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Theorem dalem57 33712
Description: Lemma for dath 33719. 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 33710 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  =  ( ( G  .\/  H )  ./\  B )
)
151dalemkelat 33607 . . . . . . . 8  |-  ( ph  ->  K  e.  Lat )
16153ad2ant1 1009 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  Lat )
171dalemkehl 33606 . . . . . . . . 9  |-  ( ph  ->  K  e.  HL )
18173ad2ant1 1009 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
191, 2, 3, 4, 5, 6, 7, 8, 9, 10dalem23 33679 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  A )
201, 2, 3, 4, 5, 6, 7, 8, 9, 11dalem29 33684 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  H  e.  A )
21 eqid 2454 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
2221, 3, 4hlatjcl 33350 . . . . . . . 8  |-  ( ( K  e.  HL  /\  G  e.  A  /\  H  e.  A )  ->  ( G  .\/  H
)  e.  ( Base `  K ) )
2318, 19, 20, 22syl3anc 1219 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  .\/  H
)  e.  ( Base `  K ) )
241, 3, 4dalempjqeb 33628 . . . . . . . 8  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
25243ad2ant1 1009 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  .\/  Q
)  e.  ( Base `  K ) )
2621, 2, 6latmle2 15367 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  ( P  .\/  Q )  e.  (
Base `  K )
)  ->  ( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( P  .\/  Q ) )
2716, 23, 25, 26syl3anc 1219 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( P  .\/  Q ) )
2814, 27eqbrtrrd 4423 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  .<_  ( P  .\/  Q
) )
291, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13dalem56 33711 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( S  .\/  T ) )  =  ( ( G  .\/  H )  ./\  B )
)
301, 3, 4dalemsjteb 33629 . . . . . . . 8  |-  ( ph  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
31303ad2ant1 1009 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( S  .\/  T
)  e.  ( Base `  K ) )
3221, 2, 6latmle2 15367 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  ( S  .\/  T )  e.  (
Base `  K )
)  ->  ( ( G  .\/  H )  ./\  ( S  .\/  T ) )  .<_  ( S  .\/  T ) )
3316, 23, 31, 32syl3anc 1219 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( S  .\/  T ) )  .<_  ( S  .\/  T ) )
3429, 33eqbrtrrd 4423 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  .<_  ( S  .\/  T
) )
351, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13dalem54 33709 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  e.  A )
3621, 4atbase 33273 . . . . . . 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 15368 . . . . . 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 1221 . . . . 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 912 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  .<_  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) ) )
41 dalem57.d . . . 4  |-  D  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )
4240, 41syl6breqr 4441 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  .<_  D )
43 hlatl 33344 . . . . 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 33645 . . . . 5  |-  ( ph  ->  D  e.  A )
46453ad2ant1 1009 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  D  e.  A )
472, 4atcmp 33295 . . . 4  |-  ( ( K  e.  AtLat  /\  (
( G  .\/  H
)  ./\  B )  e.  A  /\  D  e.  A )  ->  (
( ( G  .\/  H )  ./\  B )  .<_  D  <->  ( ( G 
.\/  H )  ./\  B )  =  D ) )
4844, 35, 46, 47syl3anc 1219 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( G 
.\/  H )  ./\  B )  .<_  D  <->  ( ( G  .\/  H )  ./\  B )  =  D ) )
4942, 48mpbid 210 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  =  D )
50 eqid 2454 . . . . 5  |-  ( LLines `  K )  =  (
LLines `  K )
511, 2, 3, 4, 5, 6, 50, 7, 8, 9, 10, 11, 12, 13dalem53 33708 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  e.  ( LLines `  K ) )
5221, 50llnbase 33492 . . . 4  |-  ( B  e.  ( LLines `  K
)  ->  B  e.  ( Base `  K )
)
5351, 52syl 16 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  e.  ( Base `  K ) )
5421, 2, 6latmle2 15367 . . 3  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  B  e.  ( Base `  K )
)  ->  ( ( G  .\/  H )  ./\  B )  .<_  B )
5516, 23, 53, 54syl3anc 1219 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  .<_  B )
5649, 55eqbrtrrd 4423 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  D  .<_  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   class class class wbr 4401   ` cfv 5527  (class class class)co 6201   Basecbs 14293   lecple 14365   joincjn 15234   meetcmee 15235   Latclat 15335   Atomscatm 33247   AtLatcal 33248   HLchlt 33334   LLinesclln 33474   LPlanesclpl 33475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-poset 15236  df-plt 15248  df-lub 15264  df-glb 15265  df-join 15266  df-meet 15267  df-p0 15329  df-lat 15336  df-clat 15398  df-oposet 33160  df-ol 33162  df-oml 33163  df-covers 33250  df-ats 33251  df-atl 33282  df-cvlat 33306  df-hlat 33335  df-llines 33481  df-lplanes 33482  df-lvols 33483
This theorem is referenced by:  dalem58  33713  dalem60  33715
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