Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dalem24 Unicode version

Theorem dalem24 29886
Description: Lemma for dath 29925. Show that auxiliary atom  G is outside of plane  Y. (Contributed by NM, 2-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
) ) ) )
dalem23.m  |-  ./\  =  ( meet `  K )
dalem23.o  |-  O  =  ( LPlanes `  K )
dalem23.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem23.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
dalem23.g  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
Assertion
Ref Expression
dalem24  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  G  .<_  Y )

Proof of Theorem dalem24
StepHypRef Expression
1 dalem23.g . . . . 5  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
21oveq1i 5868 . . . 4  |-  ( G 
./\  Y )  =  ( ( ( c 
.\/  P )  ./\  ( d  .\/  S
) )  ./\  Y
)
3 dalem.ph . . . . . . . 8  |-  ( 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 ) ) ) ) )
43dalemkehl 29812 . . . . . . 7  |-  ( ph  ->  K  e.  HL )
5 hlol 29551 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OL )
64, 5syl 15 . . . . . 6  |-  ( ph  ->  K  e.  OL )
763ad2ant1 976 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  OL )
843ad2ant1 976 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
9 dalem.ps . . . . . . . 8  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
109dalemccea 29872 . . . . . . 7  |-  ( ps 
->  c  e.  A
)
11103ad2ant3 978 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  A )
123dalempea 29815 . . . . . . 7  |-  ( ph  ->  P  e.  A )
13123ad2ant1 976 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  P  e.  A )
14 eqid 2283 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
15 dalem.j . . . . . . 7  |-  .\/  =  ( join `  K )
16 dalem.a . . . . . . 7  |-  A  =  ( Atoms `  K )
1714, 15, 16hlatjcl 29556 . . . . . 6  |-  ( ( K  e.  HL  /\  c  e.  A  /\  P  e.  A )  ->  ( c  .\/  P
)  e.  ( Base `  K ) )
188, 11, 13, 17syl3anc 1182 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  P
)  e.  ( Base `  K ) )
199dalemddea 29873 . . . . . . 7  |-  ( ps 
->  d  e.  A
)
20193ad2ant3 978 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
d  e.  A )
213dalemsea 29818 . . . . . . 7  |-  ( ph  ->  S  e.  A )
22213ad2ant1 976 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  S  e.  A )
2314, 15, 16hlatjcl 29556 . . . . . 6  |-  ( ( K  e.  HL  /\  d  e.  A  /\  S  e.  A )  ->  ( d  .\/  S
)  e.  ( Base `  K ) )
248, 20, 22, 23syl3anc 1182 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( d  .\/  S
)  e.  ( Base `  K ) )
25 dalem23.o . . . . . . 7  |-  O  =  ( LPlanes `  K )
263, 25dalemyeb 29838 . . . . . 6  |-  ( ph  ->  Y  e.  ( Base `  K ) )
27263ad2ant1 976 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  e.  ( Base `  K ) )
28 dalem23.m . . . . . 6  |-  ./\  =  ( meet `  K )
2914, 28latmmdir 29425 . . . . 5  |-  ( ( K  e.  OL  /\  ( ( c  .\/  P )  e.  ( Base `  K )  /\  (
d  .\/  S )  e.  ( Base `  K
)  /\  Y  e.  ( Base `  K )
) )  ->  (
( ( c  .\/  P )  ./\  ( d  .\/  S ) )  ./\  Y )  =  ( ( ( c  .\/  P
)  ./\  Y )  ./\  ( ( d  .\/  S )  ./\  Y )
) )
307, 18, 24, 27, 29syl13anc 1184 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( c 
.\/  P )  ./\  ( d  .\/  S
) )  ./\  Y
)  =  ( ( ( c  .\/  P
)  ./\  Y )  ./\  ( ( d  .\/  S )  ./\  Y )
) )
312, 30syl5eq 2327 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  ./\  Y
)  =  ( ( ( c  .\/  P
)  ./\  Y )  ./\  ( ( d  .\/  S )  ./\  Y )
) )
3215, 16hlatjcom 29557 . . . . . . 7  |-  ( ( K  e.  HL  /\  c  e.  A  /\  P  e.  A )  ->  ( c  .\/  P
)  =  ( P 
.\/  c ) )
338, 11, 13, 32syl3anc 1182 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  P
)  =  ( P 
.\/  c ) )
3433oveq1d 5873 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  P )  ./\  Y )  =  ( ( P 
.\/  c )  ./\  Y ) )
35 dalem.l . . . . . . . 8  |-  .<_  =  ( le `  K )
36 dalem23.y . . . . . . . 8  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
