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Theorem dalem16 30490
Description: Lemma for dath 30547. 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 2296 . . . 4  |-  ( Y 
./\  Z )  =  ( Y  ./\  Z
)
10 dalem16.f . . . 4  |-  F  =  ( ( R  .\/  P )  ./\  ( U  .\/  S ) )
111, 2, 3, 4, 5, 6, 7, 8, 9, 10dalem12 30486 . . 3  |-  ( ph  ->  F  .<_  ( Y  ./\ 
Z ) )
1211adantr 451 . 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 30484 . . . . 5  |-  ( ph  ->  D  .<_  ( Y  ./\ 
Z ) )
15 dalem16.e . . . . . 6  |-  E  =  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )
161, 2, 3, 4, 5, 6, 7, 8, 9, 15dalem11 30485 . . . . 5  |-  ( ph  ->  E  .<_  ( Y  ./\ 
Z ) )
171dalemkelat 30435 . . . . . 6  |-  ( ph  ->  K  e.  Lat )
181, 2, 3, 4, 5, 6, 7, 8, 13dalemdea 30473 . . . . . . 7  |-  ( ph  ->  D  e.  A )
19 eqid 2296 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2019, 4atbase 30101 . . . . . . 7  |-  ( D  e.  A  ->  D  e.  ( Base `  K
) )
2118, 20syl 15 . . . . . 6  |-  ( ph  ->  D  e.  ( Base `  K ) )
221, 2, 3, 4, 5, 6, 7, 8, 15dalemeea 30474 . . . . . . 7  |-  ( ph  ->  E  e.  A )
2319, 4atbase 30101 . . . . . . 7  |-  ( E  e.  A  ->  E  e.  ( Base `  K
) )
2422, 23syl 15 . . . . . 6  |-  ( ph  ->  E  e.  ( Base `  K ) )
251, 6dalemyeb 30460 . . . . . . 7  |-  ( ph  ->  Y  e.  ( Base `  K ) )
261dalemzeo 30444 . . . . . . . 8  |-  ( ph  ->  Z  e.  O )
2719, 6lplnbase 30345 . . . . . . . 8  |-  ( Z  e.  O  ->  Z  e.  ( Base `  K
) )
2826, 27syl 15 . . . . . . 7  |-  ( ph  ->  Z  e.  ( Base `  K ) )
2919, 5latmcl 14173 . . . . . . 7  |-  ( ( K  e.  Lat  /\  Y  e.  ( Base `  K )  /\  Z  e.  ( Base `  K
) )  ->  ( Y  ./\  Z )  e.  ( Base `  K
) )
3017, 25, 28, 29syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( Y  ./\  Z
)  e.  ( Base `  K ) )
3119, 2, 3latjle12 14184 . . . . . 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 1184 . . . . 5  |-  ( ph  ->  ( ( D  .<_  ( Y  ./\  Z )  /\  E  .<_  ( Y 
./\  Z ) )  <-> 
( D  .\/  E
)  .<_  ( Y  ./\  Z ) ) )
3314, 16, 32mpbi2and 887 . . . 4  |-  ( ph  ->  ( D  .\/  E
)  .<_  ( Y  ./\  Z ) )
3433adantr 451 . . 3  |-  ( (
ph  /\  Y  =/=  Z )  ->  ( D  .\/  E )  .<_  ( Y 
./\  Z ) )
351dalemkehl 30434 . . . . 5  |-  ( ph  ->  K  e.  HL )
3635adantr 451 . . . 4  |-  ( (
ph  /\  Y  =/=  Z )  ->  K  e.  HL )
371, 2, 3, 4, 5, 6, 7, 8, 13, 15dalemdnee 30477 . . . . . 6  |-  ( ph  ->  D  =/=  E )
38 eqid 2296 . . . . . . 7  |-  ( LLines `  K )  =  (
LLines `  K )
393, 4, 38llni2 30323 . . . . . 6  |-  ( ( ( K  e.  HL  /\  D  e.  A  /\  E  e.  A )  /\  D  =/=  E
)  ->  ( D  .\/  E )  e.  (
LLines `  K ) )
4035, 18, 22, 37, 39syl31anc 1185 . . . . 5  |-  ( ph  ->  ( D  .\/  E
)  e.  ( LLines `  K ) )
4140adantr 451 . . . 4  |-  ( (
ph  /\  Y  =/=  Z )  ->  ( D  .\/  E )  e.  (
LLines `  K ) )
421, 2, 3, 4, 5, 38, 6, 7, 8, 9dalem15 30489 . . . 4  |-  ( (
ph  /\  Y  =/=  Z )  ->  ( Y  ./\ 
Z )  e.  (
LLines `  K ) )
432, 38llncmp 30333 . . . 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 1182 . . 3  |-  ( (
ph  /\  Y  =/=  Z )  ->  ( ( D  .\/  E )  .<_  ( Y  ./\  Z )  <-> 
( D  .\/  E
)  =  ( Y 
./\  Z ) ) )
4534, 44mpbid 201 . 2  |-  ( (
ph  /\  Y  =/=  Z )  ->  ( D  .\/  E )  =  ( Y  ./\  Z )
)
4612, 45breqtrrd 4065 1  |-  ( (
ph  /\  Y  =/=  Z )  ->  F  .<_  ( D  .\/  E ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   meetcmee 14095   Latclat 14167   Atomscatm 30075   HLchlt 30162   LLinesclln 30302   LPlanesclpl 30303
This theorem is referenced by:  dalem63  30546
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310  df-lvols 30311
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