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Theorem dalem17 32953
Description: Lemma for dath 33009. 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 32909 . . 3  |-  ( ph  ->  C  .<_  ( R  .\/  U ) )
32adantr 466 . 2  |-  ( (
ph  /\  Y  =  Z )  ->  C  .<_  ( R  .\/  U
) )
41dalemkelat 32897 . . . . . 6  |-  ( ph  ->  K  e.  Lat )
5 dalemc.j . . . . . . 7  |-  .\/  =  ( join `  K )
6 dalemc.a . . . . . . 7  |-  A  =  ( Atoms `  K )
71, 5, 6dalempjqeb 32918 . . . . . 6  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
81, 6dalemreb 32914 . . . . . 6  |-  ( ph  ->  R  e.  ( Base `  K ) )
9 eqid 2429 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
10 dalemc.l . . . . . . 7  |-  .<_  =  ( le `  K )
119, 10, 5latlej2 16258 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  R  e.  ( Base `  K )
)  ->  R  .<_  ( ( P  .\/  Q
)  .\/  R )
)
124, 7, 8, 11syl3anc 1264 . . . . 5  |-  ( ph  ->  R  .<_  ( ( P  .\/  Q )  .\/  R ) )
13 dalem17.y . . . . 5  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
1412, 13syl6breqr 4466 . . . 4  |-  ( ph  ->  R  .<_  Y )
1514adantr 466 . . 3  |-  ( (
ph  /\  Y  =  Z )  ->  R  .<_  Y )
161, 5, 6dalemsjteb 32919 . . . . . . 7  |-  ( ph  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
171, 6dalemueb 32917 . . . . . . 7  |-  ( ph  ->  U  e.  ( Base `  K ) )
189, 10, 5latlej2 16258 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( S  .\/  T )  e.  ( Base `  K
)  /\  U  e.  ( Base `  K )
)  ->  U  .<_  ( ( S  .\/  T
)  .\/  U )
)
194, 16, 17, 18syl3anc 1264 . . . . . 6  |-  ( ph  ->  U  .<_  ( ( S  .\/  T )  .\/  U ) )
20 dalem17.z . . . . . 6  |-  Z  =  ( ( S  .\/  T )  .\/  U )
2119, 20syl6breqr 4466 . . . . 5  |-  ( ph  ->  U  .<_  Z )
2221adantr 466 . . . 4  |-  ( (
ph  /\  Y  =  Z )  ->  U  .<_  Z )
23 simpr 462 . . . 4  |-  ( (
ph  /\  Y  =  Z )  ->  Y  =  Z )
2422, 23breqtrrd 4452 . . 3  |-  ( (
ph  /\  Y  =  Z )  ->  U  .<_  Y )
25 dalem17.o . . . . . 6  |-  O  =  ( LPlanes `  K )
261, 25dalemyeb 32922 . . . . 5  |-  ( ph  ->  Y  e.  ( Base `  K ) )
279, 10, 5latjle12 16259 . . . . 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 1266 . . . 4  |-  ( ph  ->  ( ( R  .<_  Y  /\  U  .<_  Y )  <-> 
( R  .\/  U
)  .<_  Y ) )
2928adantr 466 . . 3  |-  ( (
ph  /\  Y  =  Z )  ->  (
( R  .<_  Y  /\  U  .<_  Y )  <->  ( R  .\/  U )  .<_  Y ) )
3015, 24, 29mpbi2and 929 . 2  |-  ( (
ph  /\  Y  =  Z )  ->  ( R  .\/  U )  .<_  Y )
311, 6dalemceb 32911 . . . 4  |-  ( ph  ->  C  e.  ( Base `  K ) )
321dalemkehl 32896 . . . . 5  |-  ( ph  ->  K  e.  HL )
331dalemrea 32901 . . . . 5  |-  ( ph  ->  R  e.  A )
341dalemuea 32904 . . . . 5  |-  ( ph  ->  U  e.  A )
359, 5, 6hlatjcl 32640 . . . . 5  |-  ( ( K  e.  HL  /\  R  e.  A  /\  U  e.  A )  ->  ( R  .\/  U
)  e.  ( Base `  K ) )
3632, 33, 34, 35syl3anc 1264 . . . 4  |-  ( ph  ->  ( R  .\/  U
)  e.  ( Base `  K ) )
379, 10lattr 16253 . . . 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 1266 . . 3  |-  ( ph  ->  ( ( C  .<_  ( R  .\/  U )  /\  ( R  .\/  U )  .<_  Y )  ->  C  .<_  Y )
)
3938adantr 466 . 2  |-  ( (
ph  /\  Y  =  Z )  ->  (
( C  .<_  ( R 
.\/  U )  /\  ( R  .\/  U ) 
.<_  Y )  ->  C  .<_  Y ) )
403, 30, 39mp2and 683 1  |-  ( (
ph  /\  Y  =  Z )  ->  C  .<_  Y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   Basecbs 15084   lecple 15159   joincjn 16140   Latclat 16242   Atomscatm 32537   HLchlt 32624   LPlanesclpl 32765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-poset 16142  df-lub 16171  df-glb 16172  df-join 16173  df-meet 16174  df-lat 16243  df-ats 32541  df-atl 32572  df-cvlat 32596  df-hlat 32625  df-lplanes 32772
This theorem is referenced by:  dalem19  32955  dalem25  32971
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