373, 35, 15, 16, 25, 36dalemply 29843 . . . . . . 7  |-  ( ph  ->  P  .<_  Y )
38373ad2ant1 976 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  P  .<_  Y )
399dalem-ccly 29874 . . . . . . 7  |-  ( ps 
->  -.  c  .<_  Y )
40393ad2ant3 978 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  c  .<_  Y )
4114, 35, 15, 28, 162atjm 29634 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  c  e.  A  /\  Y  e.  ( Base `  K ) )  /\  ( P  .<_  Y  /\  -.  c  .<_  Y ) )  -> 
( ( P  .\/  c )  ./\  Y
)  =  P )
428, 13, 11, 27, 38, 40, 41syl132anc 1200 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( P  .\/  c )  ./\  Y
)  =  P )
4334, 42eqtrd 2315 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  P )  ./\  Y )  =  P )
4415, 16hlatjcom 29557 . . . . . . 7  |-  ( ( K  e.  HL  /\  d  e.  A  /\  S  e.  A )  ->  ( d  .\/  S
)  =  ( S 
.\/  d ) )
458, 20, 22, 44syl3anc 1182 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( d  .\/  S
)  =  ( S 
.\/  d ) )
4645oveq1d 5873 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( d  .\/  S )  ./\  Y )  =  ( ( S 
.\/  d )  ./\  Y ) )
47 dalem23.z . . . . . . . 8  |-  Z  =  ( ( S  .\/  T )  .\/  U )
483, 35, 15, 16, 47dalemsly 29844 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z )  ->  S  .<_  Y )
49483adant3 975 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  S  .<_  Y )
509dalem-ddly 29875 . . . . . . 7  |-  ( ps 
->  -.  d  .<_  Y )
51503ad2ant3 978 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  d  .<_  Y )
5214, 35, 15, 28, 162atjm 29634 . . . . . 6  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  d  e.  A  /\  Y  e.  ( Base `  K ) )  /\  ( S  .<_  Y  /\  -.  d  .<_  Y ) )  -> 
( ( S  .\/  d )  ./\  Y
)  =  S )
538, 22, 20, 27, 49, 51, 52syl132anc 1200 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( S  .\/  d )  ./\  Y
)  =  S )
5446, 53eqtrd 2315 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( d  .\/  S )  ./\  Y )  =  S )
5543, 54oveq12d 5876 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( c 
.\/  P )  ./\  Y )  ./\  ( (
d  .\/  S )  ./\  Y ) )  =  ( P  ./\  S
) )
563, 35, 15, 16, 25, 36dalempnes 29840 . . . . 5  |-  ( ph  ->  P  =/=  S )
57 hlatl 29550 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  AtLat )
584, 57syl 15 . . . . . 6  |-  ( ph  ->  K  e.  AtLat )
59 eqid 2283 . . . . . . 7  |-  ( 0.
`  K )  =  ( 0. `  K
)
6028, 59, 16atnem0 29508 . . . . . 6  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  S  e.  A )  ->  ( P  =/=  S  <->  ( P  ./\ 
S )  =  ( 0. `  K ) ) )
6158, 12, 21, 60syl3anc 1182 . . . . 5  |-  ( ph  ->  ( P  =/=  S  <->  ( P  ./\  S )  =  ( 0. `  K ) ) )
6256, 61mpbid 201 . . . 4  |-  ( ph  ->  ( P  ./\  S
)  =  ( 0.
`  K ) )
63623ad2ant1 976 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  ./\  S
)  =  ( 0.
`  K ) )
6431, 55, 633eqtrd 2319 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  ./\  Y
)  =  ( 0.
`  K ) )
65583ad2ant1 976 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  AtLat )
663, 35, 15, 16, 9, 28, 25, 36, 47, 1dalem23 29885 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  A )
6714, 35, 28, 59, 16atnle 29507 . . 3  |-  ( ( K  e.  AtLat  /\  G  e.  A  /\  Y  e.  ( Base `  K
) )  ->  ( -.  G  .<_  Y  <->  ( G  ./\ 
Y )  =  ( 0. `  K ) ) )
6865, 66, 27, 67syl3anc 1182 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( -.  G  .<_  Y  <-> 
( G  ./\  Y
)  =  ( 0.
`  K ) ) )
6964, 68mpbird 223 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  G  .<_  Y )
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   0.cp0 14143   OLcol 29364   Atomscatm 29453   AtLatcal 29454   HLchlt 29540   LPlanesclpl 29681
This theorem is referenced by:  dalem27  29888  dalem30  29891  dalem54  29915
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-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
  Copyright terms: Public domain W3C validator