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Theorem dalem17 29869
Description: Lemma for dath 29925. 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 29825 . . 3  |-  ( ph  ->  C  .<_  ( R  .\/  U ) )
32adantr 451 . 2  |-  ( (
ph  /\  Y  =  Z )  ->  C  .<_  ( R  .\/  U
) )
41dalemkelat 29813 . . . . . 6  |-  ( ph  ->  K  e.  Lat )
5 dalemc.j . . . . . . 7  |-  .\/  =  ( join `  K )
6 dalemc.a . . . . . . 7  |-  A  =  ( Atoms `  K )
71, 5, 6dalempjqeb 29834 . . . . . 6  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
81, 6dalemreb 29830 . . . . . 6  |-  ( ph  ->  R  e.  ( Base `  K ) )
9 eqid 2283 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
10 dalemc.l . . . . . . 7  |-  .<_  =  ( le `  K )
119, 10, 5latlej2 14167 . . . . . 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 4063 . . . 4  |-  ( ph  ->  R  .<_  Y )
1514adantr 451 . . 3  |-  ( (
ph  /\  Y  =  Z )  ->  R  .<_  Y )
161, 5, 6dalemsjteb 29835 . . . . . . 7  |-  ( ph  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
171, 6dalemueb 29833 . . . . . . 7  |-  ( ph  ->  U  e.  ( Base `  K ) )
189, 10, 5latlej2 14167 . . . . . . 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 4063 . . . . 5  |-  ( ph  ->  U  .<_  Z )
2221adantr 451 . . . 4  |-  ( (
ph  /\  Y  =  Z )  ->  U  .<_  Z )
23 simpr 447 . . . 4  |-  ( (
ph  /\  Y  =  Z )  ->  Y  =  Z )
2422, 23breqtrrd 4049 . . 3  |-  ( (
ph  /\  Y  =  Z )  ->  U  .<_  Y )
25 dalem17.o . . . . . 6  |-  O  =  ( LPlanes `  K )
261, 25dalemyeb 29838 . . . . 5  |-  ( ph  ->  Y  e.  ( Base `  K ) )
279, 10, 5latjle12 14168 . . . . 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 29827 . . . 4  |-  ( ph  ->  C  e.  ( Base `  K ) )
321dalemkehl 29812 . . . . 5  |-  ( ph  ->  K  e.  HL )
331dalemrea 29817 . . . . 5  |-  ( ph  ->  R  e.  A )
341dalemuea 29820 . . . . 5  |-  ( ph  ->  U  e.  A )
359, 5, 6hlatjcl 29556 . . . . 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 14162 . . . 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 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   Latclat 14151   Atomscatm 29453   HLchlt 29540   LPlanesclpl 29681
This theorem is referenced by:  dalem19  29871  dalem25  29887
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-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-lub 14108  df-join 14110  df-lat 14152  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-lplanes 29688
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