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Theorem dalem17 30491
Description: Lemma for dath 30547. 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 30447 . . 3  |-  ( ph  ->  C  .<_  ( R  .\/  U ) )
32adantr 451 . 2  |-  ( (
ph  /\  Y  =  Z )  ->  C  .<_  ( R  .\/  U
) )
41dalemkelat 30435 . . . . . 6  |-  ( ph  ->  K  e.  Lat )
5 dalemc.j . . . . . . 7  |-  .\/  =  ( join `  K )
6 dalemc.a . . . . . . 7  |-  A  =  ( Atoms `  K )
71, 5, 6dalempjqeb 30456 . . . . . 6  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
81, 6dalemreb 30452 . . . . . 6  |-  ( ph  ->  R  e.  ( Base `  K ) )
9 eqid 2296 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
10 dalemc.l . . . . . . 7  |-  .<_  =  ( le `  K )
119, 10, 5latlej2 14183 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  R  e.  ( Base `  K )
)  ->  R  .<_  ( ( P  .\/  Q
)  .\/  R )
)
124, 7, 8, 11syl3anc 1182 . . . . 5  |-  ( ph  ->  R  .<_  ( ( P  .\/  Q )  .\/  R ) )
13 dalem17.y . . . . 5  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
1412, 13syl6breqr 4079 . . . 4  |-  ( ph  ->  R  .<_  Y )
1514adantr 451 . . 3  |-  ( (
ph  /\  Y  =  Z )  ->  R  .<_  Y )
161, 5, 6dalemsjteb 30457 . . . . . . 7  |-  ( ph  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
171, 6dalemueb 30455 . . . . . . 7  |-  ( ph  ->  U  e.  ( Base `  K ) )
189, 10, 5latlej2 14183 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( S  .\/  T )  e.  ( Base `  K
)  /\  U  e.  ( Base `  K )
)  ->  U  .<_  ( ( S  .\/  T
)  .\/  U )
)
194, 16, 17, 18syl3anc 1182 . . . . . 6  |-  ( ph  ->  U  .<_  ( ( S  .\/  T )  .\/  U ) )
20 dalem17.z . . . . . 6  |-  Z  =  ( ( S  .\/  T )  .\/  U )
2119, 20syl6breqr 4079 . . . . 5  |-  ( ph  ->  U  .<_  Z )
2221adantr 451 . . . 4  |-  ( (
ph  /\  Y  =  Z )  ->  U  .<_  Z )
23 simpr 447 . . . 4  |-  ( (
ph  /\  Y  =  Z )  ->  Y  =  Z )
2422, 23breqtrrd 4065 . . 3  |-  ( (
ph  /\  Y  =  Z )  ->  U  .<_  Y )
25 dalem17.o . . . . . 6  |-  O  =  ( LPlanes `  K )
261, 25dalemyeb 30460 . . . . 5  |-  ( ph  ->  Y  e.  ( Base `  K ) )
279, 10, 5latjle12 14184 . . . . 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 1184 . . . 4  |-  ( ph  ->  ( ( R  .<_  Y  /\  U  .<_  Y )  <-> 
( R  .\/  U
)  .<_  Y ) )
2928adantr 451 . . 3  |-  ( (
ph  /\  Y  =  Z )  ->  (
( R  .<_  Y  /\  U  .<_  Y )  <->  ( R  .\/  U )  .<_  Y ) )
3015, 24, 29mpbi2and 887 . 2  |-  ( (
ph  /\  Y  =  Z )  ->  ( R  .\/  U )  .<_  Y )
311, 6dalemceb 30449 . . . 4  |-  ( ph  ->  C  e.  ( Base `  K ) )
321dalemkehl 30434 . . . . 5  |-  ( ph  ->  K  e.  HL )
331dalemrea 30439 . . . . 5  |-  ( ph  ->  R  e.  A )
341dalemuea 30442 . . . . 5  |-  ( ph  ->  U  e.  A )
359, 5, 6hlatjcl 30178 . . . . 5  |-  ( ( K  e.  HL  /\  R  e.  A  /\  U  e.  A )  ->  ( R  .\/  U
)  e.  ( Base `  K ) )
3632, 33, 34, 35syl3anc 1182 . . . 4  |-  ( ph  ->  ( R  .\/  U
)  e.  ( Base `  K ) )
379, 10lattr 14178 . . . 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 1184 . . 3  |-  ( ph  ->  ( ( C  .<_  ( R  .\/  U )  /\  ( R  .\/  U )  .<_  Y )  ->  C  .<_  Y )
)
3938adantr 451 . 2  |-  ( (
ph  /\  Y  =  Z )  ->  (
( C  .<_  ( R 
.\/  U )  /\  ( R  .\/  U ) 
.<_  Y )  ->  C  .<_  Y ) )
403, 30, 39mp2and 660 1  |-  ( (
ph  /\  Y  =  Z )  ->  C  .<_  Y )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   Latclat 14167   Atomscatm 30075   HLchlt 30162   LPlanesclpl 30303
This theorem is referenced by:  dalem19  30493  dalem25  30509
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-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-lub 14124  df-join 14126  df-lat 14168  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-lplanes 30310
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