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Theorem dalem2 29776
Description: Lemma for dath 29851. 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 29738 . . 3  |-  ( ph  ->  K  e.  HL )
31dalempea 29741 . . 3  |-  ( ph  ->  P  e.  A )
41dalemqea 29742 . . 3  |-  ( ph  ->  Q  e.  A )
51dalemsea 29744 . . 3  |-  ( ph  ->  S  e.  A )
61dalemtea 29745 . . 3  |-  ( ph  ->  T  e.  A )
7 dalemc.j . . . 4  |-  .\/  =  ( join `  K )
8 dalemc.a . . . 4  |-  A  =  ( Atoms `  K )
97, 8hlatj4 29489 . . 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 29768 . . . 4  |-  ( ph  ->  ( P  .\/  S
)  e.  ( LLines `  K ) )
151, 11, 7, 8, 12, 13dalemqnet 29767 . . . . 5  |-  ( ph  ->  Q  =/=  T )
16 eqid 2388 . . . . . 6  |-  ( LLines `  K )  =  (
LLines `  K )
177, 8, 16llni2 29627 . . . . 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 29774 . . . 4  |-  ( ph  ->  ( P  .\/  S
)  =/=  ( Q 
.\/  T ) )
201, 11, 7, 8, 12, 13dalemcea 29775 . . . . 5  |-  ( ph  ->  C  e.  A )
211dalemclpjs 29749 . . . . 5  |-  ( ph  ->  C  .<_  ( P  .\/  S ) )
221dalemclqjt 29750 . . . . 5  |-  ( ph  ->  C  .<_  ( Q  .\/  T ) )
23 eqid 2388 . . . . . 6  |-  ( meet `  K )  =  (
meet `  K )
24 eqid 2388 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
2511, 23, 24, 8, 162llnm4 29685 . . . . 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 29639 . . . 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 29675 . . . 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 2462 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 1649    e. wcel 1717    =/= wne 2551   class class class wbr 4154   ` cfv 5395  (class class class)co 6021   Basecbs 13397   lecple 13464   joincjn 14329   meetcmee 14330   0.cp0 14394   Atomscatm 29379   HLchlt 29466   LLinesclln 29606   LPlanesclpl 29607
This theorem is referenced by:  dalemdea  29777
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-plt 14343  df-lub 14359  df-glb 14360  df-join 14361  df-meet 14362  df-p0 14396  df-lat 14403  df-clat 14465  df-oposet 29292  df-ol 29294  df-oml 29295  df-covers 29382  df-ats 29383  df-atl 29414  df-cvlat 29438  df-hlat 29467  df-llines 29613  df-lplanes 29614
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