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Theorem dalem5 34463
Description: Lemma for dath 34532. 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 2467 . 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 34420 . 2  |-  ( ph  ->  K  e.  Lat )
5 dalemc.a . . 3  |-  A  =  ( Atoms `  K )
63, 5dalemueb 34440 . 2  |-  ( ph  ->  U  e.  ( Base `  K ) )
73dalemkehl 34419 . . 3  |-  ( ph  ->  K  e.  HL )
83dalemrea 34424 . . 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 34456 . . 3  |-  ( ph  ->  C  e.  A )
131, 9, 5hlatjcl 34163 . . 3  |-  ( ( K  e.  HL  /\  R  e.  A  /\  C  e.  A )  ->  ( R  .\/  C
)  e.  ( Base `  K ) )
147, 8, 12, 13syl3anc 1228 . 2  |-  ( ph  ->  ( R  .\/  C
)  e.  ( Base `  K ) )
15 dalem5.w . . 3  |-  W  =  ( Y  .\/  C
)
163, 10dalemyeb 34445 . . . 4  |-  ( ph  ->  Y  e.  ( Base `  K ) )
173, 5dalemceb 34434 . . . 4  |-  ( ph  ->  C  e.  ( Base `  K ) )
181, 9latjcl 15534 . . . 4  |-  ( ( K  e.  Lat  /\  Y  e.  ( Base `  K )  /\  C  e.  ( Base `  K
) )  ->  ( Y  .\/  C )  e.  ( Base `  K
) )
194, 16, 17, 18syl3anc 1228 . . 3  |-  ( ph  ->  ( Y  .\/  C
)  e.  ( Base `  K ) )
2015, 19syl5eqel 2559 . 2  |-  ( ph  ->  W  e.  ( Base `  K ) )
213dalemclrju 34432 . . 3  |-  ( ph  ->  C  .<_  ( R  .\/  U ) )
223dalemuea 34427 . . . 4  |-  ( ph  ->  U  e.  A )
233dalempea 34422 . . . . 5  |-  ( ph  ->  P  e.  A )
24 simp313 1145 . . . . . 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 34175 . . . . 5  |-  ( ( K  e.  HL  /\  ( C  e.  A  /\  R  e.  A  /\  P  e.  A
)  /\  -.  C  .<_  ( R  .\/  P
) )  ->  C  =/=  R )
277, 12, 8, 23, 25, 26syl131anc 1241 . . . 4  |-  ( ph  ->  C  =/=  R )
282, 9, 5hlatexch1 34191 . . . 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 1241 . . 3  |-  ( ph  ->  ( C  .<_  ( R 
.\/  U )  ->  U  .<_  ( R  .\/  C ) ) )
3021, 29mpd 15 . 2  |-  ( ph  ->  U  .<_  ( R  .\/  C ) )
313, 9, 5dalempjqeb 34441 . . . . . 6  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
323, 5dalemreb 34437 . . . . . 6  |-  ( ph  ->  R  e.  ( Base `  K ) )
331, 2, 9latlej2 15544 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  R  e.  ( Base `  K )
)  ->  R  .<_  ( ( P  .\/  Q
)  .\/  R )
)
344, 31, 32, 33syl3anc 1228 . . . . 5  |-  ( ph  ->  R  .<_  ( ( P  .\/  Q )  .\/  R ) )
3534, 11syl6breqr 4487 . . . 4  |-  ( ph  ->  R  .<_  Y )
361, 2, 9latjlej1 15548 . . . . 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 1230 . . . 4  |-  ( ph  ->  ( R  .<_  Y  -> 
( R  .\/  C
)  .<_  ( Y  .\/  C ) ) )
3835, 37mpd 15 . . 3  |-  ( ph  ->  ( R  .\/  C
)  .<_  ( Y  .\/  C ) )
3938, 15syl6breqr 4487 . 2  |-  ( ph  ->  ( R  .\/  C
)  .<_  W )
401, 2, 4, 6, 14, 20, 30, 39lattrd 15541 1  |-  ( ph  ->  U  .<_  W )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   Basecbs 14486   lecple 14558   joincjn 15427   Latclat 15528   Atomscatm 34060   HLchlt 34147   LPlanesclpl 34288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-poset 15429  df-plt 15441  df-lub 15457  df-glb 15458  df-join 15459  df-meet 15460  df-p0 15522  df-lat 15529  df-clat 15591  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-llines 34294  df-lplanes 34295
This theorem is referenced by:  dalem6  34464  dalem8  34466
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