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Theorem dalem57 35850
Description: Lemma for dath 35857. 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 35848 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  =  ( ( G  .\/  H )  ./\  B )
)
151dalemkelat 35745 . . . . . . . 8  |-  ( ph  ->  K  e.  Lat )
16153ad2ant1 1015 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  Lat )
171dalemkehl 35744 . . . . . . . . 9  |-  ( ph  ->  K  e.  HL )
18173ad2ant1 1015 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
191, 2, 3, 4, 5, 6, 7, 8, 9, 10dalem23 35817 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  A )
201, 2, 3, 4, 5, 6, 7, 8, 9, 11dalem29 35822 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  H  e.  A )
21 eqid 2454 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
2221, 3, 4hlatjcl 35488 . . . . . . . 8  |-  ( ( K  e.  HL  /\  G  e.  A  /\  H  e.  A )  ->  ( G  .\/  H
)  e.  ( Base `  K ) )
2318, 19, 20, 22syl3anc 1226 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  .\/  H
)  e.  ( Base `  K ) )
241, 3, 4dalempjqeb 35766 . . . . . . . 8  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
25243ad2ant1 1015 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  .\/  Q
)  e.  ( Base `  K ) )
2621, 2, 6latmle2 15906 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  ( P  .\/  Q )  e.  (
Base `  K )
)  ->  ( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( P  .\/  Q ) )
2716, 23, 25, 26syl3anc 1226 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( P  .\/  Q ) )
2814, 27eqbrtrrd 4461 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  .<_  ( P  .\/  Q
) )
291, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13dalem56 35849 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( S  .\/  T ) )  =  ( ( G  .\/  H )  ./\  B )
)
301, 3, 4dalemsjteb 35767 . . . . . . . 8  |-  ( ph  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
31303ad2ant1 1015 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( S  .\/  T
)  e.  ( Base `  K ) )
3221, 2, 6latmle2 15906 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  ( S  .\/  T )  e.  (
Base `  K )
)  ->  ( ( G  .\/  H )  ./\  ( S  .\/  T ) )  .<_  ( S  .\/  T ) )
3316, 23, 31, 32syl3anc 1226 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( S  .\/  T ) )  .<_  ( S  .\/  T ) )
3429, 33eqbrtrrd 4461 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  .<_  ( S  .\/  T
) )
351, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13dalem54 35847 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  e.  A )
3621, 4atbase 35411 . . . . . . 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 15907 . . . . . 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 1228 . . . . 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 919 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  .<_  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) ) )
41 dalem57.d . . . 4  |-  D  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )
4240, 41syl6breqr 4479 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  .<_  D )
43 hlatl 35482 . . . . 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 35783 . . . . 5  |-  ( ph  ->  D  e.  A )
46453ad2ant1 1015 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  D  e.  A )
472, 4atcmp 35433 . . . 4  |-  ( ( K  e.  AtLat  /\  (
( G  .\/  H
)  ./\  B )  e.  A  /\  D  e.  A )  ->  (
( ( G  .\/  H )  ./\  B )  .<_  D  <->  ( ( G 
.\/  H )  ./\  B )  =  D ) )
4844, 35, 46, 47syl3anc 1226 . . 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 35846 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  e.  ( LLines `  K ) )
5221, 50llnbase 35630 . . . 4  |-  ( B  e.  ( LLines `  K
)  ->  B  e.  ( Base `  K )
)
5351, 52syl 16 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  e.  ( Base `  K ) )
5421, 2, 6latmle2 15906 . . 3  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  B  e.  ( Base `  K )
)  ->  ( ( G  .\/  H )  ./\  B )  .<_  B )
5516, 23, 53, 54syl3anc 1226 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  .<_  B )
5649, 55eqbrtrrd 4461 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  D  .<_  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14716   lecple 14791   joincjn 15772   meetcmee 15773   Latclat 15874   Atomscatm 35385   AtLatcal 35386   HLchlt 35472   LLinesclln 35612   LPlanesclpl 35613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-preset 15756  df-poset 15774  df-plt 15787  df-lub 15803  df-glb 15804  df-join 15805  df-meet 15806  df-p0 15868  df-lat 15875  df-clat 15937  df-oposet 35298  df-ol 35300  df-oml 35301  df-covers 35388  df-ats 35389  df-atl 35420  df-cvlat 35444  df-hlat 35473  df-llines 35619  df-lplanes 35620  df-lvols 35621
This theorem is referenced by:  dalem58  35851  dalem60  35853
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