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Theorem dalem16 33288
Description: Lemma for dath 33345. 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 2461 . . . 4  |-  ( Y 
./\  Z )  =  ( Y  ./\  Z
)
10 dalem16.f . . . 4  |-  F  =  ( ( R  .\/  P )  ./\  ( U  .\/  S ) )
111, 2, 3, 4, 5, 6, 7, 8, 9, 10dalem12 33284 . . 3  |-  ( ph  ->  F  .<_  ( Y  ./\ 
Z ) )
1211adantr 471 . 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 33282 . . . . 5  |-  ( ph  ->  D  .<_  ( Y  ./\ 
Z ) )
15 dalem16.e . . . . . 6  |-  E  =  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )
161, 2, 3, 4, 5, 6, 7, 8, 9, 15dalem11 33283 . . . . 5  |-  ( ph  ->  E  .<_  ( Y  ./\ 
Z ) )
171dalemkelat 33233 . . . . . 6  |-  ( ph  ->  K  e.  Lat )
181, 2, 3, 4, 5, 6, 7, 8, 13dalemdea 33271 . . . . . . 7  |-  ( ph  ->  D  e.  A )
19 eqid 2461 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2019, 4atbase 32899 . . . . . . 7  |-  ( D  e.  A  ->  D  e.  ( Base `  K
) )
2118, 20syl 17 . . . . . 6  |-  ( ph  ->  D  e.  ( Base `  K ) )
221, 2, 3, 4, 5, 6, 7, 8, 15dalemeea 33272 . . . . . . 7  |-  ( ph  ->  E  e.  A )
2319, 4atbase 32899 . . . . . . 7  |-  ( E  e.  A  ->  E  e.  ( Base `  K
) )
2422, 23syl 17 . . . . . 6  |-  ( ph  ->  E  e.  ( Base `  K ) )
251, 6dalemyeb 33258 . . . . . . 7  |-  ( ph  ->  Y  e.  ( Base `  K ) )
261dalemzeo 33242 . . . . . . . 8  |-  ( ph  ->  Z  e.  O )
2719, 6lplnbase 33143 . . . . . . . 8  |-  ( Z  e.  O  ->  Z  e.  ( Base `  K
) )
2826, 27syl 17 . . . . . . 7  |-  ( ph  ->  Z  e.  ( Base `  K ) )
2919, 5latmcl 16346 . . . . . . 7  |-  ( ( K  e.  Lat  /\  Y  e.  ( Base `  K )  /\  Z  e.  ( Base `  K
) )  ->  ( Y  ./\  Z )  e.  ( Base `  K
) )
3017, 25, 28, 29syl3anc 1276 . . . . . 6  |-  ( ph  ->  ( Y  ./\  Z
)  e.  ( Base `  K ) )
3119, 2, 3latjle12 16356 . . . . . 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 1278 . . . . 5  |-  ( ph  ->  ( ( D  .<_  ( Y  ./\  Z )  /\  E  .<_  ( Y 
./\  Z ) )  <-> 
( D  .\/  E
)  .<_  ( Y  ./\  Z ) ) )
3314, 16, 32mpbi2and 937 . . . 4  |-  ( ph  ->  ( D  .\/  E
)  .<_  ( Y  ./\  Z ) )
3433adantr 471 . . 3  |-  ( (
ph  /\  Y  =/=  Z )  ->  ( D  .\/  E )  .<_  ( Y 
./\  Z ) )
351dalemkehl 33232 . . . . 5  |-  ( ph  ->  K  e.  HL )
3635adantr 471 . . . 4  |-  ( (
ph  /\  Y  =/=  Z )  ->  K  e.  HL )
371, 2, 3, 4, 5, 6, 7, 8, 13, 15dalemdnee 33275 . . . . . 6  |-  ( ph  ->  D  =/=  E )
38 eqid 2461 . . . . . . 7  |-  ( LLines `  K )  =  (
LLines `  K )
393, 4, 38llni2 33121 . . . . . 6  |-  ( ( ( K  e.  HL  /\  D  e.  A  /\  E  e.  A )  /\  D  =/=  E
)  ->  ( D  .\/  E )  e.  (
LLines `  K ) )
4035, 18, 22, 37, 39syl31anc 1279 . . . . 5  |-  ( ph  ->  ( D  .\/  E
)  e.  ( LLines `  K ) )
4140adantr 471 . . . 4  |-  ( (
ph  /\  Y  =/=  Z )  ->  ( D  .\/  E )  e.  (
LLines `  K ) )
421, 2, 3, 4, 5, 38, 6, 7, 8, 9dalem15 33287 . . . 4  |-  ( (
ph  /\  Y  =/=  Z )  ->  ( Y  ./\ 
Z )  e.  (
LLines `  K ) )
432, 38llncmp 33131 . . . 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 1276 . . 3  |-  ( (
ph  /\  Y  =/=  Z )  ->  ( ( D  .\/  E )  .<_  ( Y  ./\  Z )  <-> 
( D  .\/  E
)  =  ( Y 
./\  Z ) ) )
4534, 44mpbid 215 . 2  |-  ( (
ph  /\  Y  =/=  Z )  ->  ( D  .\/  E )  =  ( Y  ./\  Z )
)
4612, 45breqtrrd 4442 1  |-  ( (
ph  /\  Y  =/=  Z )  ->  F  .<_  ( D  .\/  E ) )
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   Latclat 16339   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-3or 992  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  df-lvols 33109
This theorem is referenced by:  dalem63  33344
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