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Theorem dalem39 33029
Description: Lemma for dath 33054. 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 32941 . . . 4  |-  ( ph  ->  K  e.  HL )
323ad2ant1 1026 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
41dalemyeo 32950 . . . . 5  |-  ( ph  ->  Y  e.  O )
543ad2ant1 1026 . . . 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 33001 . . . . 5  |-  ( ps 
->  c  e.  A
)
873ad2ant3 1028 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  A )
96dalem-ccly 33003 . . . . 5  |-  ( ps 
->  -.  c  .<_  Y )
1093ad2ant3 1028 . . . 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 2420 . . . . 5  |-  ( LVols `  K )  =  (
LVols `  K )
1611, 12, 13, 14, 15lvoli3 32895 . . . 4  |-  ( ( ( K  e.  HL  /\  Y  e.  O  /\  c  e.  A )  /\  -.  c  .<_  Y )  ->  ( Y  .\/  c )  e.  (
LVols `  K ) )
173, 5, 8, 10, 16syl31anc 1267 . . 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 33024 . . 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 33014 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  A )
2511, 12, 13, 15lvolnle3at 32900 . . 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 1271 . 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 33028 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  .<_  ( ( ( G  .\/  H ) 
.\/  I )  .\/  c ) )
291dalemkelat 32942 . . . . . . . 8  |-  ( ph  ->  K  e.  Lat )
30293ad2ant1 1026 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  Lat )
311, 11, 12, 13, 6, 18, 14, 19, 20, 27dalem29 33019 . . . . . . . . 9  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  H  e.  A )
32 eqid 2420 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
3332, 12, 13hlatjcl 32685 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  G  e.  A  /\  H  e.  A )  ->  ( G  .\/  H
)  e.  ( Base `  K ) )
343, 24, 31, 33syl3anc 1264 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  .\/  H
)  e.  ( Base `  K ) )
3532, 13atbase 32608 . . . . . . . . 9  |-  ( I  e.  A  ->  I  e.  ( Base `  K
) )
3622, 35syl 17 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  e.  ( Base `  K ) )
3732, 12latjcl 16249 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  I  e.  ( Base `  K )
)  ->  ( ( G  .\/  H )  .\/  I )  e.  (
Base `  K )
)
3830, 34, 36, 37syl3anc 1264 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .\/  I )  e.  ( Base `  K
) )
396, 13dalemcceb 33007 . . . . . . . 8  |-  ( ps 
->  c  e.  ( Base `  K ) )
40393ad2ant3 1028 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  ( Base `  K ) )
4132, 11, 12latlej2 16259 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( ( G  .\/  H )  .\/  I )  e.  ( Base `  K
)  /\  c  e.  ( Base `  K )
)  ->  c  .<_  ( ( ( G  .\/  H )  .\/  I ) 
.\/  c ) )
4230, 38, 40, 41syl3anc 1264 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  .<_  ( ( ( G  .\/  H ) 
.\/  I )  .\/  c ) )
431, 14dalemyeb 32967 . . . . . . . 8  |-  ( ph  ->  Y  e.  ( Base `  K ) )
44433ad2ant1 1026 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  e.  ( Base `  K ) )
4532, 12latjcl 16249 . . . . . . . 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 1264 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( G 
.\/  H )  .\/  I )  .\/  c
)  e.  ( Base `  K ) )
4732, 11, 12latjle12 16260 . . . . . . 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 1266 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( Y  .<_  ( ( ( G  .\/  H )  .\/  I ) 
.\/  c )  /\  c  .<_  ( ( ( G  .\/  H ) 
.\/  I )  .\/  c ) )  <->  ( Y  .\/  c )  .<_  ( ( ( G  .\/  H
)  .\/  I )  .\/  c ) ) )
4928, 42, 48mpbi2and 929 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( Y  .\/  c
)  .<_  ( ( ( G  .\/  H ) 
.\/  I )  .\/  c ) )
5012, 13hlatjrot 32691 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( G  e.  A  /\  H  e.  A  /\  I  e.  A
) )  ->  (
( G  .\/  H
)  .\/  I )  =  ( ( I 
.\/  G )  .\/  H ) )
513, 24, 31, 22, 50syl13anc 1266 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .\/  I )  =  ( ( I 
.\/  G )  .\/  H ) )
5251oveq1d 6311 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( G 
.\/  H )  .\/  I )  .\/  c
)  =  ( ( ( I  .\/  G
)  .\/  H )  .\/  c ) )
5349, 52breqtrd 4441 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( Y  .\/  c
)  .<_  ( ( ( I  .\/  G ) 
.\/  H )  .\/  c ) )
5453adantr 466 . . 3  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  H  .<_  ( I 
.\/  G ) )  ->  ( Y  .\/  c )  .<_  ( ( ( I  .\/  G
)  .\/  H )  .\/  c ) )
5532, 13atbase 32608 . . . . . . 7  |-  ( H  e.  A  ->  H  e.  ( Base `  K
) )
5631, 55syl 17 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  H  e.  ( Base `  K ) )
5732, 12, 13hlatjcl 32685 . . . . . . 7  |-  ( ( K  e.  HL  /\  I  e.  A  /\  G  e.  A )  ->  ( I  .\/  G
)  e.  ( Base `  K ) )
583, 22, 24, 57syl3anc 1264 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( I  .\/  G
)  e.  ( Base `  K ) )
5932, 11, 12latleeqj2 16262 . . . . . 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 1264 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( H  .<_  ( I 
.\/  G )  <->  ( (
I  .\/  G )  .\/  H )  =  ( I  .\/  G ) ) )
6160biimpa 486 . . . 4  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  H  .<_  ( I 
.\/  G ) )  ->  ( ( I 
.\/  G )  .\/  H )  =  ( I 
.\/  G ) )
6261oveq1d 6311 . . 3  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  H  .<_  ( I 
.\/  G ) )  ->  ( ( ( I  .\/  G ) 
.\/  H )  .\/  c )  =  ( ( I  .\/  G
)  .\/  c )
)
6354, 62breqtrd 4441 . 2  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  H  .<_  ( I 
.\/  G ) )  ->  ( Y  .\/  c )  .<_  ( ( I  .\/  G ) 
.\/  c ) )
6426, 63mtand 663 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  H  .<_  ( I 
.\/  G ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867    =/= wne 2616   class class class wbr 4417   ` cfv 5592  (class class class)co 6296   Basecbs 15081   lecple 15157   joincjn 16141   meetcmee 16142   Latclat 16243   Atomscatm 32582   HLchlt 32669   LPlanesclpl 32810   LVolsclvol 32811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-preset 16125  df-poset 16143  df-plt 16156  df-lub 16172  df-glb 16173  df-join 16174  df-meet 16175  df-p0 16237  df-lat 16244  df-clat 16306  df-oposet 32495  df-ol 32497  df-oml 32498  df-covers 32585  df-ats 32586  df-atl 32617  df-cvlat 32641  df-hlat 32670  df-llines 32816  df-lplanes 32817  df-lvols 32818
This theorem is referenced by:  dalem40  33030  dalem41  33031
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