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Theorem dalem55 30538
Description: Lemma for dath 30547. Lines  G H and  P Q intersect at the auxiliary line  B (later shown to be an axis of perspectivity; see dalem60 30543). (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 30435 . . . . 5  |-  ( ph  ->  K  e.  Lat )
323ad2ant1 976 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  Lat )
41dalemkehl 30434 . . . . . 6  |-  ( ph  ->  K  e.  HL )
543ad2ant1 976 . . . . 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 30507 . . . . 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 30512 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  H  e.  A )
18 eqid 2296 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
1918, 7, 8hlatjcl 30178 . . . . 5  |-  ( ( K  e.  HL  /\  G  e.  A  /\  H  e.  A )  ->  ( G  .\/  H
)  e.  ( Base `  K ) )
205, 15, 17, 19syl3anc 1182 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  .\/  H
)  e.  ( Base `  K ) )
211, 7, 8dalempjqeb 30456 . . . . 5  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
22213ad2ant1 976 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  .\/  Q
)  e.  ( Base `  K ) )
2318, 6, 10latmle1 14198 . . . 4  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  ( P  .\/  Q )  e.  (
Base `  K )
)  ->  ( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( G  .\/  H ) )
243, 20, 22, 23syl3anc 1182 . . 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 30517 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  e.  A )
2718, 8atbase 30101 . . . . . . 7  |-  ( I  e.  A  ->  I  e.  ( Base `  K
) )
2826, 27syl 15 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  e.  ( Base `  K ) )
2918, 6, 7latlej1 14182 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  I  e.  ( Base `  K )
)  ->  ( G  .\/  H )  .<_  ( ( G  .\/  H ) 
.\/  I ) )
303, 20, 28, 29syl3anc 1182 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  .\/  H
)  .<_  ( ( G 
.\/  H )  .\/  I ) )
311, 8dalemreb 30452 . . . . . . . 8  |-  ( ph  ->  R  e.  ( Base `  K ) )
3218, 6, 7latlej1 14182 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  R  e.  ( Base `  K )
)  ->  ( P  .\/  Q )  .<_  ( ( P  .\/  Q ) 
.\/  R ) )
332, 21, 31, 32syl3anc 1182 . . . . . . 7  |-  ( ph  ->  ( P  .\/  Q
)  .<_  ( ( P 
.\/  Q )  .\/  R ) )
3433, 12syl6breqr 4079 . . . . . 6  |-  ( ph  ->  ( P  .\/  Q
)  .<_  Y )
35343ad2ant1 976 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  .\/  Q
)  .<_  Y )
361, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 25dalem42 30525 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .\/  I )  e.  O )
3718, 11lplnbase 30345 . . . . . . 7  |-  ( ( ( G  .\/  H
)  .\/  I )  e.  O  ->  ( ( G  .\/  H ) 
.\/  I )  e.  ( Base `  K
) )
3836, 37syl 15 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .\/  I )  e.  ( Base `  K
) )
391, 11dalemyeb 30460 . . . . . . 7  |-  ( ph  ->  Y  e.  ( Base `  K ) )
40393ad2ant1 976 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  e.  ( Base `  K ) )
4118, 6, 10latmlem12 14205 . . . . . 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 1191 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( G 
.\/  H )  .<_  ( ( G  .\/  H )  .\/  I )  /\  ( P  .\/  Q )  .<_  Y )  ->  ( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( ( ( G 
.\/  H )  .\/  I )  ./\  Y
) ) )
4330, 35, 42mp2and 660 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( ( ( G 
.\/  H )  .\/  I )  ./\  Y
) )
44 dalem54.b1 . . . 4  |-  B  =  ( ( ( G 
.\/  H )  .\/  I )  ./\  Y
)
4543, 44syl6breqr 4079 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  B )
4618, 10latmcl 14173 . . . . 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 1182 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  e.  ( Base `  K
) )
48 eqid 2296 . . . . . 6  |-  ( LLines `  K )  =  (
LLines `  K )
491, 6, 7, 8, 9, 10, 48, 11, 12, 13, 14, 16, 25, 44dalem53 30536 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  e.  ( LLines `  K ) )
5018, 48llnbase 30320 . . . . 5  |-  ( B  e.  ( LLines `  K
)  ->  B  e.  ( Base `  K )
)
5149, 50syl 15 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  e.  ( Base `  K ) )
5218, 6, 10latlem12 14200 . . . 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 1184 . . 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 887 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( ( G  .\/  H )  ./\  B )
)
55 hlatl 30172 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
565, 55syl 15 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  AtLat )
571, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 25dalem52 30535 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  e.  A )
581, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 25, 44dalem54 30537 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  e.  A )
596, 8atcmp 30123 . . 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 1182 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( G 
.\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( ( G  .\/  H )  ./\  B )  <->  ( ( G 
.\/  H )  ./\  ( P  .\/  Q ) )  =  ( ( G  .\/  H ) 
./\  B ) ) )
6154, 60mpbid 201 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   meetcmee 14095   Latclat 14167   Atomscatm 30075   AtLatcal 30076   HLchlt 30162   LLinesclln 30302   LPlanesclpl 30303
This theorem is referenced by:  dalem56  30539  dalem57  30540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310  df-lvols 30311
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