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Theorem dalem38 34524
Description: Lemma for dath 34550. 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 34514 . . . . . 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 34519 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Q  .<_  ( H  .\/  c ) )
142dalemkelat 34438 . . . . . . . 8  |-  ( ph  ->  K  e.  Lat )
15143ad2ant1 1017 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  Lat )
162, 5dalempeb 34453 . . . . . . . 8  |-  ( ph  ->  P  e.  ( Base `  K ) )
17163ad2ant1 1017 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  P  e.  ( Base `  K ) )
182dalemkehl 34437 . . . . . . . . 9  |-  ( ph  ->  K  e.  HL )
19183ad2ant1 1017 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
202, 3, 4, 5, 6, 7, 8, 1, 9, 10dalem23 34510 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  A )
216dalemccea 34497 . . . . . . . . 9  |-  ( ps 
->  c  e.  A
)
22213ad2ant3 1019 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  A )
23 eqid 2467 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
2423, 4, 5hlatjcl 34181 . . . . . . . 8  |-  ( ( K  e.  HL  /\  G  e.  A  /\  c  e.  A )  ->  ( G  .\/  c
)  e.  ( Base `  K ) )
2519, 20, 22, 24syl3anc 1228 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  .\/  c
)  e.  ( Base `  K ) )
262, 5dalemqeb 34454 . . . . . . . 8  |-  ( ph  ->  Q  e.  ( Base `  K ) )
27263ad2ant1 1017 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Q  e.  ( Base `  K ) )
282, 3, 4, 5, 6, 7, 8, 1, 9, 12dalem29 34515 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  H  e.  A )
2923, 4, 5hlatjcl 34181 . . . . . . . 8  |-  ( ( K  e.  HL  /\  H  e.  A  /\  c  e.  A )  ->  ( H  .\/  c
)  e.  ( Base `  K ) )
3019, 28, 22, 29syl3anc 1228 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( H  .\/  c
)  e.  ( Base `  K ) )
3123, 3, 4latjlej12 15554 . . . . . . 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 1237 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( P  .<_  ( G  .\/  c )  /\  Q  .<_  ( H 
.\/  c ) )  ->  ( P  .\/  Q )  .<_  ( ( G  .\/  c )  .\/  ( H  .\/  c ) ) ) )
3311, 13, 32mp2and 679 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  .\/  Q
)  .<_  ( ( G 
.\/  c )  .\/  ( H  .\/  c ) ) )
3423, 5atbase 34104 . . . . . . 7  |-  ( G  e.  A  ->  G  e.  ( Base `  K
) )
3520, 34syl 16 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  ( Base `  K ) )
3623, 5atbase 34104 . . . . . . 7  |-  ( H  e.  A  ->  H  e.  ( Base `  K
) )
3728, 36syl 16 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  H  e.  ( Base `  K ) )
386, 5dalemcceb 34503 . . . . . . 7  |-  ( ps 
->  c  e.  ( Base `  K ) )
39383ad2ant3 1019 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  ( Base `  K ) )
4023, 4latjjdir 15591 . . . . . 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 1230 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .\/  c )  =  ( ( G 
.\/  c )  .\/  ( H  .\/  c ) ) )
4233, 41breqtrrd 4473 . . . 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 34523 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  R  .<_  ( I  .\/  c ) )
452, 4, 5dalempjqeb 34459 . . . . . 6  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
46453ad2ant1 1017 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  .\/  Q
)  e.  ( Base `  K ) )
4723, 4, 5hlatjcl 34181 . . . . . . 7  |-  ( ( K  e.  HL  /\  G  e.  A  /\  H  e.  A )  ->  ( G  .\/  H
)  e.  ( Base `  K ) )
4819, 20, 28, 47syl3anc 1228 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  .\/  H
)  e.  ( Base `  K ) )
4923, 4latjcl 15538 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  c  e.  ( Base `  K )
)  ->  ( ( G  .\/  H )  .\/  c )  e.  (
Base `  K )
)
5015, 48, 39, 49syl3anc 1228 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .\/  c )  e.  ( Base `  K
) )
512, 5dalemreb 34455 . . . . . 6  |-  ( ph  ->  R  e.  ( Base `  K ) )
52513ad2ant1 1017 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  R  e.  ( Base `  K ) )
532, 3, 4, 5, 6, 7, 8, 1, 9, 43dalem34 34520 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  e.  A )
5423, 4, 5hlatjcl 34181 . . . . . 6  |-  ( ( K  e.  HL  /\  I  e.  A  /\  c  e.  A )  ->  ( I  .\/  c
)  e.  ( Base `  K ) )
5519, 53, 22, 54syl3anc 1228 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( I  .\/  c
)  e.  ( Base `  K ) )
5623, 3, 4latjlej12 15554 . . . . 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 1237 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( P 
.\/  Q )  .<_  ( ( G  .\/  H )  .\/  c )  /\  R  .<_  ( I 
.\/  c ) )  ->  ( ( P 
.\/  Q )  .\/  R )  .<_  ( (
( G  .\/  H
)  .\/  c )  .\/  ( I  .\/  c
) ) ) )
5842, 44, 57mp2and 679 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( P  .\/  Q )  .\/  R ) 
.<_  ( ( ( G 
.\/  H )  .\/  c )  .\/  (
I  .\/  c )
) )
5923, 5atbase 34104 . . . . 5  |-  ( I  e.  A  ->  I  e.  ( Base `  K
) )
6053, 59syl 16 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  e.  ( Base `  K ) )
6123, 4latjjdir 15591 . . . 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 1230 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( G 
.\/  H )  .\/  I )  .\/  c
)  =  ( ( ( G  .\/  H
)  .\/  c )  .\/  ( I  .\/  c
) ) )
6358, 62breqtrrd 4473 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( P  .\/  Q )  .\/  R ) 
.<_  ( ( ( G 
.\/  H )  .\/  I )  .\/  c
) )
641, 63syl5eqbr 4480 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  .<_  ( ( ( G  .\/  H ) 
.\/  I )  .\/  c ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447   ` cfv 5588  (class class class)co 6284   Basecbs 14490   lecple 14562   joincjn 15431   meetcmee 15432   Latclat 15532   Atomscatm 34078   HLchlt 34165   LPlanesclpl 34306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-poset 15433  df-plt 15445  df-lub 15461  df-glb 15462  df-join 15463  df-meet 15464  df-p0 15526  df-lat 15533  df-clat 15595  df-oposet 33991  df-ol 33993  df-oml 33994  df-covers 34081  df-ats 34082  df-atl 34113  df-cvlat 34137  df-hlat 34166  df-llines 34312  df-lplanes 34313
This theorem is referenced by:  dalem39  34525
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