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Theorem dalem9 32669
Description: Lemma for dath 32733. Since  -.  C  .<_  Y, the join  Y  .\/  C forms a 3-dimensional space. (Contributed by NM, 20-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 )
dalem9.o  |-  O  =  ( LPlanes `  K )
dalem9.v  |-  V  =  ( LVols `  K )
dalem9.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem9.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
dalem9.w  |-  W  =  ( Y  .\/  C
)
Assertion
Ref Expression
dalem9  |-  ( (
ph  /\  Y  =/=  Z )  ->  W  e.  V )

Proof of Theorem dalem9
StepHypRef Expression
1 dalem9.w . 2  |-  W  =  ( Y  .\/  C
)
2 dalema.ph . . . . 5  |-  ( 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 ) ) ) ) )
32dalemkehl 32620 . . . 4  |-  ( ph  ->  K  e.  HL )
43adantr 463 . . 3  |-  ( (
ph  /\  Y  =/=  Z )  ->  K  e.  HL )
52dalemyeo 32629 . . . 4  |-  ( ph  ->  Y  e.  O )
65adantr 463 . . 3  |-  ( (
ph  /\  Y  =/=  Z )  ->  Y  e.  O )
7 dalemc.l . . . . 5  |-  .<_  =  ( le `  K )
8 dalemc.j . . . . 5  |-  .\/  =  ( join `  K )
9 dalemc.a . . . . 5  |-  A  =  ( Atoms `  K )
10 dalem9.o . . . . 5  |-  O  =  ( LPlanes `  K )
11 dalem9.y . . . . 5  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
122, 7, 8, 9, 10, 11dalemcea 32657 . . . 4  |-  ( ph  ->  C  e.  A )
1312adantr 463 . . 3  |-  ( (
ph  /\  Y  =/=  Z )  ->  C  e.  A )
14 dalem9.z . . . 4  |-  Z  =  ( ( S  .\/  T )  .\/  U )
152, 7, 8, 9, 10, 11, 14dalem-cly 32668 . . 3  |-  ( (
ph  /\  Y  =/=  Z )  ->  -.  C  .<_  Y )
16 dalem9.v . . . 4  |-  V  =  ( LVols `  K )
177, 8, 9, 10, 16lvoli3 32574 . . 3  |-  ( ( ( K  e.  HL  /\  Y  e.  O  /\  C  e.  A )  /\  -.  C  .<_  Y )  ->  ( Y  .\/  C )  e.  V )
184, 6, 13, 15, 17syl31anc 1233 . 2  |-  ( (
ph  /\  Y  =/=  Z )  ->  ( Y  .\/  C )  e.  V
)
191, 18syl5eqel 2494 1  |-  ( (
ph  /\  Y  =/=  Z )  ->  W  e.  V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   class class class wbr 4394   ` cfv 5568  (class class class)co 6277   Basecbs 14839   lecple 14914   joincjn 15895   Atomscatm 32261   HLchlt 32348   LPlanesclpl 32489   LVolsclvol 32490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-preset 15879  df-poset 15897  df-plt 15910  df-lub 15926  df-glb 15927  df-join 15928  df-meet 15929  df-p0 15991  df-lat 15998  df-clat 16060  df-oposet 32174  df-ol 32176  df-oml 32177  df-covers 32264  df-ats 32265  df-atl 32296  df-cvlat 32320  df-hlat 32349  df-llines 32495  df-lplanes 32496  df-lvols 32497
This theorem is referenced by:  dalem13  32673  dalem14  32674
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