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Theorem dalem8 33306
Description: Lemma for dath 33372. Plane  Z belongs to the 3-dimensional space. (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 )
dalem6.o  |-  O  =  ( LPlanes `  K )
dalem6.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem6.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
dalem6.w  |-  W  =  ( Y  .\/  C
)
Assertion
Ref Expression
dalem8  |-  ( ph  ->  Z  .<_  W )

Proof of Theorem dalem8
StepHypRef Expression
1 dalem6.z . 2  |-  Z  =  ( ( S  .\/  T )  .\/  U )
2 dalema.ph . . . . 5  |-  ( 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 dalemc.l . . . . 5  |-  .<_  =  ( le `  K )
4 dalemc.j . . . . 5  |-  .\/  =  ( join `  K )
5 dalemc.a . . . . 5  |-  A  =  ( Atoms `  K )
6 dalem6.o . . . . 5  |-  O  =  ( LPlanes `  K )
7 dalem6.y . . . . 5  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
8 dalem6.w . . . . 5  |-  W  =  ( Y  .\/  C
)
92, 3, 4, 5, 6, 7, 1, 8dalem6 33304 . . . 4  |-  ( ph  ->  S  .<_  W )
102, 3, 4, 5, 6, 7, 1, 8dalem7 33305 . . . 4  |-  ( ph  ->  T  .<_  W )
112dalemkelat 33260 . . . . 5  |-  ( ph  ->  K  e.  Lat )
122, 5dalemseb 33278 . . . . 5  |-  ( ph  ->  S  e.  ( Base `  K ) )
132, 5dalemteb 33279 . . . . 5  |-  ( ph  ->  T  e.  ( Base `  K ) )
142, 6dalemyeb 33285 . . . . . . 7  |-  ( ph  ->  Y  e.  ( Base `  K ) )
152, 5dalemceb 33274 . . . . . . 7  |-  ( ph  ->  C  e.  ( Base `  K ) )
16 eqid 2471 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
1716, 4latjcl 16375 . . . . . . 7  |-  ( ( K  e.  Lat  /\  Y  e.  ( Base `  K )  /\  C  e.  ( Base `  K
) )  ->  ( Y  .\/  C )  e.  ( Base `  K
) )
1811, 14, 15, 17syl3anc 1292 . . . . . 6  |-  ( ph  ->  ( Y  .\/  C
)  e.  ( Base `  K ) )
198, 18syl5eqel 2553 . . . . 5  |-  ( ph  ->  W  e.  ( Base `  K ) )
2016, 3, 4latjle12 16386 . . . . 5  |-  ( ( K  e.  Lat  /\  ( S  e.  ( Base `  K )  /\  T  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) ) )  -> 
( ( S  .<_  W  /\  T  .<_  W )  <-> 
( S  .\/  T
)  .<_  W ) )
2111, 12, 13, 19, 20syl13anc 1294 . . . 4  |-  ( ph  ->  ( ( S  .<_  W  /\  T  .<_  W )  <-> 
( S  .\/  T
)  .<_  W ) )
229, 10, 21mpbi2and 935 . . 3  |-  ( ph  ->  ( S  .\/  T
)  .<_  W )
232, 3, 4, 5, 6, 7, 8dalem5 33303 . . 3  |-  ( ph  ->  U  .<_  W )
242, 4, 5dalemsjteb 33282 . . . 4  |-  ( ph  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
252, 5dalemueb 33280 . . . 4  |-  ( ph  ->  U  e.  ( Base `  K ) )
2616, 3, 4latjle12 16386 . . . 4  |-  ( ( K  e.  Lat  /\  ( ( S  .\/  T )  e.  ( Base `  K )  /\  U  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
) )  ->  (
( ( S  .\/  T )  .<_  W  /\  U  .<_  W )  <->  ( ( S  .\/  T )  .\/  U )  .<_  W )
)
2711, 24, 25, 19, 26syl13anc 1294 . . 3  |-  ( ph  ->  ( ( ( S 
.\/  T )  .<_  W  /\  U  .<_  W )  <-> 
( ( S  .\/  T )  .\/  U ) 
.<_  W ) )
2822, 23, 27mpbi2and 935 . 2  |-  ( ph  ->  ( ( S  .\/  T )  .\/  U ) 
.<_  W )
291, 28syl5eqbr 4429 1  |-  ( ph  ->  Z  .<_  W )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   Basecbs 15199   lecple 15275   joincjn 16267   Latclat 16369   Atomscatm 32900   HLchlt 32987   LPlanesclpl 33128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-preset 16251  df-poset 16269  df-plt 16282  df-lub 16298  df-glb 16299  df-join 16300  df-meet 16301  df-p0 16363  df-lat 16370  df-clat 16432  df-oposet 32813  df-ol 32815  df-oml 32816  df-covers 32903  df-ats 32904  df-atl 32935  df-cvlat 32959  df-hlat 32988  df-llines 33134  df-lplanes 33135
This theorem is referenced by:  dalem13  33312
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