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Theorem dalem52 35150
Description: Lemma for dath 35162. 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 35049 . . . 4  |-  ( ph  ->  K  e.  HL )
323ad2ant1 1016 . . 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 35115 . . . 4  |-  ( ps 
->  c  e.  ( Base `  K ) )
763ad2ant3 1018 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  ( Base `  K ) )
83, 7jca 532 . 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 35122 . . 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 35127 . . 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 35132 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  e.  A )
2116, 18, 203jca 1175 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  e.  A  /\  H  e.  A  /\  I  e.  A
) )
221dalempea 35052 . . . 4  |-  ( ph  ->  P  e.  A )
231dalemqea 35053 . . . 4  |-  ( ph  ->  Q  e.  A )
241dalemrea 35054 . . . 4  |-  ( ph  ->  R  e.  A )
2522, 23, 243jca 1175 . . 3  |-  ( ph  ->  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )
26253ad2ant1 1016 . 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 35140 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .\/  I )  e.  O )
281dalemyeo 35058 . . 3  |-  ( ph  ->  Y  e.  O )
29283ad2ant1 1016 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  e.  O )
301, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19dalem45 35143 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  c  .<_  ( G 
.\/  H ) )
311, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19dalem46 35144 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  c  .<_  ( H 
.\/  I ) )
321, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19dalem47 35145 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  c  .<_  ( I 
.\/  G ) )
3330, 31, 323jca 1175 . 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 35146 . . . 4  |-  ( (
ph  /\  ps )  ->  -.  c  .<_  ( P 
.\/  Q ) )
351, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19dalem49 35147 . . . 4  |-  ( (
ph  /\  ps )  ->  -.  c  .<_  ( Q 
.\/  R ) )
361, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19dalem50 35148 . . . 4  |-  ( (
ph  /\  ps )  ->  -.  c  .<_  ( R 
.\/  P ) )
3734, 35, 363jca 1175 . . 3  |-  ( (
ph  /\  ps )  ->  ( -.  c  .<_  ( P  .\/  Q )  /\  -.  c  .<_  ( Q  .\/  R )  /\  -.  c  .<_  ( R  .\/  P ) ) )
38373adant2 1014 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( -.  c  .<_  ( P  .\/  Q )  /\  -.  c  .<_  ( Q  .\/  R )  /\  -.  c  .<_  ( R  .\/  P ) ) )
391, 9, 10, 5, 4, 11, 12, 13, 14, 15dalem27 35125 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  .<_  ( G  .\/  P ) )
401, 9, 10, 5, 4, 11, 12, 13, 14, 17dalem32 35130 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  .<_  ( H  .\/  Q ) )
411, 9, 10, 5, 4, 11, 12, 13, 14, 19dalem36 35134 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  .<_  ( I  .\/  R ) )
4239, 40, 413jca 1175 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .<_  ( G 
.\/  P )  /\  c  .<_  ( H  .\/  Q )  /\  c  .<_  ( I  .\/  R ) ) )
43 biid 236 . . 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 2441 . . 3  |-  ( ( G  .\/  H ) 
.\/  I )  =  ( ( G  .\/  H )  .\/  I )
45 eqid 2441 . . 3  |-  ( ( G  .\/  H ) 
./\  ( P  .\/  Q ) )  =  ( ( G  .\/  H
)  ./\  ( P  .\/  Q ) )
4643, 9, 10, 5, 11, 12, 44, 13, 45dalemdea 35088 . 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 1257 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  e.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802    =/= wne 2636   class class class wbr 4433   ` cfv 5574  (class class class)co 6277   Basecbs 14504   lecple 14576   joincjn 15442   meetcmee 15443   Atomscatm 34690   HLchlt 34777   LPlanesclpl 34918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-preset 15426  df-poset 15444  df-plt 15457  df-lub 15473  df-glb 15474  df-join 15475  df-meet 15476  df-p0 15538  df-lat 15545  df-clat 15607  df-oposet 34603  df-ol 34605  df-oml 34606  df-covers 34693  df-ats 34694  df-atl 34725  df-cvlat 34749  df-hlat 34778  df-llines 34924  df-lplanes 34925  df-lvols 34926
This theorem is referenced by:  dalem54  35152  dalem55  35153
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