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Theorem dalem39 30570
Description: Lemma for dath 30595. Auxiliary atoms  G,  H, and  I are not colinear. (Contributed by NM, 4-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
) ) ) )
dalem38.m  |-  ./\  =  ( meet `  K )
dalem38.o  |-  O  =  ( LPlanes `  K )
dalem38.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem38.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
dalem38.g  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
dalem38.h  |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )
dalem38.i  |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )
Assertion
Ref Expression
dalem39  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  H  .<_  ( I 
.\/  G ) )

Proof of Theorem dalem39
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 30482 . . . 4  |-  ( ph  ->  K  e.  HL )
323ad2ant1 979 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
41dalemyeo 30491 . . . . 5  |-  ( ph  ->  Y  e.  O )
543ad2ant1 979 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  e.  O )
6 dalem.ps . . . . . 6  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
76dalemccea 30542 . . . . 5  |-  ( ps 
->  c  e.  A
)
873ad2ant3 981 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  A )
96dalem-ccly 30544 . . . . 5  |-  ( ps 
->  -.  c  .<_  Y )
1093ad2ant3 981 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  c  .<_  Y )
11 dalem.l . . . . 5  |-  .<_  =  ( le `  K )
12 dalem.j . . . . 5  |-  .\/  =  ( join `  K )
13 dalem.a . . . . 5  |-  A  =  ( Atoms `  K )
14 dalem38.o . . . . 5  |-  O  =  ( LPlanes `  K )
15 eqid 2438 . . . . 5  |-  ( LVols `  K )  =  (
LVols `  K )
1611, 12, 13, 14, 15lvoli3 30436 . . . 4  |-  ( ( ( K  e.  HL  /\  Y  e.  O  /\  c  e.  A )  /\  -.  c  .<_  Y )  ->  ( Y  .\/  c )  e.  (
LVols `  K ) )
173, 5, 8, 10, 16syl31anc 1188 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( Y  .\/  c
)  e.  ( LVols `  K ) )
18 dalem38.m . . . 4  |-  ./\  =  ( meet `  K )
19 dalem38.y . . . 4  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
20 dalem38.z . . . 4  |-  Z  =  ( ( S  .\/  T )  .\/  U )
21 dalem38.i . . . 4  |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )
221, 11, 12, 13, 6, 18, 14, 19, 20, 21dalem34 30565 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  e.  A )
23 dalem38.g . . . 4  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
241, 11, 12, 13, 6, 18, 14, 19, 20, 23dalem23 30555 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  A )
2511, 12, 13, 15lvolnle3at 30441 . . 3  |-  ( ( ( K  e.  HL  /\  ( Y  .\/  c
)  e.  ( LVols `  K ) )  /\  ( I  e.  A  /\  G  e.  A  /\  c  e.  A
) )  ->  -.  ( Y  .\/  c ) 
.<_  ( ( I  .\/  G )  .\/  c ) )
263, 17, 22, 24, 8, 25syl23anc 1192 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  ( Y  .\/  c
)  .<_  ( ( I 
.\/  G )  .\/  c ) )
27 dalem38.h . . . . . . 7  |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )
281, 11, 12, 13, 6, 18, 14, 19, 20, 23, 27, 21dalem38 30569 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  .<_  ( ( ( G  .\/  H ) 
.\/  I )  .\/  c ) )
291dalemkelat 30483 . . . . . . . 8  |-  ( ph  ->  K  e.  Lat )
30293ad2ant1 979 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  Lat )
311, 11, 12, 13, 6, 18, 14, 19, 20, 27dalem29 30560 . . . . . . . . 9  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  H  e.  A )
32 eqid 2438 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
3332, 12, 13hlatjcl 30226 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  G  e.  A  /\  H  e.  A )  ->  ( G  .\/  H
)  e.  ( Base `  K ) )
343, 24, 31, 33syl3anc 1185 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  .\/  H
)  e.  ( Base `  K ) )
3532, 13atbase 30149 . . . . . . . . 9  |-  ( I  e.  A  ->  I  e.  ( Base `  K
) )
3622, 35syl 16 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  e.  ( Base `  K ) )
3732, 12latjcl 14481 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  I  e.  ( Base `  K )
)  ->  ( ( G  .\/  H )  .\/  I )  e.  (
Base `  K )
)
3830, 34, 36, 37syl3anc 1185 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .\/  I )  e.  ( Base `  K
) )
396, 13dalemcceb 30548 . . . . . . . 8  |-  ( ps 
->  c  e.  ( Base `  K ) )
