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Theorem dalem52 30523
Description: Lemma for dath 30535. Lines  G H and  P Q intersect at an atom. (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
) ) ) )
dalem44.m  |-  ./\  =  ( meet `  K )
dalem44.o  |-  O  =  ( LPlanes `  K )
dalem44.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem44.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
dalem44.g  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
dalem44.h  |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )
dalem44.i  |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )
Assertion
Ref Expression
dalem52  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  e.  A )

Proof of Theorem dalem52
StepHypRef Expression
1 dalem.ph . . . . 5  |-  ( 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 ) ) ) ) )
21dalemkehl 30422 . . . 4  |-  ( ph  ->  K  e.  HL )
323ad2ant1 979 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
4 dalem.ps . . . . 5  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
5 dalem.a . . . . 5  |-  A  =  ( Atoms `  K )
64, 5dalemcceb 30488 . . . 4  |-  ( ps 
->  c  e.  ( Base `  K ) )
763ad2ant3 981 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  ( Base `  K ) )
83, 7jca 520 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( K  e.  HL  /\  c  e.  ( Base `  K ) ) )
9 dalem.l . . . 4  |-  .<_  =  ( le `  K )
10 dalem.j . . . 4  |-  .\/  =  ( join `  K )
11 dalem44.m . . . 4  |-  ./\  =  ( meet `  K )
12 dalem44.o . . . 4  |-  O  =  ( LPlanes `  K )
13 dalem44.y . . . 4  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
14 dalem44.z . . . 4  |-  Z  =  ( ( S  .\/  T )  .\/  U )
15 dalem44.g . . . 4  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
161, 9, 10, 5, 4, 11, 12, 13, 14, 15dalem23 30495 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  A )
17 dalem44.h . . . 4  |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )
181, 9, 10, 5, 4, 11, 12, 13, 14, 17dalem29 30500 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  H  e.  A )
19 dalem44.i . . . 4  |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )
201, 9, 10, 5, 4, 11, 12, 13, 14, 19dalem34 30505 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  e.  A )
2116, 18, 203jca 1135 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  e.  A  /\  H  e.  A  /\  I  e.  A
) )
221dalempea 30425 . . . 4  |-  ( ph  ->  P  e.  A )
231dalemqea 30426 . . . 4  |-  ( ph  ->  Q  e.  A )
241dalemrea 30427 . . . 4  |-  ( ph  ->  R  e.  A )
2522, 23, 243jca 1135 . . 3  |-  ( ph  ->  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )
26253ad2ant1 979 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )
271, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19dalem42 30513 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .\/  I )  e.  O )
281dalemyeo 30431 . . 3  |-  ( ph  ->  Y  e.  O )
29283ad2ant1 979 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  e.  O )
301, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19dalem45 30516 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  c  .<_  ( G 
.\/  H ) )
311, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19dalem46 30517 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  c  .<_  ( H 
.\/  I ) )
321, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19dalem47 30518 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  c  .<_  ( I 
.\/  G ) )
3330, 31, 323jca 1135 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( -.  c  .<_  ( G  .\/  H )  /\  -.  c  .<_  ( H  .\/  I )  /\  -.  c  .<_  ( I  .\/  G ) ) )
341, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19dalem48 30519 . . . 4  |-  ( (
ph  /\  ps )  ->  -.  c  .<_  ( P 
.\/  Q ) )
351, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19dalem49 30520 . . . 4  |-  ( (
ph  /\  ps )  ->  -.  c  .<_  ( Q 
.\/  R ) )
361, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19dalem50 30521 . . . 4  |-  ( (
ph  /\  ps )  ->  -.  c  .<_  ( R 
.\/  P ) )
3734, 35, 363jca 1135 . . 3  |-  ( (
ph  /\  ps )  ->  ( -.  c  .<_  ( P  .\/  Q )  /\  -.  c  .<_  ( Q  .\/  R )  /\  -.  c  .<_  ( R  .\/  P ) ) )
38373adant2 977 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( -.  c  .<_  ( P  .\/  Q )  /\  -.  c  .<_  ( Q  .\/  R )  /\  -.  c  .<_  ( R  .\/  P ) ) )
391, 9, 10, 5, 4, 11, 12, 13, 14, 15dalem27 30498 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  .<_  ( G  .\/  P ) )
401, 9, 10, 5, 4, 11, 12, 13, 14, 17dalem32 30503 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  .<_  ( H  .\/  Q ) )
411, 9, 10, 5, 4, 11, 12, 13, 14, 19dalem36 30507 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  .<_  ( I  .\/  R ) )
4239, 40, 413jca 1135 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .<_  ( G 
.\/  P )  /\  c  .<_  ( H  .\/  Q )  /\  c  .<_  ( I  .\/  R ) ) )
43 biid 229 . . 3  |-  ( ( ( ( K  e.  HL  /\  c  e.  ( Base `  K
) )  /\  ( G  e.  A  /\  H  e.  A  /\  I  e.  A )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  (
( ( G  .\/  H )  .\/  I )  e.  O  /\  Y  e.  O )  /\  (
( -.  c  .<_  ( G  .\/  H )  /\  -.  c  .<_  ( H  .\/  I )  /\  -.  c  .<_  ( I  .\/  G ) )  /\  ( -.  c  .<_  ( P  .\/  Q )  /\  -.  c  .<_  ( Q  .\/  R )  /\  -.  c  .<_  ( R  .\/  P
) )  /\  (
c  .<_  ( G  .\/  P )  /\  c  .<_  ( H  .\/  Q )  /\  c  .<_  ( I 
.\/  R ) ) ) )  <->  ( (
( K  e.  HL  /\  c  e.  ( Base `  K ) )  /\  ( G  e.  A  /\  H  e.  A  /\  I  e.  A
)  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  /\  ( ( ( G 
.\/  H )  .\/  I )  e.  O  /\  Y  e.  O
)  /\  ( ( -.  c  .<_  ( G 
.\/  H )  /\  -.  c  .<_  ( H 
.\/  I )  /\  -.  c  .<_  ( I 
.\/  G ) )  /\  ( -.  c  .<_  ( P  .\/  Q
)  /\  -.  c  .<_  ( Q  .\/  R
)  /\  -.  c  .<_  ( R  .\/  P
) )  /\  (
c  .<_  ( G  .\/  P )  /\  c  .<_  ( H  .\/  Q )  /\  c  .<_  ( I 
.\/  R ) ) ) ) )
44 eqid 2438 . . 3  |-  ( ( G  .\/  H ) 
.\/  I )  =  ( ( G  .\/  H )  .\/  I )
45 eqid 2438 . . 3  |-  ( ( G  .\/  H ) 
./\  ( P  .\/  Q ) )  =  ( ( G  .\/  H
)  ./\  ( P  .\/  Q ) )
4643, 9, 10, 5, 11, 12, 44, 13, 45dalemdea 30461 . 2  |-  ( ( ( ( K  e.  HL  /\  c  e.  ( Base `  K
) )  /\  ( G  e.  A  /\  H  e.  A  /\  I  e.  A )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  (
( ( G  .\/  H )  .\/  I )  e.  O  /\  Y  e.  O )  /\  (
( -.  c  .<_  ( G  .\/  H )  /\  -.  c  .<_  ( H  .\/  I )  /\  -.  c  .<_  ( I  .\/  G ) )  /\  ( -.  c  .<_  ( P  .\/  Q )  /\  -.  c  .<_  ( Q  .\/  R )  /\  -.  c  .<_  ( R  .\/  P
) )  /\  (
c  .<_  ( G  .\/  P )  /\  c  .<_  ( H  .\/  Q )  /\  c  .<_  ( I 
.\/  R ) ) ) )  ->  (
( G  .\/  H
)  ./\  ( P  .\/  Q ) )  e.  A )
478, 21, 26, 27, 29, 33, 38, 42, 46syl323anc 1215 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   lecple 13538   joincjn 14403   meetcmee 14404   Atomscatm 30063   HLchlt 30150   LPlanesclpl 30291
This theorem is referenced by:  dalem54  30525  dalem55  30526
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-lat 14477  df-clat 14539  df-oposet 29976  df-ol 29978  df-oml 29979  df-covers 30066  df-ats 30067  df-atl 30098  df-cvlat 30122  df-hlat 30151  df-llines 30297  df-lplanes 30298  df-lvols 30299
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