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Theorem dalem5 30464
Description: Lemma for dath 30533. 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 2436 . 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 30421 . 2  |-  ( ph  ->  K  e.  Lat )
5 dalemc.a . . 3  |-  A  =  ( Atoms `  K )
63, 5dalemueb 30441 . 2  |-  ( ph  ->  U  e.  ( Base `  K ) )
73dalemkehl 30420 . . 3  |-  ( ph  ->  K  e.  HL )
83dalemrea 30425 . . 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 30457 . . 3  |-  ( ph  ->  C  e.  A )
131, 9, 5hlatjcl 30164 . . 3  |-  ( ( K  e.  HL  /\  R  e.  A  /\  C  e.  A )  ->  ( R  .\/  C
)  e.  ( Base `  K ) )
147, 8, 12, 13syl3anc 1184 . 2  |-  ( ph  ->  ( R  .\/  C
)  e.  ( Base `  K ) )
15 dalem5.w . . 3  |-  W  =  ( Y  .\/  C
)
163, 10dalemyeb 30446 . . . 4  |-  ( ph  ->  Y  e.  ( Base `  K ) )
173, 5dalemceb 30435 . . . 4  |-  ( ph  ->  C  e.  ( Base `  K ) )
181, 9latjcl 14479 . . . 4  |-  ( ( K  e.  Lat  /\  Y  e.  ( Base `  K )  /\  C  e.  ( Base `  K
) )  ->  ( Y  .\/  C )  e.  ( Base `  K
) )
194, 16, 17, 18syl3anc 1184 . . 3  |-  ( ph  ->  ( Y  .\/  C
)  e.  ( Base `  K ) )
2015, 19syl5eqel 2520 . 2  |-  ( ph  ->  W  e.  ( Base `  K ) )
213dalemclrju 30433 . . 3  |-  ( ph  ->  C  .<_  ( R  .\/  U ) )
223dalemuea 30428 . . . 4  |-  ( ph  ->  U  e.  A )
233dalempea 30423 . . . . 5  |-  ( ph  ->  P  e.  A )
24 simp313 1106 . . . . . 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 188 . . . . 5  |-  ( ph  ->  -.  C  .<_  ( R 
.\/  P ) )
262, 9, 5atnlej1 30176 . . . . 5  |-  ( ( K  e.  HL  /\  ( C  e.  A  /\  R  e.  A  /\  P  e.  A
)  /\  -.  C  .<_  ( R  .\/  P
) )  ->  C  =/=  R )
277, 12, 8, 23, 25, 26syl131anc 1197 . . . 4  |-  ( ph  ->  C  =/=  R )
282, 9, 5hlatexch1 30192 . . . 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 1197 . . 3  |-  ( ph  ->  ( C  .<_  ( R 
.\/  U )  ->  U  .<_  ( R  .\/  C ) ) )
3021, 29mpd 15 . 2  |-  ( ph  ->  U  .<_  ( R  .\/  C ) )
313, 9, 5dalempjqeb 30442 . . . . . 6  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
323, 5dalemreb 30438 . . . . . 6  |-  ( ph  ->  R  e.  ( Base `  K ) )
331, 2, 9latlej2 14490 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  R  e.  ( Base `  K )
)  ->  R  .<_  ( ( P  .\/  Q
)  .\/  R )
)
344, 31, 32, 33syl3anc 1184 . . . . 5  |-  ( ph  ->  R  .<_  ( ( P  .\/  Q )  .\/  R ) )
3534, 11syl6breqr 4252 . . . 4  |-  ( ph  ->  R  .<_  Y )
361, 2, 9latjlej1 14494 . . . . 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 1186 . . . 4  |-  ( ph  ->  ( R  .<_  Y  -> 
( R  .\/  C
)  .<_  ( Y  .\/  C ) ) )
3835, 37mpd 15 . . 3  |-  ( ph  ->  ( R  .\/  C
)  .<_  ( Y  .\/  C ) )
3938, 15syl6breqr 4252 . 2  |-  ( ph  ->  ( R  .\/  C
)  .<_  W )
401, 2, 4, 6, 14, 20, 30, 39lattrd 14487 1  |-  ( ph  ->  U  .<_  W )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   lecple 13536   joincjn 14401   Latclat 14474   Atomscatm 30061   HLchlt 30148   LPlanesclpl 30289
This theorem is referenced by:  dalem6  30465  dalem8  30467
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-llines 30295  df-lplanes 30296
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