40393ad2ant3 981 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  ( Base `  K ) )
4132, 11, 12latlej2 14492 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( ( G  .\/  H )  .\/  I )  e.  ( Base `  K
)  /\  c  e.  ( Base `  K )
)  ->  c  .<_  ( ( ( G  .\/  H )  .\/  I ) 
.\/  c ) )
4230, 38, 40, 41syl3anc 1185 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  .<_  ( ( ( G  .\/  H ) 
.\/  I )  .\/  c ) )
431, 14dalemyeb 30508 . . . . . . . 8  |-  ( ph  ->  Y  e.  ( Base `  K ) )
44433ad2ant1 979 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  e.  ( Base `  K ) )
4532, 12latjcl 14481 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( ( G  .\/  H )  .\/  I )  e.  ( Base `  K
)  /\  c  e.  ( Base `  K )
)  ->  ( (
( G  .\/  H
)  .\/  I )  .\/  c )  e.  (
Base `  K )
)
4630, 38, 40, 45syl3anc 1185 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( G 
.\/  H )  .\/  I )  .\/  c
)  e.  ( Base `  K ) )
4732, 11, 12latjle12 14493 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( Y  e.  ( Base `  K )  /\  c  e.  ( Base `  K )  /\  (
( ( G  .\/  H )  .\/  I ) 
.\/  c )  e.  ( Base `  K
) ) )  -> 
( ( Y  .<_  ( ( ( G  .\/  H )  .\/  I ) 
.\/  c )  /\  c  .<_  ( ( ( G  .\/  H ) 
.\/  I )  .\/  c ) )  <->  ( Y  .\/  c )  .<_  ( ( ( G  .\/  H
)  .\/  I )  .\/  c ) ) )
4830, 44, 40, 46, 47syl13anc 1187 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( Y  .<_  ( ( ( G  .\/  H )  .\/  I ) 
.\/  c )  /\  c  .<_  ( ( ( G  .\/  H ) 
.\/  I )  .\/  c ) )  <->  ( Y  .\/  c )  .<_  ( ( ( G  .\/  H
)  .\/  I )  .\/  c ) ) )
4928, 42, 48mpbi2and 889 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( Y  .\/  c
)  .<_  ( ( ( G  .\/  H ) 
.\/  I )  .\/  c ) )
5012, 13hlatjrot 30232 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( G  e.  A  /\  H  e.  A  /\  I  e.  A
) )  ->  (
( G  .\/  H
)  .\/  I )  =  ( ( I 
.\/  G )  .\/  H ) )
513, 24, 31, 22, 50syl13anc 1187 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .\/  I )  =  ( ( I 
.\/  G )  .\/  H ) )
5251oveq1d 6098 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( G 
.\/  H )  .\/  I )  .\/  c
)  =  ( ( ( I  .\/  G
)  .\/  H )  .\/  c ) )
5349, 52breqtrd 4238 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( Y  .\/  c
)  .<_  ( ( ( I  .\/  G ) 
.\/  H )  .\/  c ) )
5453adantr 453 . . 3  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  H  .<_  ( I 
.\/  G ) )  ->  ( Y  .\/  c )  .<_  ( ( ( I  .\/  G
)  .\/  H )  .\/  c ) )
5532, 13atbase 30149 . . . . . . 7  |-  ( H  e.  A  ->  H  e.  ( Base `  K
) )
5631, 55syl 16 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  H  e.  ( Base `  K ) )
5732, 12, 13hlatjcl 30226 . . . . . . 7  |-  ( ( K  e.  HL  /\  I  e.  A  /\  G  e.  A )  ->  ( I  .\/  G
)  e.  ( Base `  K ) )
583, 22, 24, 57syl3anc 1185 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( I  .\/  G
)  e.  ( Base `  K ) )
5932, 11, 12latleeqj2 14495 . . . . . 6  |-  ( ( K  e.  Lat  /\  H  e.  ( Base `  K )  /\  (
I  .\/  G )  e.  ( Base `  K
) )  ->  ( H  .<_  ( I  .\/  G )  <->  ( ( I 
.\/  G )  .\/  H )  =  ( I 
.\/  G ) ) )
6030, 56, 58, 59syl3anc 1185 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( H  .<_  ( I 
.\/  G )  <->  ( (
I  .\/  G )  .\/  H )  =  ( I  .\/  G ) ) )
6160biimpa 472 . . . 4  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  H  .<_  ( I 
.\/  G ) )  ->  ( ( I 
.\/  G )  .\/  H )  =  ( I 
.\/  G ) )
6261oveq1d 6098 . . 3  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  H  .<_  ( I 
.\/  G ) )  ->  ( ( ( I  .\/  G ) 
.\/  H )  .\/  c )  =  ( ( I  .\/  G
)  .\/  c )
)
6354, 62breqtrd 4238 . 2  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  H  .<_  ( I 
.\/  G ) )  ->  ( Y  .\/  c )  .<_  ( ( I  .\/  G ) 
.\/  c ) )
6426, 63mtand 642 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  H  .<_  ( I 
.\/  G ) )
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   Latclat 14476   Atomscatm 30123   HLchlt 30210   LPlanesclpl 30351   LVolsclvol 30352
This theorem is referenced by:  dalem40  30571  dalem41  30572
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 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-llines 30357  df-lplanes 30358  df-lvols 30359
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