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Theorem dalem38 30521
Description: Lemma for dath 30547. 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 30511 . . . . . 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 30516 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Q  .<_  ( H  .\/  c ) )
142dalemkelat 30435 . . . . . . . 8  |-  ( ph  ->  K  e.  Lat )
15143ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  Lat )
162, 5dalempeb 30450 . . . . . . . 8  |-  ( ph  ->  P  e.  ( Base `  K ) )
17163ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  P  e.  ( Base `  K ) )
182dalemkehl 30434 . . . . . . . . 9  |-  ( ph  ->  K  e.  HL )
19183ad2ant1 976 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
202, 3, 4, 5, 6, 7, 8, 1, 9, 10dalem23 30507 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  A )
216dalemccea 30494 . . . . . . . . 9  |-  ( ps 
->  c  e.  A
)
22213ad2ant3 978 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  A )
23 eqid 2296 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
2423, 4, 5hlatjcl 30178 . . . . . . . 8  |-  ( ( K  e.  HL  /\  G  e.  A  /\  c  e.  A )  ->  ( G  .\/  c
)  e.  ( Base `  K ) )
2519, 20, 22, 24syl3anc 1182 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  .\/  c
)  e.  ( Base `  K ) )
262, 5dalemqeb 30451 . . . . . . . 8  |-  ( ph  ->  Q  e.  ( Base `  K ) )
27263ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Q  e.  ( Base `  K ) )
282, 3, 4, 5, 6, 7, 8, 1, 9, 12dalem29 30512 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  H  e.  A )
2923, 4, 5hlatjcl 30178 . . . . . . . 8  |-  ( ( K  e.  HL  /\  H  e.  A  /\  c  e.  A )  ->  ( H  .\/  c
)  e.  ( Base `  K ) )
3019, 28, 22, 29syl3anc 1182 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( H  .\/  c
)  e.  ( Base `  K ) )
3123, 3, 4latjlej12 14189 . . . . . . 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 1191 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( P  .<_  ( G  .\/  c )  /\  Q  .<_  ( H 
.\/  c ) )  ->  ( P  .\/  Q )  .<_  ( ( G  .\/  c )  .\/  ( H  .\/  c ) ) ) )
3311, 13, 32mp2and 660 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  .\/  Q
)  .<_  ( ( G 
.\/  c )  .\/  ( H  .\/  c ) ) )
3423, 5atbase 30101 . . . . . . 7  |-  ( G  e.  A  ->  G  e.  ( Base `  K
) )
3520, 34syl 15 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  ( Base `  K ) )
3623, 5atbase 30101 . . . . . . 7  |-  ( H  e.  A  ->  H  e.  ( Base `  K
) )
3728, 36syl 15 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  H  e.  ( Base `  K ) )
386, 5dalemcceb 30500 . . . . . . 7  |-  ( ps 
->  c  e.  ( Base `  K ) )
39383ad2ant3 978 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  ( Base `  K ) )
4023, 4latjjdir 14226 . . . . . 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 1184 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .\/  c )  =  ( ( G 
.\/  c )  .\/  ( H  .\/  c ) ) )
4233, 41breqtrrd 4065 . . . 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 30520 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  R  .<_  ( I  .\/  c ) )
452, 4, 5dalempjqeb 30456 . . . . . 6  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
46453ad2ant1 976 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  .\/  Q
)  e.  ( Base `  K ) )
4723, 4, 5hlatjcl 30178 . . . . . . 7  |-  ( ( K  e.  HL  /\  G  e.  A  /\  H  e.  A )  ->  ( G  .\/  H
)  e.  ( Base `  K ) )
4819, 20, 28, 47syl3anc 1182 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  .\/  H
)  e.  ( Base `  K ) )
4923, 4latjcl 14172 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  c  e.  ( Base `  K )
)  ->  ( ( G  .\/  H )  .\/  c )  e.  (
Base `  K )
)
5015, 48, 39, 49syl3anc 1182 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .\/  c )  e.  ( Base `  K
) )
512, 5dalemreb 30452 . . . . . 6  |-  ( ph  ->  R  e.  ( Base `  K ) )
52513ad2ant1 976 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  R  e.  ( Base `  K ) )
532, 3, 4, 5, 6, 7, 8, 1, 9, 43dalem34 30517 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  e.  A )
5423, 4, 5hlatjcl 30178 . . . . . 6  |-  ( ( K  e.  HL  /\  I  e.  A  /\  c  e.  A )  ->  ( I  .\/  c
)  e.  ( Base `  K ) )
5519, 53, 22, 54syl3anc 1182 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( I  .\/  c
)  e.  ( Base `  K ) )
5623, 3, 4latjlej12 14189 . . . . 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 1191 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( P 
.\/  Q )  .<_  ( ( G  .\/  H )  .\/  c )  /\  R  .<_  ( I 
.\/  c ) )  ->  ( ( P 
.\/  Q )  .\/  R )  .<_  ( (
( G  .\/  H
)  .\/  c )  .\/  ( I  .\/  c
) ) ) )
5842, 44, 57mp2and 660 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( P  .\/  Q )  .\/  R ) 
.<_  ( ( ( G 
.\/  H )  .\/  c )  .\/  (
I  .\/  c )
) )
5923, 5atbase 30101 . . . . 5  |-  ( I  e.  A  ->  I  e.  ( Base `  K
) )
6053, 59syl 15 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  e.  ( Base `  K ) )
6123, 4latjjdir 14226 . . . 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 1184 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( G 
.\/  H )  .\/  I )  .\/  c
)  =  ( ( ( G  .\/  H
)  .\/  c )  .\/  ( I  .\/  c
) ) )
6358, 62breqtrrd 4065 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( P  .\/  Q )  .\/  R ) 
.<_  ( ( ( G 
.\/  H )  .\/  I )  .\/  c
) )
641, 63syl5eqbr 4072 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  .<_  ( ( ( G  .\/  H ) 
.\/  I )  .\/  c ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   meetcmee 14095   Latclat 14167   Atomscatm 30075   HLchlt 30162   LPlanesclpl 30303
This theorem is referenced by:  dalem39  30522
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310
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