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Theorem dalem14 33629
Description: Lemma for dath 33688. Planes  Y and 
Z form a 3-dimensional space (when they are different). (Contributed by NM, 22-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 )
dalem14.o  |-  O  =  ( LPlanes `  K )
dalem14.v  |-  V  =  ( LVols `  K )
dalem14.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem14.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
dalem14.w  |-  W  =  ( Y  .\/  C
)
Assertion
Ref Expression
dalem14  |-  ( (
ph  /\  Y  =/=  Z )  ->  ( Y  .\/  Z )  e.  V
)

Proof of Theorem dalem14
StepHypRef Expression
1 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 ) ) ) ) )
2 dalemc.l . . 3  |-  .<_  =  ( le `  K )
3 dalemc.j . . 3  |-  .\/  =  ( join `  K )
4 dalemc.a . . 3  |-  A  =  ( Atoms `  K )
5 dalem14.o . . 3  |-  O  =  ( LPlanes `  K )
6 dalem14.y . . 3  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
7 dalem14.z . . 3  |-  Z  =  ( ( S  .\/  T )  .\/  U )
8 dalem14.w . . 3  |-  W  =  ( Y  .\/  C
)
91, 2, 3, 4, 5, 6, 7, 8dalem13 33628 . 2  |-  ( (
ph  /\  Y  =/=  Z )  ->  ( Y  .\/  Z )  =  W )
10 dalem14.v . . 3  |-  V  =  ( LVols `  K )
111, 2, 3, 4, 5, 10, 6, 7, 8dalem9 33624 . 2  |-  ( (
ph  /\  Y  =/=  Z )  ->  W  e.  V )
129, 11eqeltrd 2539 1  |-  ( (
ph  /\  Y  =/=  Z )  ->  ( Y  .\/  Z )  e.  V
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   class class class wbr 4392   ` cfv 5518  (class class class)co 6192   Basecbs 14278   lecple 14349   joincjn 15218   Atomscatm 33216   HLchlt 33303   LPlanesclpl 33444   LVolsclvol 33445
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 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-poset 15220  df-plt 15232  df-lub 15248  df-glb 15249  df-join 15250  df-meet 15251  df-p0 15313  df-lat 15320  df-clat 15382  df-oposet 33129  df-ol 33131  df-oml 33132  df-covers 33219  df-ats 33220  df-atl 33251  df-cvlat 33275  df-hlat 33304  df-llines 33450  df-lplanes 33451  df-lvols 33452
This theorem is referenced by:  dalem15  33630
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