Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dalem16 Structured version   Unicode version

Theorem dalem16 34876
Description: Lemma for dath 34933. The atoms  D,  E, and  F form a line of perspectivity. This is Desargue's Theorem for the special case where planes  Y and  Z are different. (Contributed by NM, 7-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 )
dalem16.m  |-  ./\  =  ( meet `  K )
dalem16.o  |-  O  =  ( LPlanes `  K )
dalem16.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem16.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
dalem16.d  |-  D  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )
dalem16.e  |-  E  =  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )
dalem16.f  |-  F  =  ( ( R  .\/  P )  ./\  ( U  .\/  S ) )
Assertion
Ref Expression
dalem16  |-  ( (
ph  /\  Y  =/=  Z )  ->  F  .<_  ( D  .\/  E ) )

Proof of Theorem dalem16
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 ) ) ) ) )
2 dalemc.l . . . 4  |-  .<_  =  ( le `  K )
3 dalemc.j . . . 4  |-  .\/  =  ( join `  K )
4 dalemc.a . . . 4  |-  A  =  ( Atoms `  K )
5 dalem16.m . . . 4  |-  ./\  =  ( meet `  K )
6 dalem16.o . . . 4  |-  O  =  ( LPlanes `  K )
7 dalem16.y . . . 4  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
8 dalem16.z . . . 4  |-  Z  =  ( ( S  .\/  T )  .\/  U )
9 eqid 2467 . . . 4  |-  ( Y 
./\  Z )  =  ( Y  ./\  Z
)
10 dalem16.f . . . 4  |-  F  =  ( ( R  .\/  P )  ./\  ( U  .\/  S ) )
111, 2, 3, 4, 5, 6, 7, 8, 9, 10dalem12 34872 . . 3  |-  ( ph  ->  F  .<_  ( Y  ./\ 
Z ) )
1211adantr 465 . 2  |-  ( (
ph  /\  Y  =/=  Z )  ->  F  .<_  ( Y  ./\  Z )
)
13 dalem16.d . . . . . 6  |-  D  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )
141, 2, 3, 4, 5, 6, 7, 8, 9, 13dalem10 34870 . . . . 5  |-  ( ph  ->  D  .<_  ( Y  ./\ 
Z ) )
15 dalem16.e . . . . . 6  |-  E  =  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )
161, 2, 3, 4, 5, 6, 7, 8, 9, 15dalem11 34871 . . . . 5  |-  ( ph  ->  E  .<_  ( Y  ./\ 
Z ) )
171dalemkelat 34821 . . . . . 6  |-  ( ph  ->  K  e.  Lat )
181, 2, 3, 4, 5, 6, 7, 8, 13dalemdea 34859 . . . . . . 7  |-  ( ph  ->  D  e.  A )
19 eqid 2467 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2019, 4atbase 34487 . . . . . . 7  |-  ( D  e.  A  ->  D  e.  ( Base `  K
) )
2118, 20syl 16 . . . . . 6  |-  ( ph  ->  D  e.  ( Base `  K ) )
221, 2, 3, 4, 5, 6, 7, 8, 15dalemeea 34860 . . . . . . 7  |-  ( ph  ->  E  e.  A )
2319, 4atbase 34487 . . . . . . 7  |-  ( E  e.  A  ->  E  e.  ( Base `  K
) )
2422, 23syl 16 . . . . . 6  |-  ( ph  ->  E  e.  ( Base `  K ) )
251, 6dalemyeb 34846 . . . . . . 7  |-  ( ph  ->  Y  e.  ( Base `  K ) )
261dalemzeo 34830 . . . . . . . 8  |-  ( ph  ->  Z  e.  O )
2719, 6lplnbase 34731 . . . . . . . 8  |-  ( Z  e.  O  ->  Z  e.  ( Base `  K
) )
