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Theorem dalem8 32687
Description: Lemma for dath 32753. 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 32685 . . . 4  |-  ( ph  ->  S  .<_  W )
102, 3, 4, 5, 6, 7, 1, 8dalem7 32686 . . . 4  |-  ( ph  ->  T  .<_  W )
112dalemkelat 32641 . . . . 5  |-  ( ph  ->  K  e.  Lat )
122, 5dalemseb 32659 . . . . 5  |-  ( ph  ->  S  e.  ( Base `  K ) )
132, 5dalemteb 32660 . . . . 5  |-  ( ph  ->  T  e.  ( Base `  K ) )
142, 6dalemyeb 32666 . . . . . . 7  |-  ( ph  ->  Y  e.  ( Base `  K ) )
152, 5dalemceb 32655 . . . . . . 7  |-  ( ph  ->  C  e.  ( Base `  K ) )
16 eqid 2402 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
1716, 4latjcl 16005 . . . . . . 7  |-  ( ( K  e.  Lat  /\  Y  e.  ( Base `  K )  /\  C  e.  ( Base `  K
) )  ->  ( Y  .\/  C )  e.  ( Base `  K
) )
1811, 14, 15, 17syl3anc 1230 . . . . . 6  |-  ( ph  ->  ( Y  .\/  C
)  e.  ( Base `  K ) )
198, 18syl5eqel 2494 . . . . 5  |-  ( ph  ->  W  e.  ( Base `  K ) )
2016, 3, 4latjle12 16016 . . . . 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 1232 . . . 4  |-  ( ph  ->  ( ( S  .<_  W  /\  T  .<_  W )  <-> 
( S  .\/  T
)  .<_  W ) )
229, 10, 21mpbi2and 922 . . 3  |-  ( ph  ->  ( S  .\/  T
)  .<_  W )
232, 3, 4, 5, 6, 7, 8dalem5 32684 . . 3  |-  ( ph  ->  U  .<_  W )
242, 4, 5dalemsjteb 32663 . . . 4  |-  ( ph  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
252, 5dalemueb 32661 . . . 4  |-  ( ph  ->  U  e.  ( Base `  K ) )
2616, 3, 4latjle12 16016 . . . 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 1232 . . 3  |-  ( ph  ->  ( ( ( S 
.\/  T )  .<_  W  /\  U  .<_  W )  <-> 
( ( S  .\/  T )  .\/  U ) 
.<_  W ) )
2822, 23, 27mpbi2and 922 . 2  |-  ( ph  ->  ( ( S  .\/  T )  .\/  U ) 
.<_  W )
291, 28syl5eqbr 4428 1  |-  ( ph  ->  Z  .<_  W )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   class class class wbr 4395   ` cfv 5569  (class class class)co 6278   Basecbs 14841   lecple 14916   joincjn 15897   Latclat 15999   Atomscatm 32281   HLchlt 32368   LPlanesclpl 32509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-preset 15881  df-poset 15899  df-plt 15912  df-lub 15928  df-glb 15929  df-join 15930  df-meet 15931  df-p0 15993  df-lat 16000  df-clat 16062  df-oposet 32194  df-ol 32196  df-oml 32197  df-covers 32284  df-ats 32285  df-atl 32316  df-cvlat 32340  df-hlat 32369  df-llines 32515  df-lplanes 32516
This theorem is referenced by:  dalem13  32693
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