Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dalem57 Structured version   Unicode version

Theorem dalem57 30588
Description: Lemma for dath 30595. Axis of perspectivity point  D is on the auxiliary line  B. (Contributed by NM, 9-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
) ) ) )
dalem57.m  |-  ./\  =  ( meet `  K )
dalem57.o  |-  O  =  ( LPlanes `  K )
dalem57.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem57.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
dalem57.d  |-  D  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )
dalem57.g  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
dalem57.h  |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )
dalem57.i  |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )
dalem57.b1  |-  B  =  ( ( ( G 
.\/  H )  .\/  I )  ./\  Y
)
Assertion
Ref Expression
dalem57  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  D  .<_  B )

Proof of Theorem dalem57
StepHypRef Expression
1 dalem.ph . . . . . . 7  |-  ( 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 . . . . . . 7  |-  .<_  =  ( le `  K )
3 dalem.j . . . . . . 7  |-  .\/  =  ( join `  K )
4 dalem.a . . . . . . 7  |-  A  =  ( Atoms `  K )
5 dalem.ps . . . . . . 7  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
6 dalem57.m . . . . . . 7  |-  ./\  =  ( meet `  K )
7 dalem57.o . . . . . . 7  |-  O  =  ( LPlanes `  K )
8 dalem57.y . . . . . . 7  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
9 dalem57.z . . . . . . 7  |-  Z  =  ( ( S  .\/  T )  .\/  U )
10 dalem57.g . . . . . . 7  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
11 dalem57.h . . . . . . 7  |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )
12 dalem57.i . . . . . . 7  |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )
13 dalem57.b1 . . . . . . 7  |-  B  =  ( ( ( G 
.\/  H )  .\/  I )  ./\  Y
)
141, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13dalem55 30586 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  =  ( ( G  .\/  H )  ./\  B )
)
151dalemkelat 30483 . . . . . . . 8  |-  ( ph  ->  K  e.  Lat )
16153ad2ant1 979 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  Lat )
171dalemkehl 30482 . . . . . . . . 9  |-  ( ph  ->  K  e.  HL )
18173ad2ant1 979 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
191, 2, 3, 4, 5, 6, 7, 8, 9, 10dalem23 30555 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  A )
201, 2, 3, 4, 5, 6, 7, 8, 9, 11dalem29 30560 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  H  e.  A )
21 eqid 2438 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
2221, 3, 4hlatjcl 30226 . . . . . . . 8  |-  ( ( K  e.  HL  /\  G  e.  A  /\  H  e.  A )  ->  ( G  .\/  H
)  e.  ( Base `  K ) )
2318, 19, 20, 22syl3anc 1185 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  .\/  H
)  e.  ( Base `  K ) )
241, 3, 4dalempjqeb 30504 . . . . . . . 8  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
25243ad2ant1 979 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  .\/  Q
)  e.  ( Base `  K ) )
2621, 2, 6latmle2 14508 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  ( P  .\/  Q )  e.  (
Base `  K )
)  ->  ( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( P  .\/  Q ) )
2716, 23, 25, 26syl3anc 1185 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( P  .\/  Q ) )
2814, 27eqbrtrrd 4236 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  .<_  ( P  .\/  Q
) )
291, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13dalem56 30587 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( S  .\/  T ) )  =  ( ( G  .\/  H )  ./\  B )
)
301, 3, 4dalemsjteb 30505 . . . . . . . 8  |-  ( ph  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
31303ad2ant1 979 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( S  .\/  T
)  e.  ( Base `  K ) )
3221, 2, 6latmle2 14508 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  ( S  .\/  T )  e.  (
Base `  K )
)  ->  ( ( G  .\/  H )  ./\  ( S  .\/  T ) )  .<_  ( S  .\/  T ) )
3316, 23, 31, 32syl3anc 1185 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( S  .\/  T ) )  .<_  ( S  .\/  T ) )
3429, 33eqbrtrrd 4236 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  .<_  ( S  .\/  T
) )
351, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13dalem54 30585 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  e.  A )
3621, 4atbase 30149 . . . . . . 7  |-  ( ( ( G  .\/  H
)  ./\  B )  e.  A  ->  ( ( G  .\/  H ) 
./\  B )  e.  ( Base `  K
) )
3735, 36syl 16 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  e.  ( Base `  K
) )
3821, 2, 6latlem12 14509 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( ( ( G 
.\/  H )  ./\  B )  e.  ( Base `  K )  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  ( S  .\/  T )  e.  (
Base `  K )
) )  ->  (
( ( ( G 
.\/  H )  ./\  B )  .<_  ( P  .\/  Q )  /\  (
( G  .\/  H
)  ./\  B )  .<_  ( S  .\/  T
) )  <->  ( ( G  .\/  H )  ./\  B )  .<_  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) ) ) )
3916, 37, 25, 31, 38syl13anc 1187 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( ( G  .\/  H ) 
./\  B )  .<_  ( P  .\/  Q )  /\  ( ( G 
.\/  H )  ./\  B )  .<_  ( S  .\/  T ) )  <->  ( ( G  .\/  H )  ./\  B )  .<_  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) ) ) )
4028, 34, 39mpbi2and 889 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  .<_  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) ) )
41 dalem57.d . . . 4  |-  D  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )
4240, 41syl6breqr 4254 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  .<_  D )
43 hlatl 30220 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
4418, 43syl 16 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  AtLat )
451, 2, 3, 4, 6, 7, 8, 9, 41dalemdea 30521 . . . . 5  |-  ( ph  ->  D  e.  A )
46453ad2ant1 979 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  D  e.  A )
472, 4atcmp 30171 . . . 4  |-  ( ( K  e.  AtLat  /\  (
( G  .\/  H
)  ./\  B )  e.  A  /\  D  e.  A )  ->  (
( ( G  .\/  H )  ./\  B )  .<_  D  <->  ( ( G 
.\/  H )  ./\  B )  =  D ) )
4844, 35, 46, 47syl3anc 1185 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( G 
.\/  H )  ./\  B )  .<_  D  <->  ( ( G  .\/  H )  ./\  B )  =  D ) )
4942, 48mpbid 203 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  =  D )
50 eqid 2438 . . . . 5  |-  ( LLines `  K )  =  (
LLines `  K )
511, 2, 3, 4, 5, 6, 50, 7, 8, 9, 10, 11, 12, 13dalem53 30584 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  e.  ( LLines `  K ) )
5221, 50llnbase 30368 . . . 4  |-  ( B  e.  ( LLines `  K
)  ->  B  e.  ( Base `  K )
)
5351, 52syl 16 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  e.  ( Base `  K ) )
5421, 2, 6latmle2 14508 . . 3  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  B  e.  ( Base `  K )
)  ->  ( ( G  .\/  H )  ./\  B )  .<_  B )
5516, 23, 53, 54syl3anc 1185 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  .<_  B )
5649, 55eqbrtrrd 4236 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  D  .<_  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   lecple 13538   joincjn 14403   meetcmee 14404   Latclat 14476   Atomscatm 30123   AtLatcal 30124   HLchlt 30210   LLinesclln 30350   LPlanesclpl 30351
This theorem is referenced by:  dalem58  30589  dalem60  30591
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-lat 14477  df-clat 14539  df-oposet 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-llines 30357  df-lplanes 30358  df-lvols 30359
  Copyright terms: Public domain W3C validator