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Theorem dalem2 30359
Description: Lemma for dath 30434. 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 30321 . . 3  |-  ( ph  ->  K  e.  HL )
31dalempea 30324 . . 3  |-  ( ph  ->  P  e.  A )
41dalemqea 30325 . . 3  |-  ( ph  ->  Q  e.  A )
51dalemsea 30327 . . 3  |-  ( ph  ->  S  e.  A )
61dalemtea 30328 . . 3  |-  ( ph  ->  T  e.  A )
7 dalemc.j . . . 4  |-  .\/  =  ( join `  K )
8 dalemc.a . . . 4  |-  A  =  ( Atoms `  K )
97, 8hlatj4 30072 . . 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 1193 . 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 30351 . . . 4  |-  ( ph  ->  ( P  .\/  S
)  e.  ( LLines `  K ) )
151, 11, 7, 8, 12, 13dalemqnet 30350 . . . . 5  |-  ( ph  ->  Q  =/=  T )
16 eqid 2435 . . . . . 6  |-  ( LLines `  K )  =  (
LLines `  K )
177, 8, 16llni2 30210 . . . . 5  |-  ( ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  /\  Q  =/=  T
)  ->  ( Q  .\/  T )  e.  (
LLines `  K ) )
182, 4, 6, 15, 17syl31anc 1187 . . . 4  |-  ( ph  ->  ( Q  .\/  T
)  e.  ( LLines `  K ) )
191, 11, 7, 8, 12, 13dalem1 30357 . . . 4  |-  ( ph  ->  ( P  .\/  S
)  =/=  ( Q 
.\/  T ) )
201, 11, 7, 8, 12, 13dalemcea 30358 . . . . 5  |-  ( ph  ->  C  e.  A )
211dalemclpjs 30332 . . . . 5  |-  ( ph  ->  C  .<_  ( P  .\/  S ) )
221dalemclqjt 30333 . . . . 5  |-  ( ph  ->  C  .<_  ( Q  .\/  T ) )
23 eqid 2435 . . . . . 6  |-  ( meet `  K )  =  (
meet `  K )
24 eqid 2435 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
2511, 23, 24, 8, 162llnm4 30268 . . . . 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 1202 . . . 4  |-  ( ph  ->  ( ( P  .\/  S ) ( meet `  K
) ( Q  .\/  T ) )  =/=  ( 0. `  K ) )
2723, 24, 8, 162llnmat 30222 . . . 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 1192 . . 3  |-  ( ph  ->  ( ( P  .\/  S ) ( meet `  K
) ( Q  .\/  T ) )  e.  A
)
297, 23, 8, 16, 122llnmj 30258 . . . 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 1184 . . 3  |-  ( ph  ->  ( ( ( P 
.\/  S ) (
meet `  K )
( Q  .\/  T
) )  e.  A  <->  ( ( P  .\/  S
)  .\/  ( Q  .\/  T ) )  e.  O ) )
3128, 30mpbid 202 . 2  |-  ( ph  ->  ( ( P  .\/  S )  .\/  ( Q 
.\/  T ) )  e.  O )
3210, 31eqeltrd 2509 1  |-  ( ph  ->  ( ( P  .\/  Q )  .\/  ( S 
.\/  T ) )  e.  O )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13459   lecple 13526   joincjn 14391   meetcmee 14392   0.cp0 14456   Atomscatm 29962   HLchlt 30049   LLinesclln 30189   LPlanesclpl 30190
This theorem is referenced by:  dalemdea  30360
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14393  df-plt 14405  df-lub 14421  df-glb 14422  df-join 14423  df-meet 14424  df-p0 14458  df-lat 14465  df-clat 14527  df-oposet 29875  df-ol 29877  df-oml 29878  df-covers 29965  df-ats 29966  df-atl 29997  df-cvlat 30021  df-hlat 30050  df-llines 30196  df-lplanes 30197
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