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Theorem dalem63 30630
Description: Lemma for dath 30631. Combine the cases where  Y and  Z are different planes with the case where  Y and 
Z are the same plane. (Contributed by NM, 11-Aug-2012.)
Hypotheses
Ref Expression
dalem62.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 ) ) ) ) )
dalem62.l  |-  .<_  =  ( le `  K )
dalem62.j  |-  .\/  =  ( join `  K )
dalem62.a  |-  A  =  ( Atoms `  K )
dalem62.m  |-  ./\  =  ( meet `  K )
dalem62.o  |-  O  =  ( LPlanes `  K )
dalem62.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem62.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
dalem62.d  |-  D  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )
dalem62.e  |-  E  =  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )
dalem62.f  |-  F  =  ( ( R  .\/  P )  ./\  ( U  .\/  S ) )
Assertion
Ref Expression
dalem63  |-  ( ph  ->  F  .<_  ( D  .\/  E ) )

Proof of Theorem dalem63
StepHypRef Expression
1 dalem62.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 dalem62.l . . 3  |-  .<_  =  ( le `  K )
3 dalem62.j . . 3  |-  .\/  =  ( join `  K )
4 dalem62.a . . 3  |-  A  =  ( Atoms `  K )
5 dalem62.m . . 3  |-  ./\  =  ( meet `  K )
6 dalem62.o . . 3  |-  O  =  ( LPlanes `  K )
7 dalem62.y . . 3  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
8 dalem62.z . . 3  |-  Z  =  ( ( S  .\/  T )  .\/  U )
9 dalem62.d . . 3  |-  D  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )
10 dalem62.e . . 3  |-  E  =  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )
11 dalem62.f . . 3  |-  F  =  ( ( R  .\/  P )  ./\  ( U  .\/  S ) )
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11dalem62 30629 . 2  |-  ( (
ph  /\  Y  =  Z )  ->  F  .<_  ( D  .\/  E
) )
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11dalem16 30574 . 2  |-  ( (
ph  /\  Y  =/=  Z )  ->  F  .<_  ( D  .\/  E ) )
1412, 13pm2.61dane 2688 1  |-  ( ph  ->  F  .<_  ( D  .\/  E ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727   class class class wbr 4237   ` cfv 5483  (class class class)co 6110   Basecbs 13500   lecple 13567   joincjn 14432   meetcmee 14433   Atomscatm 30159   HLchlt 30246   LPlanesclpl 30387
This theorem is referenced by:  dath  30631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-undef 6572  df-riota 6578  df-poset 14434  df-plt 14446  df-lub 14462  df-glb 14463  df-join 14464  df-meet 14465  df-p0 14499  df-lat 14506  df-clat 14568  df-oposet 30072  df-ol 30074  df-oml 30075  df-covers 30162  df-ats 30163  df-atl 30194  df-cvlat 30218  df-hlat 30247  df-llines 30393  df-lplanes 30394  df-lvols 30395
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