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Theorem dalem16 29868
Description: Lemma for dath 29925. 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 2283 . . . 4  |-  ( Y 
./\  Z )  =  ( Y  ./\  Z
)
10 dalem16.f . . . 4  |-  F  =  ( ( R  .\/  P )  ./\  ( U  .\/  S ) )
111, 2, 3, 4, 5, 6, 7, 8, 9, 10dalem12 29864 . . 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 29862 . . . . 5  |-  ( ph  ->  D  .<_  ( Y  ./\ 
Z ) )
15 dalem16.e . . . . . 6  |-  E  =  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )
161, 2, 3, 4, 5, 6, 7, 8, 9, 15dalem11 29863 . . . . 5  |-  ( ph  ->  E  .<_  ( Y  ./\ 
Z ) )
171dalemkelat 29813 . . . . . 6  |-  ( ph  ->  K  e.  Lat )
181, 2, 3, 4, 5, 6, 7, 8, 13dalemdea 29851 . . . . . . 7  |-  ( ph  ->  D  e.  A )
19 eqid 2283 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2019, 4atbase 29479 . . . . . . 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 29852 . . . . . . 7  |-  ( ph  ->  E  e.  A )
2319, 4atbase 29479 . . . . . . 7  |-  ( E  e.  A  ->  E  e.  ( Base `  K
) )
2422, 23syl 15 . . . . . 6  |-  ( ph  ->  E  e.  ( Base `  K ) )
251, 6dalemyeb 29838 . . . . . . 7  |-  ( ph  ->  Y  e.  ( Base `  K ) )
261dalemzeo 29822 . . . . . . . 8  |-  ( ph  ->  Z  e.  O )
2719, 6lplnbase 29723 . . . . . . . 8  |-  ( Z  e.  O  ->  Z  e.  ( Base `  K
) )
2826, 27syl 15 . . . . . . 7  |-  ( ph  ->  Z  e.  ( Base `  K ) )
2919, 5latmcl 14157 . . . . . . 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 14168 . . . . . 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 29812 . . . . 5  |-  ( ph  ->  K  e.  HL )
3635adantr 451 . . . 4  |-  ( (
ph  /\  Y  =/=  Z )  ->  K  e.  HL )
371, 2, 3, 4, 5, 6, 7, 8, 13, 15dalemdnee 29855 . . . . . 6  |-  ( ph  ->  D  =/=  E )
38 eqid 2283 . . . . . . 7  |-  ( LLines `  K )  =  (
LLines `  K )
393, 4, 38llni2 29701 . . . . . 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 29867 . . . 4  |-  ( (
ph  /\  Y  =/=  Z )  ->  ( Y  ./\ 
Z )  e.  (
LLines `  K ) )
432, 38llncmp 29711 . . . 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 4049 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 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Latclat 14151   Atomscatm 29453   HLchlt 29540   LLinesclln 29680   LPlanesclpl 29681
This theorem is referenced by:  dalem63  29924
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689
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