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Theorem dalem5 33276
Description: Lemma for dath 33345. Atom  U (in plane  Z  =  S T U) belongs to the 3-dimensional volume formed by  Y and 
C. (Contributed by NM, 21-Jul-2012.)
Hypotheses
Ref Expression
dalema.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 ) ) ) ) )
dalemc.l  |-  .<_  =  ( le `  K )
dalemc.j  |-  .\/  =  ( join `  K )
dalemc.a  |-  A  =  ( Atoms `  K )
dalem5.o  |-  O  =  ( LPlanes `  K )
dalem5.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem5.w  |-  W  =  ( Y  .\/  C
)
Assertion
Ref Expression
dalem5  |-  ( ph  ->  U  .<_  W )

Proof of Theorem dalem5
StepHypRef Expression
1 eqid 2461 . 2  |-  ( Base `  K )  =  (
Base `  K )
2 dalemc.l . 2  |-  .<_  =  ( le `  K )
3 dalema.ph . . 3  |-  ( 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 ) ) ) ) )
43dalemkelat 33233 . 2  |-  ( ph  ->  K  e.  Lat )
5 dalemc.a . . 3  |-  A  =  ( Atoms `  K )
63, 5dalemueb 33253 . 2  |-  ( ph  ->  U  e.  ( Base `  K ) )
73dalemkehl 33232 . . 3  |-  ( ph  ->  K  e.  HL )
83dalemrea 33237 . . 3  |-  ( ph  ->  R  e.  A )
9 dalemc.j . . . 4  |-  .\/  =  ( join `  K )
10 dalem5.o . . . 4  |-  O  =  ( LPlanes `  K )
11 dalem5.y . . . 4  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
123, 2, 9, 5, 10, 11dalemcea 33269 . . 3  |-  ( ph  ->  C  e.  A )
131, 9, 5hlatjcl 32976 . . 3  |-  ( ( K  e.  HL  /\  R  e.  A  /\  C  e.  A )  ->  ( R  .\/  C
)  e.  ( Base `  K ) )
147, 8, 12, 13syl3anc 1276 . 2  |-  ( ph  ->  ( R  .\/  C
)  e.  ( Base `  K ) )
15 dalem5.w . . 3  |-  W  =  ( Y  .\/  C
)
163, 10dalemyeb 33258 . . . 4  |-  ( ph  ->  Y  e.  ( Base `  K ) )
173, 5dalemceb 33247 . . . 4  |-  ( ph  ->  C  e.  ( Base `  K ) )
181, 9latjcl 16345 . . . 4  |-  ( ( K  e.  Lat  /\  Y  e.  ( Base `  K )  /\  C  e.  ( Base `  K
) )  ->  ( Y  .\/  C )  e.  ( Base `  K
) )
194, 16, 17, 18syl3anc 1276 . . 3  |-  ( ph  ->  ( Y  .\/  C
)  e.  ( Base `  K ) )
2015, 19syl5eqel 2543 . 2  |-  ( ph  ->  W  e.  ( Base `  K ) )
213dalemclrju 33245 . . 3  |-  ( ph  ->  C  .<_  ( R  .\/  U ) )
223dalemuea 33240 . . . 4  |-  ( ph  ->  U  e.  A )
233dalempea 33235 . . . . 5  |-  ( ph  ->  P  e.  A )
24 simp313 1163 . . . . . 6  |-  ( ( ( ( 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 ) ) ) )  ->  -.  C  .<_  ( R  .\/  P ) )
253, 24sylbi 200 . . . . 5  |-  ( ph  ->  -.  C  .<_  ( R 
.\/  P ) )
262, 9, 5atnlej1 32988 . . . . 5  |-  ( ( K  e.  HL  /\  ( C  e.  A  /\  R  e.  A  /\  P  e.  A
)  /\  -.  C  .<_  ( R  .\/  P
) )  ->  C  =/=  R )
277, 12, 8, 23, 25, 26syl131anc 1289 . . . 4  |-  ( ph  ->  C  =/=  R )
282, 9, 5hlatexch1 33004 . . . 4  |-  ( ( K  e.  HL  /\  ( C  e.  A  /\  U  e.  A  /\  R  e.  A
)  /\  C  =/=  R )  ->  ( C  .<_  ( R  .\/  U
)  ->  U  .<_  ( R  .\/  C ) ) )
297, 12, 22, 8, 27, 28syl131anc 1289 . . 3  |-  ( ph  ->  ( C  .<_  ( R 
.\/  U )  ->  U  .<_  ( R  .\/  C ) ) )
3021, 29mpd 15 . 2  |-  ( ph  ->  U  .<_  ( R  .\/  C ) )
313, 9, 5dalempjqeb 33254 . . . . . 6  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
323, 5dalemreb 33250 . . . . . 6  |-  ( ph  ->  R  e.  ( Base `  K ) )
331, 2, 9latlej2 16355 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  R  e.  ( Base `  K )
)  ->  R  .<_  ( ( P  .\/  Q
)  .\/  R )
)
344, 31, 32, 33syl3anc 1276 . . . . 5  |-  ( ph  ->  R  .<_  ( ( P  .\/  Q )  .\/  R ) )
3534, 11syl6breqr 4456 . . . 4  |-  ( ph  ->  R  .<_  Y )
361, 2, 9latjlej1 16359 . . . . 5  |-  ( ( K  e.  Lat  /\  ( R  e.  ( Base `  K )  /\  Y  e.  ( Base `  K )  /\  C  e.  ( Base `  K
) ) )  -> 
( R  .<_  Y  -> 
( R  .\/  C
)  .<_  ( Y  .\/  C ) ) )
374, 32, 16, 17, 36syl13anc 1278 . . . 4  |-  ( ph  ->  ( R  .<_  Y  -> 
( R  .\/  C
)  .<_  ( Y  .\/  C ) ) )
3835, 37mpd 15 . . 3  |-  ( ph  ->  ( R  .\/  C
)  .<_  ( Y  .\/  C ) )
3938, 15syl6breqr 4456 . 2  |-  ( ph  ->  ( R  .\/  C
)  .<_  W )
401, 2, 4, 6, 14, 20, 30, 39lattrd 16352 1  |-  ( ph  ->  U  .<_  W )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 991    = wceq 1454    e. wcel 1897    =/= wne 2632   class class class wbr 4415   ` cfv 5600  (class class class)co 6314   Basecbs 15169   lecple 15245   joincjn 16237   Latclat 16339   Atomscatm 32873   HLchlt 32960   LPlanesclpl 33101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-reu 2755  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6276  df-ov 6317  df-oprab 6318  df-preset 16221  df-poset 16239  df-plt 16252  df-lub 16268  df-glb 16269  df-join 16270  df-meet 16271  df-p0 16333  df-lat 16340  df-clat 16402  df-oposet 32786  df-ol 32788  df-oml 32789  df-covers 32876  df-ats 32877  df-atl 32908  df-cvlat 32932  df-hlat 32961  df-llines 33107  df-lplanes 33108
This theorem is referenced by:  dalem6  33277  dalem8  33279
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