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Theorem dalem57 29918
Description: Lemma for dath 29925. 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 29916 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  =  ( ( G  .\/  H )  ./\  B )
)
151dalemkelat 29813 . . . . . . . 8  |-  ( ph  ->  K  e.  Lat )
16153ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  Lat )
171dalemkehl 29812 . . . . . . . . 9  |-  ( ph  ->  K  e.  HL )
18173ad2ant1 976 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
191, 2, 3, 4, 5, 6, 7, 8, 9, 10dalem23 29885 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  A )
201, 2, 3, 4, 5, 6, 7, 8, 9, 11dalem29 29890 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  H  e.  A )
21 eqid 2283 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
2221, 3, 4hlatjcl 29556 . . . . . . . 8  |-  ( ( K  e.  HL  /\  G  e.  A  /\  H  e.  A )  ->  ( G  .\/  H
)  e.  ( Base `  K ) )
2318, 19, 20, 22syl3anc 1182 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  .\/  H
)  e.  ( Base `  K ) )
241, 3, 4dalempjqeb 29834 . . . . . . . 8  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
25243ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  .\/  Q
)  e.  ( Base `  K ) )
2621, 2, 6latmle2 14183 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  ( P  .\/  Q )  e.  (
Base `  K )
)  ->  ( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( P  .\/  Q ) )
2716, 23, 25, 26syl3anc 1182 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( P  .\/  Q ) )
2814, 27eqbrtrrd 4045 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  .<_  ( P  .\/  Q
) )
291, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13dalem56 29917 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( S  .\/  T ) )  =  ( ( G  .\/  H )  ./\  B )
)
301, 3, 4dalemsjteb 29835 . . . . . . . 8  |-  ( ph  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
31303ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( S  .\/  T
)  e.  ( Base `  K ) )
3221, 2, 6latmle2 14183 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  ( S  .\/  T )  e.  (
Base `  K )
)  ->  ( ( G  .\/  H )  ./\  ( S  .\/  T ) )  .<_  ( S  .\/  T ) )
3316, 23, 31, 32syl3anc 1182 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( S  .\/  T ) )  .<_  ( S  .\/  T ) )
3429, 33eqbrtrrd 4045 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  .<_  ( S  .\/  T
) )
351, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13dalem54 29915 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  e.  A )
3621, 4atbase 29479 . . . . . . 7  |-  ( ( ( G  .\/  H
)  ./\  B )  e.  A  ->  ( ( G  .\/  H ) 
./\  B )  e.  ( Base `  K
) )
3735, 36syl 15 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  e.  ( Base `  K
) )
3821, 2, 6latlem12 14184 . . . . . 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 1184 . . . . 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 887 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  .<_  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) ) )
41 dalem57.d . . . 4  |-  D  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )
4240, 41syl6breqr 4063 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  .<_  D )
43 hlatl 29550 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
4418, 43syl 15 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  AtLat )
451, 2, 3, 4, 6, 7, 8, 9, 41dalemdea 29851 . . . . 5  |-  ( ph  ->  D  e.  A )
46453ad2ant1 976 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  D  e.  A )
472, 4atcmp 29501 . . . 4  |-  ( ( K  e.  AtLat  /\  (
( G  .\/  H
)  ./\  B )  e.  A  /\  D  e.  A )  ->  (
( ( G  .\/  H )  ./\  B )  .<_  D  <->  ( ( G 
.\/  H )  ./\  B )  =  D ) )
4844, 35, 46, 47syl3anc 1182 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( G 
.\/  H )  ./\  B )  .<_  D  <->  ( ( G  .\/  H )  ./\  B )  =  D ) )
4942, 48mpbid 201 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  =  D )
50 eqid 2283 . . . . 5  |-  ( LLines `  K )  =  (
LLines `  K )
511, 2, 3, 4, 5, 6, 50, 7, 8, 9, 10, 11, 12, 13dalem53 29914 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  e.  ( LLines `  K ) )
5221, 50llnbase 29698 . . . 4  |-  ( B  e.  ( LLines `  K
)  ->  B  e.  ( Base `  K )
)
5351, 52syl 15 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  e.  ( Base `  K ) )
5421, 2, 6latmle2 14183 . . 3  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  B  e.  ( Base `  K )
)  ->  ( ( G  .\/  H )  ./\  B )  .<_  B )
5516, 23, 53, 54syl3anc 1182 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  .<_  B )
5649, 55eqbrtrrd 4045 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  D  .<_  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Latclat 14151   Atomscatm 29453   AtLatcal 29454   HLchlt 29540   LLinesclln 29680   LPlanesclpl 29681
This theorem is referenced by:  dalem58  29919  dalem60  29921
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689
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