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Theorem dalem22 33304
Description: Lemma for dath 33345. 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 2461 . . 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 33303 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  d ) ( meet `  K ) ( P 
.\/  S ) )  e.  A )
111dalemkehl 33232 . . . . 5  |-  ( ph  ->  K  e.  HL )
1211adantr 471 . . . 4  |-  ( (
ph  /\  ps )  ->  K  e.  HL )
131, 2, 3, 4, 5dalemcjden 33301 . . . 4  |-  ( (
ph  /\  ps )  ->  ( c  .\/  d
)  e.  ( LLines `  K ) )
141, 2, 3, 4, 7, 8dalempjsen 33262 . . . . 5  |-  ( ph  ->  ( P  .\/  S
)  e.  ( LLines `  K ) )
1514adantr 471 . . . 4  |-  ( (
ph  /\  ps )  ->  ( P  .\/  S
)  e.  ( LLines `  K ) )
16 eqid 2461 . . . . 5  |-  ( LLines `  K )  =  (
LLines `  K )
173, 6, 4, 16, 72llnmj 33169 . . . 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 1276 . . 3  |-  ( (
ph  /\  ps )  ->  ( ( ( c 
.\/  d ) (
meet `  K )
( P  .\/  S
) )  e.  A  <->  ( ( c  .\/  d
)  .\/  ( P  .\/  S ) )  e.  O ) )
19183adant2 1033 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( c 
.\/  d ) (
meet `  K )
( P  .\/  S
) )  e.  A  <->  ( ( c  .\/  d
)  .\/  ( P  .\/  S ) )  e.  O ) )
2010, 19mpbid 215 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  d )  .\/  ( P  .\/  S ) )  e.  O )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 991    = wceq 1454    e. wcel 1897    =/= wne 2632   class class class wbr 4415   ` cfv 5600  (class class class)co 6314   Basecbs 15169   lecple 15245   joincjn 16237   meetcmee 16238   Atomscatm 32873   HLchlt 32960   LLinesclln 33100   LPlanesclpl 33101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-reu 2755  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6276  df-ov 6317  df-oprab 6318  df-preset 16221  df-poset 16239  df-plt 16252  df-lub 16268  df-glb 16269  df-join 16270  df-meet 16271  df-p0 16333  df-lat 16340  df-clat 16402  df-oposet 32786  df-ol 32788  df-oml 32789  df-covers 32876  df-ats 32877  df-atl 32908  df-cvlat 32932  df-hlat 32961  df-llines 33107  df-lplanes 33108
This theorem is referenced by:  dalem23  33305
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