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Theorem dalem17 34476
Description: Lemma for dath 34532. 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 34432 . . 3  |-  ( ph  ->  C  .<_  ( R  .\/  U ) )
32adantr 465 . 2  |-  ( (
ph  /\  Y  =  Z )  ->  C  .<_  ( R  .\/  U
) )
41dalemkelat 34420 . . . . . 6  |-  ( ph  ->  K  e.  Lat )
5 dalemc.j . . . . . . 7  |-  .\/  =  ( join `  K )
6 dalemc.a . . . . . . 7  |-  A  =  ( Atoms `  K )
71, 5, 6dalempjqeb 34441 . . . . . 6  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
81, 6dalemreb 34437 . . . . . 6  |-  ( ph  ->  R  e.  ( Base `  K ) )
9 eqid 2467 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
10 dalemc.l . . . . . . 7  |-  .<_  =  ( le `  K )
119, 10, 5latlej2 15541 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  R  e.  ( Base `  K )
)  ->  R  .<_  ( ( P  .\/  Q
)  .\/  R )
)
124, 7, 8, 11syl3anc 1228 . . . . 5  |-  ( ph  ->  R  .<_  ( ( P  .\/  Q )  .\/  R ) )
13 dalem17.y . . . . 5  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
1412, 13syl6breqr 4487 . . . 4  |-  ( ph  ->  R  .<_  Y )
1514adantr 465 . . 3  |-  ( (
ph  /\  Y  =  Z )  ->  R  .<_  Y )
161, 5, 6dalemsjteb 34442 . . . . . . 7  |-  ( ph  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
171, 6dalemueb 34440 . . . . . . 7  |-  ( ph  ->  U  e.  ( Base `  K ) )
189, 10, 5latlej2 15541 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( S  .\/  T )  e.  ( Base `  K
)  /\  U  e.  ( Base `  K )
)  ->  U  .<_  ( ( S  .\/  T
)  .\/  U )
)
194, 16, 17, 18syl3anc 1228 . . . . . 6  |-  ( ph  ->  U  .<_  ( ( S  .\/  T )  .\/  U ) )
20 dalem17.z . . . . . 6  |-  Z  =  ( ( S  .\/  T )  .\/  U )
2119, 20syl6breqr 4487 . . . . 5  |-  ( ph  ->  U  .<_  Z )
2221adantr 465 . . . 4  |-  ( (
ph  /\  Y  =  Z )  ->  U  .<_  Z )
23 simpr 461 . . . 4  |-  ( (
ph  /\  Y  =  Z )  ->  Y  =  Z )
2422, 23breqtrrd 4473 . . 3  |-  ( (
ph  /\  Y  =  Z )  ->  U  .<_  Y )
25 dalem17.o . . . . . 6  |-  O  =  ( LPlanes `  K )
261, 25dalemyeb 34445 . . . . 5  |-  ( ph  ->  Y  e.  ( Base `  K ) )
279, 10, 5latjle12 15542 . . . . 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 1230 . . . 4  |-  ( ph  ->  ( ( R  .<_  Y  /\  U  .<_  Y )  <-> 
( R  .\/  U
)  .<_  Y ) )
2928adantr 465 . . 3  |-  ( (
ph  /\  Y  =  Z )  ->  (
( R  .<_  Y  /\  U  .<_  Y )  <->  ( R  .\/  U )  .<_  Y ) )
3015, 24, 29mpbi2and 919 . 2  |-  ( (
ph  /\  Y  =  Z )  ->  ( R  .\/  U )  .<_  Y )
311, 6dalemceb 34434 . . . 4  |-  ( ph  ->  C  e.  ( Base `  K ) )
321dalemkehl 34419 . . . . 5  |-  ( ph  ->  K  e.  HL )
331dalemrea 34424 . . . . 5  |-  ( ph  ->  R  e.  A )
341dalemuea 34427 . . . . 5  |-  ( ph  ->  U  e.  A )
359, 5, 6hlatjcl 34163 . . . . 5  |-  ( ( K  e.  HL  /\  R  e.  A  /\  U  e.  A )  ->  ( R  .\/  U
)  e.  ( Base `  K ) )
3632, 33, 34, 35syl3anc 1228 . . . 4  |-  ( ph  ->  ( R  .\/  U
)  e.  ( Base `  K ) )
379, 10lattr 15536 . . . 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 1230 . . 3  |-  ( ph  ->  ( ( C  .<_  ( R  .\/  U )  /\  ( R  .\/  U )  .<_  Y )  ->  C  .<_  Y )
)
3938adantr 465 . 2  |-  ( (
ph  /\  Y  =  Z )  ->  (
( C  .<_  ( R 
.\/  U )  /\  ( R  .\/  U ) 
.<_  Y )  ->  C  .<_  Y ) )
403, 30, 39mp2and 679 1  |-  ( (
ph  /\  Y  =  Z )  ->  C  .<_  Y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   Basecbs 14483   lecple 14555   joincjn 15424   Latclat 15525   Atomscatm 34060   HLchlt 34147   LPlanesclpl 34288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-poset 15426  df-lub 15454  df-glb 15455  df-join 15456  df-meet 15457  df-lat 15526  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-lplanes 34295
This theorem is referenced by:  dalem19  34478  dalem25  34494
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