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Theorem dalem39 29900
Description: Lemma for dath 29925. 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 29812 . . . 4  |-  ( ph  ->  K  e.  HL )
323ad2ant1 976 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
41dalemyeo 29821 . . . . 5  |-  ( ph  ->  Y  e.  O )
543ad2ant1 976 . . . 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 29872 . . . . 5  |-  ( ps 
->  c  e.  A
)
873ad2ant3 978 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  A )
96dalem-ccly 29874 . . . . 5  |-  ( ps 
->  -.  c  .<_  Y )
1093ad2ant3 978 . . . 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 2283 . . . . 5  |-  ( LVols `  K )  =  (
LVols `  K )
1611, 12, 13, 14, 15lvoli3 29766 . . . 4  |-  ( ( ( K  e.  HL  /\  Y  e.  O  /\  c  e.  A )  /\  -.  c  .<_  Y )  ->  ( Y  .\/  c )  e.  (
LVols `  K ) )
173, 5, 8, 10, 16syl31anc 1185 . . 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 29895 . . 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 29885 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  A )
2511, 12, 13, 15lvolnle3at 29771 . . 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 1189 . 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 29899 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  .<_  ( ( ( G  .\/  H ) 
.\/  I )  .\/  c ) )
291dalemkelat 29813 . . . . . . . 8  |-  ( ph  ->  K  e.  Lat )
30293ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  Lat )
311, 11, 12, 13, 6, 18, 14, 19, 20, 27dalem29 29890 . . . . . . . . 9  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  H  e.  A )
32 eqid 2283 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
3332, 12, 13hlatjcl 29556 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  G  e.  A  /\  H  e.  A )  ->  ( G  .\/  H
)  e.  ( Base `  K ) )
343, 24, 31, 33syl3anc 1182 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  .\/  H
)  e.  ( Base `  K ) )
3532, 13atbase 29479 . . . . . . . . 9  |-  ( I  e.  A  ->  I  e.  ( Base `  K
) )
3622, 35syl 15 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  e.  ( Base `  K ) )
3732, 12latjcl 14156 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  I  e.  ( Base `  K )
)  ->  ( ( G  .\/  H )  .\/  I )  e.  (
Base `  K )
)
3830, 34, 36, 37syl3anc 1182 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .\/  I )  e.  ( Base `  K
) )
396, 13dalemcceb 29878 . . . . . . . 8  |-  ( ps 
->  c  e.  ( Base `  K ) )
40393ad2ant3 978 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  ( Base `  K ) )
4132, 11, 12latlej2 14167 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( ( G  .\/  H )  .\/  I )  e.  ( Base `  K
)  /\  c  e.  ( Base `  K )
)  ->  c  .<_  ( ( ( G  .\/  H )  .\/  I ) 
.\/  c ) )
4230, 38, 40, 41syl3anc 1182 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  .<_  ( ( ( G  .\/  H ) 
.\/  I )  .\/  c ) )
431, 14dalemyeb 29838 . . . . . . . 8  |-  ( ph  ->  Y  e.  ( Base `  K ) )
44433ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  e.  ( Base `  K ) )
4532, 12latjcl 14156 . . . . . . . 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 1182 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( G 
.\/  H )  .\/  I )  .\/  c
)  e.  ( Base `  K ) )
4732, 11, 12latjle12 14168 . . . . . . 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 1184 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( Y  .<_  ( ( ( G  .\/  H )  .\/  I ) 
.\/  c )  /\  c  .<_  ( ( ( G  .\/  H ) 
.\/  I )  .\/  c ) )  <->  ( Y  .\/  c )  .<_  ( ( ( G  .\/  H
)  .\/  I )  .\/  c ) ) )
4928, 42, 48mpbi2and 887 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( Y  .\/  c
)  .<_  ( ( ( G  .\/  H ) 
.\/  I )  .\/  c ) )
5012, 13hlatjrot 29562 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( G  e.  A  /\  H  e.  A  /\  I  e.  A
) )  ->  (
( G  .\/  H
)  .\/  I )  =  ( ( I 
.\/  G )  .\/  H ) )
513, 24, 31, 22, 50syl13anc 1184 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .\/  I )  =  ( ( I 
.\/  G )  .\/  H ) )
5251oveq1d 5873 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( G 
.\/  H )  .\/  I )  .\/  c
)  =  ( ( ( I  .\/  G
)  .\/  H )  .\/  c ) )
5349, 52breqtrd 4047 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( Y  .\/  c
)  .<_  ( ( ( I  .\/  G ) 
.\/  H )  .\/  c ) )
5453adantr 451 . . 3  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  H  .<_  ( I 
.\/  G ) )  ->  ( Y  .\/  c )  .<_  ( ( ( I  .\/  G
)  .\/  H )  .\/  c ) )
5532, 13atbase 29479 . . . . . . 7  |-  ( H  e.  A  ->  H  e.  ( Base `  K
) )
5631, 55syl 15 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  H  e.  ( Base `  K ) )
5732, 12, 13hlatjcl 29556 . . . . . . 7  |-  ( ( K  e.  HL  /\  I  e.  A  /\  G  e.  A )  ->  ( I  .\/  G
)  e.  ( Base `  K ) )
583, 22, 24, 57syl3anc 1182 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( I  .\/  G
)  e.  ( Base `  K ) )
5932, 11, 12latleeqj2 14170 . . . . . 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 1182 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( H  .<_  ( I 
.\/  G )  <->  ( (
I  .\/  G )  .\/  H )  =  ( I  .\/  G ) ) )
6160biimpa 470 . . . 4  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  H  .<_  ( I 
.\/  G ) )  ->  ( ( I 
.\/  G )  .\/  H )  =  ( I 
.\/  G ) )
6261oveq1d 5873 . . 3  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  H  .<_  ( I 
.\/  G ) )  ->  ( ( ( I  .\/  G ) 
.\/  H )  .\/  c )  =  ( ( I  .\/  G
)  .\/  c )
)
6354, 62breqtrd 4047 . 2  |-  ( ( ( ph  /\  Y  =  Z  /\  ps )  /\  H  .<_  ( I 
.\/  G ) )  ->  ( Y  .\/  c )  .<_  ( ( I  .\/  G ) 
.\/  c ) )
6426, 63mtand 640 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  H  .<_  ( I 
.\/  G ) )
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   Latclat 14151   Atomscatm 29453   HLchlt 29540   LPlanesclpl 29681   LVolsclvol 29682
This theorem is referenced by:  dalem40  29901  dalem41  29902
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  df-lvols 29689
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