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Theorem dalem17 30477
Description: Lemma for dath 30533. When planes  Y and 
Z are equal, the center of perspectivity  C is in  Y. (Contributed by NM, 1-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 )
dalem17.o  |-  O  =  ( LPlanes `  K )
dalem17.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem17.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
Assertion
Ref Expression
dalem17  |-  ( (
ph  /\  Y  =  Z )  ->  C  .<_  Y )

Proof of Theorem dalem17
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 ) ) ) ) )
21dalemclrju 30433 . . 3  |-  ( ph  ->  C  .<_  ( R  .\/  U ) )
32adantr 452 . 2  |-  ( (
ph  /\  Y  =  Z )  ->  C  .<_  ( R  .\/  U
) )
41dalemkelat 30421 . . . . . 6  |-  ( ph  ->  K  e.  Lat )
5 dalemc.j . . . . . . 7  |-  .\/  =  ( join `  K )
6 dalemc.a . . . . . . 7  |-  A  =  ( Atoms `  K )
71, 5, 6dalempjqeb 30442 . . . . . 6  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
81, 6dalemreb 30438 . . . . . 6  |-  ( ph  ->  R  e.  ( Base `  K ) )
9 eqid 2436 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
10 dalemc.l . . . . . . 7  |-  .<_  =  ( le `  K )
119, 10, 5latlej2 14490 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  R  e.  ( Base `  K )
)  ->  R  .<_  ( ( P  .\/  Q
)  .\/  R )
)
124, 7, 8, 11syl3anc 1184 . . . . 5  |-  ( ph  ->  R  .<_  ( ( P  .\/  Q )  .\/  R ) )
13 dalem17.y . . . . 5  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
1412, 13syl6breqr 4252 . . . 4  |-  ( ph  ->  R  .<_  Y )
1514adantr 452 . . 3  |-  ( (
ph  /\  Y  =  Z )  ->  R  .<_  Y )
161, 5, 6dalemsjteb 30443 . . . . . . 7  |-  ( ph  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
171, 6dalemueb 30441 . . . . . . 7  |-  ( ph  ->  U  e.  ( Base `  K ) )
189, 10, 5latlej2 14490 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( S  .\/  T )  e.  ( Base `  K
)  /\  U  e.  ( Base `  K )
)  ->  U  .<_  ( ( S  .\/  T
)  .\/  U )
)
194, 16, 17, 18syl3anc 1184 . . . . . 6  |-  ( ph  ->  U  .<_  ( ( S  .\/  T )  .\/  U ) )
20 dalem17.z . . . . . 6  |-  Z  =  ( ( S  .\/  T )  .\/  U )
2119, 20syl6breqr 4252 . . . . 5  |-  ( ph  ->  U  .<_  Z )
2221adantr 452 . . . 4  |-  ( (
ph  /\  Y  =  Z )  ->  U  .<_  Z )
23 simpr 448 . . . 4  |-  ( (
ph  /\  Y  =  Z )  ->  Y  =  Z )
2422, 23breqtrrd 4238 . . 3  |-  ( (
ph  /\  Y  =  Z )  ->  U  .<_  Y )
25 dalem17.o . . . . . 6  |-  O  =  ( LPlanes `  K )
261, 25dalemyeb 30446 . . . . 5  |-  ( ph  ->  Y  e.  ( Base `  K ) )
279, 10, 5latjle12 14491 . . . . 5  |-  ( ( K  e.  Lat  /\  ( R  e.  ( Base `  K )  /\  U  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) ) )  -> 
( ( R  .<_  Y  /\  U  .<_  Y )  <-> 
( R  .\/  U
)  .<_  Y ) )
284, 8, 17, 26, 27syl13anc 1186 . . . 4  |-  ( ph  ->  ( ( R  .<_  Y  /\  U  .<_  Y )  <-> 
( R  .\/  U
)  .<_  Y ) )
2928adantr 452 . . 3  |-  ( (
ph  /\  Y  =  Z )  ->  (
( R  .<_  Y  /\  U  .<_  Y )  <->  ( R  .\/  U )  .<_  Y ) )
3015, 24, 29mpbi2and 888 . 2  |-  ( (
ph  /\  Y  =  Z )  ->  ( R  .\/  U )  .<_  Y )
311, 6dalemceb 30435 . . . 4  |-  ( ph  ->  C  e.  ( Base `  K ) )
321dalemkehl 30420 . . . . 5  |-  ( ph  ->  K  e.  HL )
331dalemrea 30425 . . . . 5  |-  ( ph  ->  R  e.  A )
341dalemuea 30428 . . . . 5  |-  ( ph  ->  U  e.  A )
359, 5, 6hlatjcl 30164 . . . . 5  |-  ( ( K  e.  HL  /\  R  e.  A  /\  U  e.  A )  ->  ( R  .\/  U
)  e.  ( Base `  K ) )
3632, 33, 34, 35syl3anc 1184 . . . 4  |-  ( ph  ->  ( R  .\/  U
)  e.  ( Base `  K ) )
379, 10lattr 14485 . . . 4  |-  ( ( K  e.  Lat  /\  ( C  e.  ( Base `  K )  /\  ( R  .\/  U )  e.  ( Base `  K
)  /\  Y  e.  ( Base `  K )
) )  ->  (
( C  .<_  ( R 
.\/  U )  /\  ( R  .\/  U ) 
.<_  Y )  ->  C  .<_  Y ) )
384, 31, 36, 26, 37syl13anc 1186 . . 3  |-  ( ph  ->  ( ( C  .<_  ( R  .\/  U )  /\  ( R  .\/  U )  .<_  Y )  ->  C  .<_  Y )
)
3938adantr 452 . 2  |-  ( (
ph  /\  Y  =  Z )  ->  (
( C  .<_  ( R 
.\/  U )  /\  ( R  .\/  U ) 
.<_  Y )  ->  C  .<_  Y ) )
403, 30, 39mp2and 661 1  |-  ( (
ph  /\  Y  =  Z )  ->  C  .<_  Y )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   lecple 13536   joincjn 14401   Latclat 14474   Atomscatm 30061   HLchlt 30148   LPlanesclpl 30289
This theorem is referenced by:  dalem19  30479  dalem25  30495
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-lub 14431  df-join 14433  df-lat 14475  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-lplanes 30296
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