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Theorem dalem55 29916
Description: Lemma for dath 29925. Lines  G H and  P Q intersect at the auxiliary line  B (later shown to be an axis of perspectivity; see dalem60 29921). (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 29813 . . . . 5  |-  ( ph  ->  K  e.  Lat )
323ad2ant1 976 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  Lat )
41dalemkehl 29812 . . . . . 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 29885 . . . . 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 29890 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  H  e.  A )
18 eqid 2283 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
1918, 7, 8hlatjcl 29556 . . . . 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 29834 . . . . 5  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
22213ad2ant1 976 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  .\/  Q
)  e.  ( Base `  K ) )
2318, 6, 10latmle1 14182 . . . 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 29895 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  e.  A )
2718, 8atbase 29479 . . . . . . 7  |-  ( I  e.  A  ->  I  e.  ( Base `  K
) )
2826, 27syl 15 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  e.  ( Base `  K ) )
2918, 6, 7latlej1 14166 . . . . . 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 29830 . . . . . . . 8  |-  ( ph  ->  R  e.  ( Base `  K ) )
3218, 6, 7latlej1 14166 . . . . . . . 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 4063 . . . . . 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 29903 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .\/  I )  e.  O )
3718, 11lplnbase 29723 . . . . . . 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 29838 . . . . . . 7  |-  ( ph  ->  Y  e.  ( Base `  K ) )
40393ad2ant1 976 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  e.  ( Base `  K ) )
4118, 6, 10latmlem12 14189 . . . . . 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 4063 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  B )
4618, 10latmcl 14157 . . . . 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 2283 . . . . . 6  |-  ( LLines `  K )  =  (
LLines `  K )
491, 6, 7, 8, 9, 10, 48, 11, 12, 13, 14, 16, 25, 44dalem53 29914 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  e.  ( LLines `  K ) )
5018, 48llnbase 29698 . . . . 5  |-  ( B  e.  ( LLines `  K
)  ->  B  e.  ( Base `  K )
)
5149, 50syl 15 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  e.  ( Base `  K ) )
5218, 6, 10latlem12 14184 . . . 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 29550 . . . 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 29913 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  e.  A )
581, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 25, 44dalem54 29915 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  e.  A )
596, 8atcmp 29501 . . 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 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:  dalem56  29917  dalem57  29918
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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