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Theorem dalem2 35798
Description: Lemma for dath 35873. 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 35760 . . 3  |-  ( ph  ->  K  e.  HL )
31dalempea 35763 . . 3  |-  ( ph  ->  P  e.  A )
41dalemqea 35764 . . 3  |-  ( ph  ->  Q  e.  A )
51dalemsea 35766 . . 3  |-  ( ph  ->  S  e.  A )
61dalemtea 35767 . . 3  |-  ( ph  ->  T  e.  A )
7 dalemc.j . . . 4  |-  .\/  =  ( join `  K )
8 dalemc.a . . . 4  |-  A  =  ( Atoms `  K )
97, 8hlatj4 35511 . . 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 1235 . 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 35790 . . . 4  |-  ( ph  ->  ( P  .\/  S
)  e.  ( LLines `  K ) )
151, 11, 7, 8, 12, 13dalemqnet 35789 . . . . 5  |-  ( ph  ->  Q  =/=  T )
16 eqid 2382 . . . . . 6  |-  ( LLines `  K )  =  (
LLines `  K )
177, 8, 16llni2 35649 . . . . 5  |-  ( ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  /\  Q  =/=  T
)  ->  ( Q  .\/  T )  e.  (
LLines `  K ) )
182, 4, 6, 15, 17syl31anc 1229 . . . 4  |-  ( ph  ->  ( Q  .\/  T
)  e.  ( LLines `  K ) )
191, 11, 7, 8, 12, 13dalem1 35796 . . . 4  |-  ( ph  ->  ( P  .\/  S
)  =/=  ( Q 
.\/  T ) )
201, 11, 7, 8, 12, 13dalemcea 35797 . . . . 5  |-  ( ph  ->  C  e.  A )
211dalemclpjs 35771 . . . . 5  |-  ( ph  ->  C  .<_  ( P  .\/  S ) )
221dalemclqjt 35772 . . . . 5  |-  ( ph  ->  C  .<_  ( Q  .\/  T ) )
23 eqid 2382 . . . . . 6  |-  ( meet `  K )  =  (
meet `  K )
24 eqid 2382 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
2511, 23, 24, 8, 162llnm4 35707 . . . . 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 1244 . . . 4  |-  ( ph  ->  ( ( P  .\/  S ) ( meet `  K
) ( Q  .\/  T ) )  =/=  ( 0. `  K ) )
2723, 24, 8, 162llnmat 35661 . . . 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 1234 . . 3  |-  ( ph  ->  ( ( P  .\/  S ) ( meet `  K
) ( Q  .\/  T ) )  e.  A
)
297, 23, 8, 16, 122llnmj 35697 . . . 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 1226 . . 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 2470 1  |-  ( ph  ->  ( ( P  .\/  Q )  .\/  ( S 
.\/  T ) )  e.  O )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826    =/= wne 2577   class class class wbr 4367   ` cfv 5496  (class class class)co 6196   Basecbs 14634   lecple 14709   joincjn 15690   meetcmee 15691   0.cp0 15784   Atomscatm 35401   HLchlt 35488   LLinesclln 35628   LPlanesclpl 35629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-preset 15674  df-poset 15692  df-plt 15705  df-lub 15721  df-glb 15722  df-join 15723  df-meet 15724  df-p0 15786  df-lat 15793  df-clat 15855  df-oposet 35314  df-ol 35316  df-oml 35317  df-covers 35404  df-ats 35405  df-atl 35436  df-cvlat 35460  df-hlat 35489  df-llines 35635  df-lplanes 35636
This theorem is referenced by:  dalemdea  35799
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