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Theorem dalem5 33630
Description: Lemma for dath 33699. 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 2452 . 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 33587 . 2  |-  ( ph  ->  K  e.  Lat )
5 dalemc.a . . 3  |-  A  =  ( Atoms `  K )
63, 5dalemueb 33607 . 2  |-  ( ph  ->  U  e.  ( Base `  K ) )
73dalemkehl 33586 . . 3  |-  ( ph  ->  K  e.  HL )
83dalemrea 33591 . . 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 33623 . . 3  |-  ( ph  ->  C  e.  A )
131, 9, 5hlatjcl 33330 . . 3  |-  ( ( K  e.  HL  /\  R  e.  A  /\  C  e.  A )  ->  ( R  .\/  C
)  e.  ( Base `  K ) )
147, 8, 12, 13syl3anc 1219 . 2  |-  ( ph  ->  ( R  .\/  C
)  e.  ( Base `  K ) )
15 dalem5.w . . 3  |-  W  =  ( Y  .\/  C
)
163, 10dalemyeb 33612 . . . 4  |-  ( ph  ->  Y  e.  ( Base `  K ) )
173, 5dalemceb 33601 . . . 4  |-  ( ph  ->  C  e.  ( Base `  K ) )
181, 9latjcl 15335 . . . 4  |-  ( ( K  e.  Lat  /\  Y  e.  ( Base `  K )  /\  C  e.  ( Base `  K
) )  ->  ( Y  .\/  C )  e.  ( Base `  K
) )
194, 16, 17, 18syl3anc 1219 . . 3  |-  ( ph  ->  ( Y  .\/  C
)  e.  ( Base `  K ) )
2015, 19syl5eqel 2544 . 2  |-  ( ph  ->  W  e.  ( Base `  K ) )
213dalemclrju 33599 . . 3  |-  ( ph  ->  C  .<_  ( R  .\/  U ) )
223dalemuea 33594 . . . 4  |-  ( ph  ->  U  e.  A )
233dalempea 33589 . . . . 5  |-  ( ph  ->  P  e.  A )
24 simp313 1137 . . . . . 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 195 . . . . 5  |-  ( ph  ->  -.  C  .<_  ( R 
.\/  P ) )
262, 9, 5atnlej1 33342 . . . . 5  |-  ( ( K  e.  HL  /\  ( C  e.  A  /\  R  e.  A  /\  P  e.  A
)  /\  -.  C  .<_  ( R  .\/  P
) )  ->  C  =/=  R )
277, 12, 8, 23, 25, 26syl131anc 1232 . . . 4  |-  ( ph  ->  C  =/=  R )
282, 9, 5hlatexch1 33358 . . . 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 1232 . . 3  |-  ( ph  ->  ( C  .<_  ( R 
.\/  U )  ->  U  .<_  ( R  .\/  C ) ) )
3021, 29mpd 15 . 2  |-  ( ph  ->  U  .<_  ( R  .\/  C ) )
313, 9, 5dalempjqeb 33608 . . . . . 6  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
323, 5dalemreb 33604 . . . . . 6  |-  ( ph  ->  R  e.  ( Base `  K ) )
331, 2, 9latlej2 15345 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  R  e.  ( Base `  K )
)  ->  R  .<_  ( ( P  .\/  Q
)  .\/  R )
)
344, 31, 32, 33syl3anc 1219 . . . . 5  |-  ( ph  ->  R  .<_  ( ( P  .\/  Q )  .\/  R ) )
3534, 11syl6breqr 4435 . . . 4  |-  ( ph  ->  R  .<_  Y )
361, 2, 9latjlej1 15349 . . . . 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 1221 . . . 4  |-  ( ph  ->  ( R  .<_  Y  -> 
( R  .\/  C
)  .<_  ( Y  .\/  C ) ) )
3835, 37mpd 15 . . 3  |-  ( ph  ->  ( R  .\/  C
)  .<_  ( Y  .\/  C ) )
3938, 15syl6breqr 4435 . 2  |-  ( ph  ->  ( R  .\/  C
)  .<_  W )
401, 2, 4, 6, 14, 20, 30, 39lattrd 15342 1  |-  ( ph  ->  U  .<_  W )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2645   class class class wbr 4395   ` cfv 5521  (class class class)co 6195   Basecbs 14287   lecple 14359   joincjn 15228   Latclat 15329   Atomscatm 33227   HLchlt 33314   LPlanesclpl 33455
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 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
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 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-poset 15230  df-plt 15242  df-lub 15258  df-glb 15259  df-join 15260  df-meet 15261  df-p0 15323  df-lat 15330  df-clat 15392  df-oposet 33140  df-ol 33142  df-oml 33143  df-covers 33230  df-ats 33231  df-atl 33262  df-cvlat 33286  df-hlat 33315  df-llines 33461  df-lplanes 33462
This theorem is referenced by:  dalem6  33631  dalem8  33633
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