2826, 27syl 16 . . . . . . 7  |-  ( ph  ->  Z  e.  ( Base `  K ) )
2919, 5latmcl 15556 . . . . . . 7  |-  ( ( K  e.  Lat  /\  Y  e.  ( Base `  K )  /\  Z  e.  ( Base `  K
) )  ->  ( Y  ./\  Z )  e.  ( Base `  K
) )
3017, 25, 28, 29syl3anc 1228 . . . . . 6  |-  ( ph  ->  ( Y  ./\  Z
)  e.  ( Base `  K ) )
3119, 2, 3latjle12 15566 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( D  e.  ( Base `  K )  /\  E  e.  ( Base `  K )  /\  ( Y  ./\  Z )  e.  ( Base `  K
) ) )  -> 
( ( D  .<_  ( Y  ./\  Z )  /\  E  .<_  ( Y 
./\  Z ) )  <-> 
( D  .\/  E
)  .<_  ( Y  ./\  Z ) ) )
3217, 21, 24, 30, 31syl13anc 1230 . . . . 5  |-  ( ph  ->  ( ( D  .<_  ( Y  ./\  Z )  /\  E  .<_  ( Y 
./\  Z ) )  <-> 
( D  .\/  E
)  .<_  ( Y  ./\  Z ) ) )
3314, 16, 32mpbi2and 919 . . . 4  |-  ( ph  ->  ( D  .\/  E
)  .<_  ( Y  ./\  Z ) )
3433adantr 465 . . 3  |-  ( (
ph  /\  Y  =/=  Z )  ->  ( D  .\/  E )  .<_  ( Y 
./\  Z ) )
351dalemkehl 34820 . . . . 5  |-  ( ph  ->  K  e.  HL )
3635adantr 465 . . . 4  |-  ( (
ph  /\  Y  =/=  Z )  ->  K  e.  HL )
371, 2, 3, 4, 5, 6, 7, 8, 13, 15dalemdnee 34863 . . . . . 6  |-  ( ph  ->  D  =/=  E )
38 eqid 2467 . . . . . . 7  |-  ( LLines `  K )  =  (
LLines `  K )
393, 4, 38llni2 34709 . . . . . 6  |-  ( ( ( K  e.  HL  /\  D  e.  A  /\  E  e.  A )  /\  D  =/=  E
)  ->  ( D  .\/  E )  e.  (
LLines `  K ) )
4035, 18, 22, 37, 39syl31anc 1231 . . . . 5  |-  ( ph  ->  ( D  .\/  E
)  e.  ( LLines `  K ) )
4140adantr 465 . . . 4  |-  ( (
ph  /\  Y  =/=  Z )  ->  ( D  .\/  E )  e.  (
LLines `  K ) )
421, 2, 3, 4, 5, 38, 6, 7, 8, 9dalem15 34875 . . . 4  |-  ( (
ph  /\  Y  =/=  Z )  ->  ( Y  ./\ 
Z )  e.  (
LLines `  K ) )
432, 38llncmp 34719 . . . 4  |-  ( ( K  e.  HL  /\  ( D  .\/  E )  e.  ( LLines `  K
)  /\  ( Y  ./\ 
Z )  e.  (
LLines `  K ) )  ->  ( ( D 
.\/  E )  .<_  ( Y  ./\  Z )  <-> 
( D  .\/  E
)  =  ( Y 
./\  Z ) ) )
4436, 41, 42, 43syl3anc 1228 . . 3  |-  ( (
ph  /\  Y  =/=  Z )  ->  ( ( D  .\/  E )  .<_  ( Y  ./\  Z )  <-> 
( D  .\/  E
)  =  ( Y 
./\  Z ) ) )
4534, 44mpbid 210 . 2  |-  ( (
ph  /\  Y  =/=  Z )  ->  ( D  .\/  E )  =  ( Y  ./\  Z )
)
4612, 45breqtrrd 4479 1  |-  ( (
ph  /\  Y  =/=  Z )  ->  F  .<_  ( D  .\/  E ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   Basecbs 14507   lecple 14579   joincjn 15448   meetcmee 15449   Latclat 15549   Atomscatm 34461   HLchlt 34548   LLinesclln 34688   LPlanesclpl 34689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-poset 15450  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-lat 15550  df-clat 15612  df-oposet 34374  df-ol 34376  df-oml 34377  df-covers 34464  df-ats 34465  df-atl 34496  df-cvlat 34520  df-hlat 34549  df-llines 34695  df-lplanes 34696  df-lvols 34697
This theorem is referenced by:  dalem63  34932
  Copyright terms: Public domain W3C validator