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Theorem dalem5 35807
Description: Lemma for dath 35876. 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 2454 . 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 35764 . 2  |-  ( ph  ->  K  e.  Lat )
5 dalemc.a . . 3  |-  A  =  ( Atoms `  K )
63, 5dalemueb 35784 . 2  |-  ( ph  ->  U  e.  ( Base `  K ) )
73dalemkehl 35763 . . 3  |-  ( ph  ->  K  e.  HL )
83dalemrea 35768 . . 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 35800 . . 3  |-  ( ph  ->  C  e.  A )
131, 9, 5hlatjcl 35507 . . 3  |-  ( ( K  e.  HL  /\  R  e.  A  /\  C  e.  A )  ->  ( R  .\/  C
)  e.  ( Base `  K ) )
147, 8, 12, 13syl3anc 1226 . 2  |-  ( ph  ->  ( R  .\/  C
)  e.  ( Base `  K ) )
15 dalem5.w . . 3  |-  W  =  ( Y  .\/  C
)
163, 10dalemyeb 35789 . . . 4  |-  ( ph  ->  Y  e.  ( Base `  K ) )
173, 5dalemceb 35778 . . . 4  |-  ( ph  ->  C  e.  ( Base `  K ) )
181, 9latjcl 15883 . . . 4  |-  ( ( K  e.  Lat  /\  Y  e.  ( Base `  K )  /\  C  e.  ( Base `  K
) )  ->  ( Y  .\/  C )  e.  ( Base `  K
) )
194, 16, 17, 18syl3anc 1226 . . 3  |-  ( ph  ->  ( Y  .\/  C
)  e.  ( Base `  K ) )
2015, 19syl5eqel 2546 . 2  |-  ( ph  ->  W  e.  ( Base `  K ) )
213dalemclrju 35776 . . 3  |-  ( ph  ->  C  .<_  ( R  .\/  U ) )
223dalemuea 35771 . . . 4  |-  ( ph  ->  U  e.  A )
233dalempea 35766 . . . . 5  |-  ( ph  ->  P  e.  A )
24 simp313 1143 . . . . . 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 35519 . . . . 5  |-  ( ( K  e.  HL  /\  ( C  e.  A  /\  R  e.  A  /\  P  e.  A
)  /\  -.  C  .<_  ( R  .\/  P
) )  ->  C  =/=  R )
277, 12, 8, 23, 25, 26syl131anc 1239 . . . 4  |-  ( ph  ->  C  =/=  R )
282, 9, 5hlatexch1 35535 . . . 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 1239 . . 3  |-  ( ph  ->  ( C  .<_  ( R 
.\/  U )  ->  U  .<_  ( R  .\/  C ) ) )
3021, 29mpd 15 . 2  |-  ( ph  ->  U  .<_  ( R  .\/  C ) )
313, 9, 5dalempjqeb 35785 . . . . . 6  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
323, 5dalemreb 35781 . . . . . 6  |-  ( ph  ->  R  e.  ( Base `  K ) )
331, 2, 9latlej2 15893 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  R  e.  ( Base `  K )
)  ->  R  .<_  ( ( P  .\/  Q
)  .\/  R )
)
344, 31, 32, 33syl3anc 1226 . . . . 5  |-  ( ph  ->  R  .<_  ( ( P  .\/  Q )  .\/  R ) )
3534, 11syl6breqr 4479 . . . 4  |-  ( ph  ->  R  .<_  Y )
361, 2, 9latjlej1 15897 . . . . 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 1228 . . . 4  |-  ( ph  ->  ( R  .<_  Y  -> 
( R  .\/  C
)  .<_  ( Y  .\/  C ) ) )
3835, 37mpd 15 . . 3  |-  ( ph  ->  ( R  .\/  C
)  .<_  ( Y  .\/  C ) )
3938, 15syl6breqr 4479 . 2  |-  ( ph  ->  ( R  .\/  C
)  .<_  W )
401, 2, 4, 6, 14, 20, 30, 39lattrd 15890 1  |-  ( ph  ->  U  .<_  W )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14719   lecple 14794   joincjn 15775   Latclat 15877   Atomscatm 35404   HLchlt 35491   LPlanesclpl 35632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-preset 15759  df-poset 15777  df-plt 15790  df-lub 15806  df-glb 15807  df-join 15808  df-meet 15809  df-p0 15871  df-lat 15878  df-clat 15940  df-oposet 35317  df-ol 35319  df-oml 35320  df-covers 35407  df-ats 35408  df-atl 35439  df-cvlat 35463  df-hlat 35492  df-llines 35638  df-lplanes 35639
This theorem is referenced by:  dalem6  35808  dalem8  35810
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