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Theorem dalem60 30466
Description: Lemma for dath 30470. 
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 30463 . . 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 30464 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  E  .<_  B )
181dalemkelat 30358 . . . . 5  |-  ( ph  ->  K  e.  Lat )
19183ad2ant1 978 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  Lat )
201, 2, 3, 4, 6, 7, 8, 9, 10dalemdea 30396 . . . . . 6  |-  ( ph  ->  D  e.  A )
21 eqid 2435 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
2221, 4atbase 30024 . . . . . 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 30397 . . . . . 6  |-  ( ph  ->  E  e.  A )
2621, 4atbase 30024 . . . . . 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 2435 . . . . . 6  |-  ( LLines `  K )  =  (
LLines `  K )
301, 2, 3, 4, 5, 6, 29, 7, 8, 9, 11, 12, 13, 14dalem53 30459 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  e.  ( LLines `  K ) )
3121, 29llnbase 30243 . . . . 5  |-  ( B  e.  ( LLines `  K
)  ->  B  e.  ( Base `  K )
)
3230, 31syl 16 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  e.  ( Base `  K ) )
3321, 2, 3latjle12 14483 . . . 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 30357 . . . 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 30400 . . . . 5  |-  ( ph  ->  D  =/=  E )
393, 4, 29llni2 30246 . . . . 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 30256 . . 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 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   lecple 13528   joincjn 14393   meetcmee 14394   Latclat 14466   Atomscatm 29998   HLchlt 30085   LLinesclln 30225   LPlanesclpl 30226
This theorem is referenced by:  dalem61  30467
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-lat 14467  df-clat 14529  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-llines 30232  df-lplanes 30233  df-lvols 30234
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