . . 5  |-  ( ph  ->  C  .<_  ( P  .\/  S ) )
221dalemclqjt 34832 . . . . 5  |-  ( ph  ->  C  .<_  ( Q  .\/  T ) )
23 eqid 2467 . . . . . 6  |-  ( meet `  K )  =  (
meet `  K )
24 eqid 2467 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
2511, 23, 24, 8, 162llnm4 34767 . . . . 5  |-  ( ( K  e.  HL  /\  ( C  e.  A  /\  ( P  .\/  S
)  e.  ( LLines `  K )  /\  ( Q  .\/  T )  e.  ( LLines `  K )
)  /\  ( C  .<_  ( P  .\/  S
)  /\  C  .<_  ( Q  .\/  T ) ) )  ->  (
( P  .\/  S
) ( meet `  K
) ( Q  .\/  T ) )  =/=  ( 0. `  K ) )
262, 20, 14, 18, 21, 22, 25syl132anc 1246 . . . 4  |-  ( ph  ->  ( ( P  .\/  S ) ( meet `  K
) ( Q  .\/  T ) )  =/=  ( 0. `  K ) )
2723, 24, 8, 162llnmat 34721 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  .\/  S
)  e.  ( LLines `  K )  /\  ( Q  .\/  T )  e.  ( LLines `  K )
)  /\  ( ( P  .\/  S )  =/=  ( Q  .\/  T
)  /\  ( ( P  .\/  S ) (
meet `  K )
( Q  .\/  T
) )  =/=  ( 0. `  K ) ) )  ->  ( ( P  .\/  S ) (
meet `  K )
( Q  .\/  T
) )  e.  A
)
282, 14, 18, 19, 26, 27syl32anc 1236 . . 3  |-  ( ph  ->  ( ( P  .\/  S ) ( meet `  K
) ( Q  .\/  T ) )  e.  A
)
297, 23, 8, 16, 122llnmj 34757 . . . 4  |-  ( ( K  e.  HL  /\  ( P  .\/  S )  e.  ( LLines `  K
)  /\  ( Q  .\/  T )  e.  (
LLines `  K ) )  ->  ( ( ( P  .\/  S ) ( meet `  K
) ( Q  .\/  T ) )  e.  A  <->  ( ( P  .\/  S
)  .\/  ( Q  .\/  T ) )  e.  O ) )
302, 14, 18, 29syl3anc 1228 . . 3  |-  ( ph  ->  ( ( ( P 
.\/  S ) (
meet `  K )
( Q  .\/  T
) )  e.  A  <->  ( ( P  .\/  S
)  .\/  ( Q  .\/  T ) )  e.  O ) )
3128, 30mpbid 210 . 2  |-  ( ph  ->  ( ( P  .\/  S )  .\/  ( Q 
.\/  T ) )  e.  O )
3210, 31eqeltrd 2555 1  |-  ( ph  ->  ( ( P  .\/  Q )  .\/  ( S 
.\/  T ) )  e.  O )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   Basecbs 14507   lecple 14579   joincjn 15448   meetcmee 15449   0.cp0 15541   Atomscatm 34461   HLchlt 34548   LLinesclln 34688   LPlanesclpl 34689
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-poset 15450  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-lat 15550  df-clat 15612  df-oposet 34374  df-ol 34376  df-oml 34377  df-covers 34464  df-ats 34465  df-atl 34496  df-cvlat 34520  df-hlat 34549  df-llines 34695  df-lplanes 34696
This theorem is referenced by:  dalemdea  34859
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