Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dalem38 Unicode version

Theorem dalem38 29826
Description: Lemma for dath 29852. Plane  Y belongs to the 3-dimensional volume  G H I c. (Contributed by NM, 5-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
dalem38  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  .<_  ( ( ( G  .\/  H ) 
.\/  I )  .\/  c ) )

Proof of Theorem dalem38
StepHypRef Expression
1 dalem38.y . 2  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
2 dalem.ph . . . . . . 7  |-  ( 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 ) ) ) ) )
3 dalem.l . . . . . . 7  |-  .<_  =  ( le `  K )
4 dalem.j . . . . . . 7  |-  .\/  =  ( join `  K )
5 dalem.a . . . . . . 7  |-  A  =  ( Atoms `  K )
6 dalem.ps . . . . . . 7  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
7 dalem38.m . . . . . . 7  |-  ./\  =  ( meet `  K )
8 dalem38.o . . . . . . 7  |-  O  =  ( LPlanes `  K )
9 dalem38.z . . . . . . 7  |-  Z  =  ( ( S  .\/  T )  .\/  U )
10 dalem38.g . . . . . . 7  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
112, 3, 4, 5, 6, 7, 8, 1, 9, 10dalem28 29816 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  P  .<_  ( G  .\/  c ) )
12 dalem38.h . . . . . . 7  |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )
132, 3, 4, 5, 6, 7, 8, 1, 9, 12dalem33 29821 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Q  .<_  ( H  .\/  c ) )
142dalemkelat 29740 . . . . . . . 8  |-  ( ph  ->  K  e.  Lat )
15143ad2ant1 978 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  Lat )
162, 5dalempeb 29755 . . . . . . . 8  |-  ( ph  ->  P  e.  ( Base `  K ) )
17163ad2ant1 978 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  P  e.  ( Base `  K ) )
182dalemkehl 29739 . . . . . . . . 9  |-  ( ph  ->  K  e.  HL )
19183ad2ant1 978 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
202, 3, 4, 5, 6, 7, 8, 1, 9, 10dalem23 29812 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  A )
216dalemccea 29799 . . . . . . . . 9  |-  ( ps 
->  c  e.  A
)
22213ad2ant3 980 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  A )
23 eqid 2389 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
2423, 4, 5hlatjcl 29483 . . . . . . . 8  |-  ( ( K  e.  HL  /\  G  e.  A  /\  c  e.  A )  ->  ( G  .\/  c
)  e.  ( Base `  K ) )
2519, 20, 22, 24syl3anc 1184 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  .\/  c
)  e.  ( Base `  K ) )
262, 5dalemqeb 29756 . . . . . . . 8  |-  ( ph  ->  Q  e.  ( Base `  K ) )
27263ad2ant1 978 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Q  e.  ( Base `  K ) )
282, 3, 4, 5, 6, 7, 8, 1, 9, 12dalem29 29817 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  H  e.  A )
2923, 4, 5hlatjcl 29483 . . . . . . . 8  |-  ( ( K  e.  HL  /\  H  e.  A  /\  c  e.  A )  ->  ( H  .\/  c
)  e.  ( Base `  K ) )
3019, 28, 22, 29syl3anc 1184 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( H  .\/  c
)  e.  ( Base `  K ) )
3123, 3, 4latjlej12 14425 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  ( G  .\/  c )  e.  ( Base `  K
) )  /\  ( Q  e.  ( Base `  K )  /\  ( H  .\/  c )  e.  ( Base `  K
) ) )  -> 
( ( P  .<_  ( G  .\/  c )  /\  Q  .<_  ( H 
.\/  c ) )  ->  ( P  .\/  Q )  .<_  ( ( G  .\/  c )  .\/  ( H  .\/  c ) ) ) )
3215, 17, 25, 27, 30, 31syl122anc 1193 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( P  .<_  ( G  .\/  c )  /\  Q  .<_  ( H 
.\/  c ) )  ->  ( P  .\/  Q )  .<_  ( ( G  .\/  c )  .\/  ( H  .\/  c ) ) ) )
3311, 13, 32mp2and 661 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  .\/  Q
)  .<_  ( ( G 
.\/  c )  .\/  ( H  .\/  c ) ) )
3423, 5atbase 29406 . . . . . . 7  |-  ( G  e.  A  ->  G  e.  ( Base `  K
) )
3520, 34syl 16 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  ( Base `  K ) )
3623, 5atbase 29406 . . . . . . 7  |-  ( H  e.  A  ->  H  e.  ( Base `  K
) )
3728, 36syl 16 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  H  e.  ( Base `  K ) )
386, 5dalemcceb 29805 . . . . . . 7  |-  ( ps 
->  c  e.  ( Base `  K ) )
39383ad2ant3 980 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  ( Base `  K ) )
4023, 4latjjdir 14462 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( G  e.  ( Base `  K )  /\  H  e.  ( Base `  K )  /\  c  e.  ( Base `  K
