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Theorem dalem22 30177
Description: Lemma for dath 30218. Show that lines  c d and  P S determine a plane. (Contributed by NM, 2-Aug-2012.)
Hypotheses
Ref Expression
dalem.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 ) ) ) ) )
dalem.l  |-  .<_  =  ( le `  K )
dalem.j  |-  .\/  =  ( join `  K )
dalem.a  |-  A  =  ( Atoms `  K )
dalem.ps  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
dalem22.o  |-  O  =  ( LPlanes `  K )
dalem22.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem22.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
Assertion
Ref Expression
dalem22  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  d )  .\/  ( P  .\/  S ) )  e.  O )

Proof of Theorem dalem22
StepHypRef Expression
1 dalem.ph . . 3  |-  ( 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 ) ) ) ) )
2 dalem.l . . 3  |-  .<_  =  ( le `  K )
3 dalem.j . . 3  |-  .\/  =  ( join `  K )
4 dalem.a . . 3  |-  A  =  ( Atoms `  K )
5 dalem.ps . . 3  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
6 eqid 2404 . . 3  |-  ( meet `  K )  =  (
meet `  K )
7 dalem22.o . . 3  |-  O  =  ( LPlanes `  K )
8 dalem22.y . . 3  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
9 dalem22.z . . 3  |-  Z  =  ( ( S  .\/  T )  .\/  U )
101, 2, 3, 4, 5, 6, 7, 8, 9dalem21 30176 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  d ) ( meet `  K ) ( P 
.\/  S ) )  e.  A )
111dalemkehl 30105 . . . . 5  |-  ( ph  ->  K  e.  HL )
1211adantr 452 . . . 4  |-  ( (
ph  /\  ps )  ->  K  e.  HL )
131, 2, 3, 4, 5dalemcjden 30174 . . . 4  |-  ( (
ph  /\  ps )  ->  ( c  .\/  d
)  e.  ( LLines `  K ) )
141, 2, 3, 4, 7, 8dalempjsen 30135 . . . . 5  |-  ( ph  ->  ( P  .\/  S
)  e.  ( LLines `  K ) )
1514adantr 452 . . . 4  |-  ( (
ph  /\  ps )  ->  ( P  .\/  S
)  e.  ( LLines `  K ) )
16 eqid 2404 . . . . 5  |-  ( LLines `  K )  =  (
LLines `  K )
173, 6, 4, 16, 72llnmj 30042 . . . 4  |-  ( ( K  e.  HL  /\  ( c  .\/  d
)  e.  ( LLines `  K )  /\  ( P  .\/  S )  e.  ( LLines `  K )
)  ->  ( (
( c  .\/  d
) ( meet `  K
) ( P  .\/  S ) )  e.  A  <->  ( ( c  .\/  d
)  .\/  ( P  .\/  S ) )  e.  O ) )
1812, 13, 15, 17syl3anc 1184 . . 3  |-  ( (
ph  /\  ps )  ->  ( ( ( c 
.\/  d ) (
meet `  K )
( P  .\/  S
) )  e.  A  <->  ( ( c  .\/  d
)  .\/  ( P  .\/  S ) )  e.  O ) )
19183adant2 976 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( c 
.\/  d ) (
meet `  K )
( P  .\/  S
) )  e.  A  <->  ( ( c  .\/  d
)  .\/  ( P  .\/  S ) )  e.  O ) )
2010, 19mpbid 202 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  d )  .\/  ( P  .\/  S ) )  e.  O )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   Atomscatm 29746   HLchlt 29833   LLinesclln 29973   LPlanesclpl 29974
This theorem is referenced by:  dalem23  30178
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981
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