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Theorem dalem55 30461
Description: Lemma for dath 30470. Lines  G H and  P Q intersect at the auxiliary line  B (later shown to be an axis of perspectivity; see dalem60 30466). (Contributed by NM, 8-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
) ) ) )
dalem54.m  |-  ./\  =  ( meet `  K )
dalem54.o  |-  O  =  ( LPlanes `  K )
dalem54.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem54.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
dalem54.g  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
dalem54.h  |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )
dalem54.i  |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )
dalem54.b1  |-  B  =  ( ( ( G 
.\/  H )  .\/  I )  ./\  Y
)
Assertion
Ref Expression
dalem55  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  =  ( ( G  .\/  H )  ./\  B )
)

Proof of Theorem dalem55
StepHypRef Expression
1 dalem.ph . . . . . 6  |-  ( 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 ) ) ) ) )
21dalemkelat 30358 . . . . 5  |-  ( ph  ->  K  e.  Lat )
323ad2ant1 978 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  Lat )
41dalemkehl 30357 . . . . . 6  |-  ( ph  ->  K  e.  HL )
543ad2ant1 978 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
6 dalem.l . . . . . 6  |-  .<_  =  ( le `  K )
7 dalem.j . . . . . 6  |-  .\/  =  ( join `  K )
8 dalem.a . . . . . 6  |-  A  =  ( Atoms `  K )
9 dalem.ps . . . . . 6  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
10 dalem54.m . . . . . 6  |-  ./\  =  ( meet `  K )
11 dalem54.o . . . . . 6  |-  O  =  ( LPlanes `  K )
12 dalem54.y . . . . . 6  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
13 dalem54.z . . . . . 6  |-  Z  =  ( ( S  .\/  T )  .\/  U )
14 dalem54.g . . . . . 6  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
151, 6, 7, 8, 9, 10, 11, 12, 13, 14dalem23 30430 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  A )
16 dalem54.h . . . . . 6  |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )
171, 6, 7, 8, 9, 10, 11, 12, 13, 16dalem29 30435 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  H  e.  A )
18 eqid 2435 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
1918, 7, 8hlatjcl 30101 . . . . 5  |-  ( ( K  e.  HL  /\  G  e.  A  /\  H  e.  A )  ->  ( G  .\/  H
)  e.  ( Base `  K ) )
205, 15, 17, 19syl3anc 1184 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  .\/  H
)  e.  ( Base `  K ) )
211, 7, 8dalempjqeb 30379 . . . . 5  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
22213ad2ant1 978 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  .\/  Q
)  e.  ( Base `  K ) )
2318, 6, 10latmle1 14497 . . . 4  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  ( P  .\/  Q )  e.  (
Base `  K )
)  ->  ( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( G  .\/  H ) )
243, 20, 22, 23syl3anc 1184 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( G  .\/  H ) )
25 dalem54.i . . . . . . . 8  |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )
261, 6, 7, 8, 9, 10, 11, 12, 13, 25dalem34 30440 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  e.  A )
2718, 8atbase 30024 . . . . . . 7  |-  ( I  e.  A  ->  I  e.  ( Base `  K
) )
2826, 27syl 16 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  e.  ( Base `  K ) )
2918, 6, 7latlej1 14481 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  I  e.  ( Base `  K )
)  ->  ( G  .\/  H )  .<_  ( ( G  .\/  H ) 
.\/  I ) )
303, 20, 28, 29syl3anc 1184 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  .\/  H
)  .<_  ( ( G 
.\/  H )  .\/  I ) )
311, 8dalemreb 30375 . . . . . . . 8  |-  ( ph  ->  R  e.  ( Base `  K ) )
3218, 6, 7latlej1 14481 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  R  e.  ( Base `  K )
)  ->  ( P  .\/  Q )  .<_  ( ( P  .\/  Q ) 
.\/  R ) )
332, 21, 31, 32syl3anc 1184 . . . . . . 7  |-  ( ph  ->  ( P  .\/  Q
)  .<_  ( ( P 
.\/  Q )  .\/  R ) )
3433, 12syl6breqr 4244 . . . . . 6  |-  ( ph  ->  ( P  .\/  Q
)  .<_  Y )
35343ad2ant1 978 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  .\/  Q
)  .<_  Y )
361, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 25dalem42 30448 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .\/  I )  e.  O )
