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Theorem dalem16 33686
Description: Lemma for dath 33743. 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 2454 . . . 4  |-  ( Y 
./\  Z )  =  ( Y  ./\  Z
)
10 dalem16.f . . . 4  |-  F  =  ( ( R  .\/  P )  ./\  ( U  .\/  S ) )
111, 2, 3, 4, 5, 6, 7, 8, 9, 10dalem12 33682 . . 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 33680 . . . . 5  |-  ( ph  ->  D  .<_  ( Y  ./\ 
Z ) )
15 dalem16.e . . . . . 6  |-  E  =  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )
161, 2, 3, 4, 5, 6, 7, 8, 9, 15dalem11 33681 . . . . 5  |-  ( ph  ->  E  .<_  ( Y  ./\ 
Z ) )
171dalemkelat 33631 . . . . . 6  |-  ( ph  ->  K  e.  Lat )
181, 2, 3, 4, 5, 6, 7, 8, 13dalemdea 33669 . . . . . . 7  |-  ( ph  ->  D  e.  A )
19 eqid 2454 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2019, 4atbase 33297 . . . . . . 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 33670 . . . . . . 7  |-  ( ph  ->  E  e.  A )
2319, 4atbase 33297 . . . . . . 7  |-  ( E  e.  A  ->  E  e.  ( Base `  K
) )
2422, 23syl 16 . . . . . 6  |-  ( ph  ->  E  e.  ( Base `  K ) )
251, 6dalemyeb 33656 . . . . . . 7  |-  ( ph  ->  Y  e.  ( Base `  K ) )
261dalemzeo 33640 . . . . . . . 8  |-  ( ph  ->  Z  e.  O )
2719, 6lplnbase 33541 . . . . . . . 8  |-  ( Z  e.  O  ->  Z  e.  ( Base `  K
) )
2826, 27syl 16 . . . . . . 7  |-  ( ph  ->  Z  e.  ( Base `  K ) )
2919, 5latmcl 15345 . . . . . . 7  |-  ( ( K  e.  Lat  /\  Y  e.  ( Base `  K )  /\  Z  e.  ( Base `  K
) )  ->  ( Y  ./\  Z )  e.  ( Base `  K
) )
3017, 25, 28, 29syl3anc 1219 . . . . . 6  |-  ( ph  ->  ( Y  ./\  Z
)  e.  ( Base `  K ) )
3119, 2, 3latjle12 15355 . . . . . 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 1221 . . . . 5  |-  ( ph  ->  ( ( D  .<_  ( Y  ./\  Z )  /\  E  .<_  ( Y 
./\  Z ) )  <-> 
( D  .\/  E
)  .<_  ( Y  ./\  Z ) ) )
3314, 16, 32mpbi2and 912 . . . 4  |-  ( ph  ->  ( D  .\/  E
)  .<_  ( Y  ./\  Z ) )
3433adantr 465 . . 3  |-  ( (
ph  /\  Y  =/=  Z )  ->  ( D  .\/  E )  .<_  ( Y 
./\  Z ) )
351dalemkehl 33630 . . . . 5  |-  ( ph  ->  K  e.  HL )
3635adantr 465 . . . 4  |-  ( (
ph  /\  Y  =/=  Z )  ->  K  e.  HL )
371, 2, 3, 4, 5, 6, 7, 8, 13, 15dalemdnee 33673 . . . . . 6  |-  ( ph  ->  D  =/=  E )
38 eqid 2454 . . . . . . 7  |-  ( LLines `  K )  =  (
LLines `  K )
393, 4, 38llni2 33519 . . . . . 6  |-  ( ( ( K  e.  HL  /\  D  e.  A  /\  E  e.  A )  /\  D  =/=  E
)  ->  ( D  .\/  E )  e.  (
LLines `  K ) )
4035, 18, 22, 37, 39syl31anc 1222 . . . . 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 33685 . . . 4  |-  ( (
ph  /\  Y  =/=  Z )  ->  ( Y  ./\ 
Z )  e.  (
LLines `  K ) )
432, 38llncmp 33529 . . . 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 1219 . . 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 4429 1  |-  ( (
ph  /\  Y  =/=  Z )  ->  F  .<_  ( D  .\/  E ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   Basecbs 14296   lecple 14368   joincjn 15237   meetcmee 15238   Latclat 15338   Atomscatm 33271   HLchlt 33358   LLinesclln 33498   LPlanesclpl 33499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-poset 15239  df-plt 15251  df-lub 15267  df-glb 15268  df-join 15269  df-meet 15270  df-p0 15332  df-lat 15339  df-clat 15401  df-oposet 33184  df-ol 33186  df-oml 33187  df-covers 33274  df-ats 33275  df-atl 33306  df-cvlat 33330  df-hlat 33359  df-llines 33505  df-lplanes 33506  df-lvols 33507
This theorem is referenced by:  dalem63  33742
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