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Theorem dalem8 33623
Description: Lemma for dath 33689. 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 33621 . . . 4  |-  ( ph  ->  S  .<_  W )
102, 3, 4, 5, 6, 7, 1, 8dalem7 33622 . . . 4  |-  ( ph  ->  T  .<_  W )
112dalemkelat 33577 . . . . 5  |-  ( ph  ->  K  e.  Lat )
122, 5dalemseb 33595 . . . . 5  |-  ( ph  ->  S  e.  ( Base `  K ) )
132, 5dalemteb 33596 . . . . 5  |-  ( ph  ->  T  e.  ( Base `  K ) )
142, 6dalemyeb 33602 . . . . . . 7  |-  ( ph  ->  Y  e.  ( Base `  K ) )
152, 5dalemceb 33591 . . . . . . 7  |-  ( ph  ->  C  e.  ( Base `  K ) )
16 eqid 2451 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
1716, 4latjcl 15332 . . . . . . 7  |-  ( ( K  e.  Lat  /\  Y  e.  ( Base `  K )  /\  C  e.  ( Base `  K
) )  ->  ( Y  .\/  C )  e.  ( Base `  K
) )
1811, 14, 15, 17syl3anc 1219 . . . . . 6  |-  ( ph  ->  ( Y  .\/  C
)  e.  ( Base `  K ) )
198, 18syl5eqel 2543 . . . . 5  |-  ( ph  ->  W  e.  ( Base `  K ) )
2016, 3, 4latjle12 15343 . . . . 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 1221 . . . 4  |-  ( ph  ->  ( ( S  .<_  W  /\  T  .<_  W )  <-> 
( S  .\/  T
)  .<_  W ) )
229, 10, 21mpbi2and 912 . . 3  |-  ( ph  ->  ( S  .\/  T
)  .<_  W )
232, 3, 4, 5, 6, 7, 8dalem5 33620 . . 3  |-  ( ph  ->  U  .<_  W )
242, 4, 5dalemsjteb 33599 . . . 4  |-  ( ph  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
252, 5dalemueb 33597 . . . 4  |-  ( ph  ->  U  e.  ( Base `  K ) )
2616, 3, 4latjle12 15343 . . . 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 1221 . . 3  |-  ( ph  ->  ( ( ( S 
.\/  T )  .<_  W  /\  U  .<_  W )  <-> 
( ( S  .\/  T )  .\/  U ) 
.<_  W ) )
2822, 23, 27mpbi2and 912 . 2  |-  ( ph  ->  ( ( S  .\/  T )  .\/  U ) 
.<_  W )
291, 28syl5eqbr 4426 1  |-  ( ph  ->  Z  .<_  W )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   class class class wbr 4393   ` cfv 5519  (class class class)co 6193   Basecbs 14285   lecple 14356   joincjn 15225   Latclat 15326   Atomscatm 33217   HLchlt 33304   LPlanesclpl 33445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-poset 15227  df-plt 15239  df-lub 15255  df-glb 15256  df-join 15257  df-meet 15258  df-p0 15320  df-lat 15327  df-clat 15389  df-oposet 33130  df-ol 33132  df-oml 33133  df-covers 33220  df-ats 33221  df-atl 33252  df-cvlat 33276  df-hlat 33305  df-llines 33451  df-lplanes 33452
This theorem is referenced by:  dalem13  33629
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