3718, 11lplnbase 30268 . . . . . . 7  |-  ( ( ( G  .\/  H
)  .\/  I )  e.  O  ->  ( ( G  .\/  H ) 
.\/  I )  e.  ( Base `  K
) )
3836, 37syl 16 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .\/  I )  e.  ( Base `  K
) )
391, 11dalemyeb 30383 . . . . . . 7  |-  ( ph  ->  Y  e.  ( Base `  K ) )
40393ad2ant1 978 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  e.  ( Base `  K ) )
4118, 6, 10latmlem12 14504 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( ( G  .\/  H )  e.  ( Base `  K )  /\  (
( G  .\/  H
)  .\/  I )  e.  ( Base `  K
) )  /\  (
( P  .\/  Q
)  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) ) )  -> 
( ( ( G 
.\/  H )  .<_  ( ( G  .\/  H )  .\/  I )  /\  ( P  .\/  Q )  .<_  Y )  ->  ( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( ( ( G 
.\/  H )  .\/  I )  ./\  Y
) ) )
423, 20, 38, 22, 40, 41syl122anc 1193 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( G 
.\/  H )  .<_  ( ( G  .\/  H )  .\/  I )  /\  ( P  .\/  Q )  .<_  Y )  ->  ( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( ( ( G 
.\/  H )  .\/  I )  ./\  Y
) ) )
4330, 35, 42mp2and 661 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( ( ( G 
.\/  H )  .\/  I )  ./\  Y
) )
44 dalem54.b1 . . . 4  |-  B  =  ( ( ( G 
.\/  H )  .\/  I )  ./\  Y
)
4543, 44syl6breqr 4244 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  B )
4618, 10latmcl 14472 . . . . 5  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  ( P  .\/  Q )  e.  (
Base `  K )
)  ->  ( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  e.  ( Base `  K ) )
473, 20, 22, 46syl3anc 1184 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  e.  ( Base `  K
) )
48 eqid 2435 . . . . . 6  |-  ( LLines `  K )  =  (
LLines `  K )
491, 6, 7, 8, 9, 10, 48, 11, 12, 13, 14, 16, 25, 44dalem53 30459 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  e.  ( LLines `  K ) )
5018, 48llnbase 30243 . . . . 5  |-  ( B  e.  ( LLines `  K
)  ->  B  e.  ( Base `  K )
)
5149, 50syl 16 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  e.  ( Base `  K ) )
5218, 6, 10latlem12 14499 . . . 4  |-  ( ( K  e.  Lat  /\  ( ( ( G 
.\/  H )  ./\  ( P  .\/  Q ) )  e.  ( Base `  K )  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  B  e.  ( Base `  K )
) )  ->  (
( ( ( G 
.\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( G  .\/  H )  /\  (
( G  .\/  H
)  ./\  ( P  .\/  Q ) )  .<_  B )  <->  ( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( ( G  .\/  H )  ./\  B ) ) )
533, 47, 20, 51, 52syl13anc 1186 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( ( G  .\/  H ) 
./\  ( P  .\/  Q ) )  .<_  ( G 
.\/  H )  /\  ( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  B )  <->  ( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( ( G  .\/  H )  ./\  B ) ) )
5424, 45, 53mpbi2and 888 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( ( G  .\/  H )  ./\  B )
)
55 hlatl 30095 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
565, 55syl 16 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  AtLat )
571, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 25dalem52 30458 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  e.  A )
581, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 25, 44dalem54 30460 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  e.  A )
596, 8atcmp 30046 . . 3  |-  ( ( K  e.  AtLat  /\  (
( G  .\/  H
)  ./\  ( P  .\/  Q ) )  e.  A  /\  ( ( G  .\/  H ) 
./\  B )  e.  A )  ->  (
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( ( G  .\/  H )  ./\  B )  <->  ( ( G  .\/  H
)  ./\  ( P  .\/  Q ) )  =  ( ( G  .\/  H )  ./\  B )
) )
6056, 57, 58, 59syl3anc 1184 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( G 
.\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( ( G  .\/  H )  ./\  B )  <->  ( ( G 
.\/  H )  ./\  ( P  .\/  Q ) )  =  ( ( G  .\/  H ) 
./\  B ) ) )
6154, 60mpbid 202 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  =  ( ( G  .\/  H )  ./\  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   AtLatcal 29999   HLchlt 30085   LLinesclln 30225   LPlanesclpl 30226
This theorem is referenced by:  dalem56  30462  dalem57  30463
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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