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

Theorem dalem38 32984
Description: Lemma for dath 33010. 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 32974 . . . . . 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 32979 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Q  .<_  ( H  .\/  c ) )
142dalemkelat 32898 . . . . . . . 8  |-  ( ph  ->  K  e.  Lat )
15143ad2ant1 1026 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  Lat )
162, 5dalempeb 32913 . . . . . . . 8  |-  ( ph  ->  P  e.  ( Base `  K ) )
17163ad2ant1 1026 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  P  e.  ( Base `  K ) )
182dalemkehl 32897 . . . . . . . . 9  |-  ( ph  ->  K  e.  HL )
19183ad2ant1 1026 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
202, 3, 4, 5, 6, 7, 8, 1, 9, 10dalem23 32970 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  A )
216dalemccea 32957 . . . . . . . . 9  |-  ( ps 
->  c  e.  A
)
22213ad2ant3 1028 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  A )
23 eqid 2429 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
2423, 4, 5hlatjcl 32641 . . . . . . . 8  |-  ( ( K  e.  HL  /\  G  e.  A  /\  c  e.  A )  ->  ( G  .\/  c
)  e.  ( Base `  K ) )
2519, 20, 22, 24syl3anc 1264 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  .\/  c
)  e.  ( Base `  K ) )
262, 5dalemqeb 32914 . . . . . . . 8  |-  ( ph  ->  Q  e.  ( Base `  K ) )
27263ad2ant1 1026 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Q  e.  ( Base `  K ) )
282, 3, 4, 5, 6, 7, 8, 1, 9, 12dalem29 32975 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  H  e.  A )
2923, 4, 5hlatjcl 32641 . . . . . . . 8  |-  ( ( K  e.  HL  /\  H  e.  A  /\  c  e.  A )  ->  ( H  .\/  c
)  e.  ( Base `  K ) )
3019, 28, 22, 29syl3anc 1264 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( H  .\/  c
)  e.  ( Base `  K ) )
3123, 3, 4latjlej12 16264 . . . . . . 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 1273 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( P  .<_  ( G  .\/  c )  /\  Q  .<_  ( H 
.\/  c ) )  ->  ( P  .\/  Q )  .<_  ( ( G  .\/  c )  .\/  ( H  .\/  c ) ) ) )
3311, 13, 32mp2and 683 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  .\/  Q
)  .<_  ( ( G 
.\/  c )  .\/  ( H  .\/  c ) ) )
3423, 5atbase 32564 . . . . . . 7  |-  ( G  e.  A  ->  G  e.  ( Base `  K
) )
3520, 34syl 17 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  ( Base `  K ) )
3623, 5atbase 32564 . . . . . . 7  |-  ( H  e.  A  ->  H  e.  ( Base `  K
) )
3728, 36syl 17 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  H  e.  ( Base `  K ) )
386, 5dalemcceb 32963 . . . . . . 7  |-  ( ps 
->  c  e.  ( Base `  K ) )
39383ad2ant3 1028 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  ( Base `  K ) )
4023, 4latjjdir 16301 . . . . . 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 1266 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .\/  c )  =  ( ( G 
.\/  c )  .\/  ( H  .\/  c ) ) )
4233, 41breqtrrd 4452 . . . 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 32983 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  R  .<_  ( I  .\/  c ) )
452, 4, 5dalempjqeb 32919 . . . . . 6  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
46453ad2ant1 1026 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  .\/  Q
)  e.  ( Base `  K ) )
4723, 4, 5hlatjcl 32641 . . . . . . 7  |-  ( ( K  e.  HL  /\  G  e.  A  /\  H  e.  A )  ->  ( G  .\/  H
)  e.  ( Base `  K ) )
4819, 20, 28, 47syl3anc 1264 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  .\/  H
)  e.  ( Base `  K ) )
4923, 4latjcl 16248 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  c  e.  ( Base `  K )
)  ->  ( ( G  .\/  H )  .\/  c )  e.  (
Base `  K )
)
5015, 48, 39, 49syl3anc 1264 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .\/  c )  e.  ( Base `  K
) )
512, 5dalemreb 32915 . . . . . 6  |-  ( ph  ->  R  e.  ( Base `  K ) )
52513ad2ant1 1026 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  R  e.  ( Base `  K ) )
532, 3, 4, 5, 6, 7, 8, 1, 9, 43dalem34 32980 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  e.  A )
5423, 4, 5hlatjcl 32641 . . . . . 6  |-  ( ( K  e.  HL  /\  I  e.  A  /\  c  e.  A )  ->  ( I  .\/  c
)  e.  ( Base `  K ) )
5519, 53, 22, 54syl3anc 1264 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( I  .\/  c
)  e.  ( Base `  K ) )
5623, 3, 4latjlej12 16264 . . . . 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 1273 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( P 
.\/  Q )  .<_  ( ( G  .\/  H )  .\/  c )  /\  R  .<_  ( I 
.\/  c ) )  ->  ( ( P 
.\/  Q )  .\/  R )  .<_  ( (
( G  .\/  H
)  .\/  c )  .\/  ( I  .\/  c
) ) ) )
5842, 44, 57mp2and 683 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( P  .\/  Q )  .\/  R ) 
.<_  ( ( ( G 
.\/  H )  .\/  c )  .\/  (
I  .\/  c )
) )
5923, 5atbase 32564 . . . . 5  |-  ( I  e.  A  ->  I  e.  ( Base `  K
) )
6053, 59syl 17 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  e.  ( Base `  K ) )
6123, 4latjjdir 16301 . . . 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 1266 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( G 
.\/  H )  .\/  I )  .\/  c
)  =  ( ( ( G  .\/  H
)  .\/  c )  .\/  ( I  .\/  c
) ) )
6358, 62breqtrrd 4452 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( P  .\/  Q )  .\/  R ) 
.<_  ( ( ( G 
.\/  H )  .\/  I )  .\/  c
) )
641, 63syl5eqbr 4459 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  .<_  ( ( ( G  .\/  H ) 
.\/  I )  .\/  c ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   Basecbs 15084   lecple 15159   joincjn 16140   meetcmee 16141   Latclat 16242   Atomscatm 32538   HLchlt 32625   LPlanesclpl 32766
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 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
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 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-preset 16124  df-poset 16142  df-plt 16155  df-lub 16171  df-glb 16172  df-join 16173  df-meet 16174  df-p0 16236  df-lat 16243  df-clat 16305  df-oposet 32451  df-ol 32453  df-oml 32454  df-covers 32541  df-ats 32542  df-atl 32573  df-cvlat 32597  df-hlat 32626  df-llines 32772  df-lplanes 32773
This theorem is referenced by:  dalem39  32985
  Copyright terms: Public domain W3C validator