) ) )  -> 
( ( G  .\/  H )  .\/  c )  =  ( ( G 
.\/  c )  .\/  ( H  .\/  c ) ) )
4115, 35, 37, 39, 40syl13anc 1186 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .\/  c )  =  ( ( G 
.\/  c )  .\/  ( H  .\/  c ) ) )
4233, 41breqtrrd 4181 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  .\/  Q
)  .<_  ( ( G 
.\/  H )  .\/  c ) )
43 dalem38.i . . . . 5  |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )
442, 3, 4, 5, 6, 7, 8, 1, 9, 43dalem37 29825 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  R  .<_  ( I  .\/  c ) )
452, 4, 5dalempjqeb 29761 . . . . . 6  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
46453ad2ant1 978 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  .\/  Q
)  e.  ( Base `  K ) )
4723, 4, 5hlatjcl 29483 . . . . . . 7  |-  ( ( K  e.  HL  /\  G  e.  A  /\  H  e.  A )  ->  ( G  .\/  H
)  e.  ( Base `  K ) )
4819, 20, 28, 47syl3anc 1184 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  .\/  H
)  e.  ( Base `  K ) )
4923, 4latjcl 14408 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  c  e.  ( Base `  K )
)  ->  ( ( G  .\/  H )  .\/  c )  e.  (
Base `  K )
)
5015, 48, 39, 49syl3anc 1184 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .\/  c )  e.  ( Base `  K
) )
512, 5dalemreb 29757 . . . . . 6  |-  ( ph  ->  R  e.  ( Base `  K ) )
52513ad2ant1 978 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  R  e.  ( Base `  K ) )
532, 3, 4, 5, 6, 7, 8, 1, 9, 43dalem34 29822 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  e.  A )
5423, 4, 5hlatjcl 29483 . . . . . 6  |-  ( ( K  e.  HL  /\  I  e.  A  /\  c  e.  A )  ->  ( I  .\/  c
)  e.  ( Base `  K ) )
5519, 53, 22, 54syl3anc 1184 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( I  .\/  c
)  e.  ( Base `  K ) )
5623, 3, 4latjlej12 14425 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( P  .\/  Q )  e.  ( Base `  K )  /\  (
( G  .\/  H
)  .\/  c )  e.  ( Base `  K
) )  /\  ( R  e.  ( Base `  K )  /\  (
I  .\/  c )  e.  ( Base `  K
) ) )  -> 
( ( ( P 
.\/  Q )  .<_  ( ( G  .\/  H )  .\/  c )  /\  R  .<_  ( I 
.\/  c ) )  ->  ( ( P 
.\/  Q )  .\/  R )  .<_  ( (
( G  .\/  H
)  .\/  c )  .\/  ( I  .\/  c
) ) ) )
5715, 46, 50, 52, 55, 56syl122anc 1193 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( P 
.\/  Q )  .<_  ( ( G  .\/  H )  .\/  c )  /\  R  .<_  ( I 
.\/  c ) )  ->  ( ( P 
.\/  Q )  .\/  R )  .<_  ( (
( G  .\/  H
)  .\/  c )  .\/  ( I  .\/  c
) ) ) )
5842, 44, 57mp2and 661 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( P  .\/  Q )  .\/  R ) 
.<_  ( ( ( G 
.\/  H )  .\/  c )  .\/  (
I  .\/  c )
) )
5923, 5atbase 29406 . . . . 5  |-  ( I  e.  A  ->  I  e.  ( Base `  K
) )
6053, 59syl 16 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  e.  ( Base `  K ) )
6123, 4latjjdir 14462 . . . 4  |-  ( ( K  e.  Lat  /\  ( ( G  .\/  H )  e.  ( Base `  K )  /\  I  e.  ( Base `  K
)  /\  c  e.  ( Base `  K )
) )  ->  (
( ( G  .\/  H )  .\/  I ) 
.\/  c )  =  ( ( ( G 
.\/  H )  .\/  c )  .\/  (
I  .\/  c )
) )
6215, 48, 60, 39, 61syl13anc 1186 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( G 
.\/  H )  .\/  I )  .\/  c
)  =  ( ( ( G  .\/  H
)  .\/  c )  .\/  ( I  .\/  c
) ) )
6358, 62breqtrrd 4181 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( P  .\/  Q )  .\/  R ) 
.<_  ( ( ( G 
.\/  H )  .\/  I )  .\/  c
) )
641, 63syl5eqbr 4188 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  .<_  ( ( ( G  .\/  H ) 
.\/  I )  .\/  c ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2552   class class class wbr 4155   ` cfv 5396  (class class class)co 6022   Basecbs 13398   lecple 13465   joincjn 14330   meetcmee 14331   Latclat 14403   Atomscatm 29380   HLchlt 29467   LPlanesclpl 29608
This theorem is referenced by:  dalem39  29827
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-undef 6481  df-riota 6487  df-poset 14332  df-plt 14344  df-lub 14360  df-glb 14361  df-join 14362  df-meet 14363  df-p0 14397  df-lat 14404  df-clat 14466  df-oposet 29293  df-ol 29295  df-oml 29296  df-covers 29383  df-ats 29384  df-atl 29415  df-cvlat 29439  df-hlat 29468  df-llines 29614  df-lplanes 29615
  Copyright terms: Public domain W3C validator