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Theorem dalem60 29846
Description: Lemma for dath 29850. 
B is an axis of perspectivity (almost). (Contributed by NM, 11-Aug-2012.)
Hypotheses
Ref Expression
dalem.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 ) ) ) ) )
dalem.l  |-  .<_  =  ( le `  K )
dalem.j  |-  .\/  =  ( join `  K )
dalem.a  |-  A  =  ( Atoms `  K )
dalem.ps  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
dalem60.m  |-  ./\  =  ( meet `  K )
dalem60.o  |-  O  =  ( LPlanes `  K )
dalem60.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem60.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
dalem60.d  |-  D  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )
dalem60.e  |-  E  =  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )
dalem60.g  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
dalem60.h  |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )
dalem60.i  |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )
dalem60.b1  |-  B  =  ( ( ( G 
.\/  H )  .\/  I )  ./\  Y
)
Assertion
Ref Expression
dalem60  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( D  .\/  E
)  =  B )

Proof of Theorem dalem60
StepHypRef Expression
1 dalem.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 ) ) ) ) )
2 dalem.l . . . 4  |-  .<_  =  ( le `  K )
3 dalem.j . . . 4  |-  .\/  =  ( join `  K )
4 dalem.a . . . 4  |-  A  =  ( Atoms `  K )
5 dalem.ps . . . 4  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
6 dalem60.m . . . 4  |-  ./\  =  ( meet `  K )
7 dalem60.o . . . 4  |-  O  =  ( LPlanes `  K )
8 dalem60.y . . . 4  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
9 dalem60.z . . . 4  |-  Z  =  ( ( S  .\/  T )  .\/  U )
10 dalem60.d . . . 4  |-  D  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )
11 dalem60.g . . . 4  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
12 dalem60.h . . . 4  |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )
13 dalem60.i . . . 4  |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )
14 dalem60.b1 . . . 4  |-  B  =  ( ( ( G 
.\/  H )  .\/  I )  ./\  Y
)
151, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14dalem57 29843 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  D  .<_  B )
16 dalem60.e . . . 4  |-  E  =  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )
171, 2, 3, 4, 5, 6, 7, 8, 9, 16, 11, 12, 13, 14dalem58 29844 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  E  .<_  B )
181dalemkelat 29738 . . . . 5  |-  ( ph  ->  K  e.  Lat )
19183ad2ant1 978 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  Lat )
201, 2, 3, 4, 6, 7, 8, 9, 10dalemdea 29776 . . . . . 6  |-  ( ph  ->  D  e.  A )
21 eqid 2387 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
2221, 4atbase 29404 . . . . . 6  |-  ( D  e.  A  ->  D  e.  ( Base `  K
) )
2320, 22syl 16 . . . . 5  |-  ( ph  ->  D  e.  ( Base `  K ) )
24233ad2ant1 978 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  D  e.  ( Base `  K ) )
251, 2, 3, 4, 6, 7, 8, 9, 16dalemeea 29777 . . . . . 6  |-  ( ph  ->  E  e.  A )
2621, 4atbase 29404 . . . . . 6  |-  ( E  e.  A  ->  E  e.  ( Base `  K
) )
2725, 26syl 16 . . . . 5  |-  ( ph  ->  E  e.  ( Base `  K ) )
28273ad2ant1 978 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  E  e.  ( Base `  K ) )
29 eqid 2387 . . . . . 6  |-  ( LLines `  K )  =  (
LLines `  K )
301, 2, 3, 4, 5, 6, 29, 7, 8, 9, 11, 12, 13, 14dalem53 29839 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  e.  ( LLines `  K ) )
3121, 29llnbase 29623 . . . . 5  |-  ( B  e.  ( LLines `  K
)  ->  B  e.  ( Base `  K )
)
3230, 31syl 16 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  e.  ( Base `  K ) )
3321, 2, 3latjle12 14418 . . . 4  |-  ( ( K  e.  Lat  /\  ( D  e.  ( Base `  K )  /\  E  e.  ( Base `  K )  /\  B  e.  ( Base `  K
) ) )  -> 
( ( D  .<_  B  /\  E  .<_  B )  <-> 
( D  .\/  E
)  .<_  B ) )
3419, 24, 28, 32, 33syl13anc 1186 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( D  .<_  B  /\  E  .<_  B )  <-> 
( D  .\/  E
)  .<_  B ) )
3515, 17, 34mpbi2and 888 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( D  .\/  E
)  .<_  B )
361dalemkehl 29737 . . . 4  |-  ( ph  ->  K  e.  HL )
37363ad2ant1 978 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
381, 2, 3, 4, 6, 7, 8, 9, 10, 16dalemdnee 29780 . . . . 5  |-  ( ph  ->  D  =/=  E )
393, 4, 29llni2 29626 . . . . 5  |-  ( ( ( K  e.  HL  /\  D  e.  A  /\  E  e.  A )  /\  D  =/=  E
)  ->  ( D  .\/  E )  e.  (
LLines `  K ) )
4036, 20, 25, 38, 39syl31anc 1187 . . . 4  |-  ( ph  ->  ( D  .\/  E
)  e.  ( LLines `  K ) )
41403ad2ant1 978 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( D  .\/  E
)  e.  ( LLines `  K ) )
422, 29llncmp 29636 . . 3  |-  ( ( K  e.  HL  /\  ( D  .\/  E )  e.  ( LLines `  K
)  /\  B  e.  ( LLines `  K )
)  ->  ( ( D  .\/  E )  .<_  B 
<->  ( D  .\/  E
)  =  B ) )
4337, 41, 30, 42syl3anc 1184 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( D  .\/  E )  .<_  B  <->  ( D  .\/  E )  =  B ) )
4435, 43mpbid 202 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( D  .\/  E
)  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550   class class class wbr 4153   ` cfv 5394  (class class class)co 6020   Basecbs 13396   lecple 13463   joincjn 14328   meetcmee 14329   Latclat 14401   Atomscatm 29378   HLchlt 29465   LLinesclln 29605   LPlanesclpl 29606
This theorem is referenced by:  dalem61  29847
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-undef 6479  df-riota 6485  df-poset 14330  df-plt 14342  df-lub 14358  df-glb 14359  df-join 14360  df-meet 14361  df-p0 14395  df-lat 14402  df-clat 14464  df-oposet 29291  df-ol 29293  df-oml 29294  df-covers 29381  df-ats 29382  df-atl 29413  df-cvlat 29437  df-hlat 29466  df-llines 29612  df-lplanes 29613  df-lvols 29614
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