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Theorem dalem15 34349
Description: Lemma for dath 34407. The axis of perspectivity  X is a line. (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 )
dalem15.m  |-  ./\  =  ( meet `  K )
dalem15.n  |-  N  =  ( LLines `  K )
dalem15.o  |-  O  =  ( LPlanes `  K )
dalem15.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem15.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
dalem15.x  |-  X  =  ( Y  ./\  Z
)
Assertion
Ref Expression
dalem15  |-  ( (
ph  /\  Y  =/=  Z )  ->  X  e.  N )

Proof of Theorem dalem15
StepHypRef Expression
1 dalem15.x . 2  |-  X  =  ( Y  ./\  Z
)
2 dalema.ph . . . 4  |-  ( 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 ) ) ) ) )
3 dalemc.l . . . 4  |-  .<_  =  ( le `  K )
4 dalemc.j . . . 4  |-  .\/  =  ( join `  K )
5 dalemc.a . . . 4  |-  A  =  ( Atoms `  K )
6 dalem15.o . . . 4  |-  O  =  ( LPlanes `  K )
7 eqid 2460 . . . 4  |-  ( LVols `  K )  =  (
LVols `  K )
8 dalem15.y . . . 4  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
9 dalem15.z . . . 4  |-  Z  =  ( ( S  .\/  T )  .\/  U )
10 eqid 2460 . . . 4  |-  ( Y 
.\/  C )  =  ( Y  .\/  C
)
112, 3, 4, 5, 6, 7, 8, 9, 10dalem14 34348 . . 3  |-  ( (
ph  /\  Y  =/=  Z )  ->  ( Y  .\/  Z )  e.  (
LVols `  K ) )
122dalemkehl 34294 . . . . 5  |-  ( ph  ->  K  e.  HL )
132dalemyeo 34303 . . . . 5  |-  ( ph  ->  Y  e.  O )
142dalemzeo 34304 . . . . 5  |-  ( ph  ->  Z  e.  O )
15 dalem15.m . . . . . 6  |-  ./\  =  ( meet `  K )
16 dalem15.n . . . . . 6  |-  N  =  ( LLines `  K )
174, 15, 16, 6, 72lplnmj 34293 . . . . 5  |-  ( ( K  e.  HL  /\  Y  e.  O  /\  Z  e.  O )  ->  ( ( Y  ./\  Z )  e.  N  <->  ( Y  .\/  Z )  e.  (
LVols `  K ) ) )
1812, 13, 14, 17syl3anc 1223 . . . 4  |-  ( ph  ->  ( ( Y  ./\  Z )  e.  N  <->  ( Y  .\/  Z )  e.  (
LVols `  K ) ) )
1918adantr 465 . . 3  |-  ( (
ph  /\  Y  =/=  Z )  ->  ( ( Y  ./\  Z )  e.  N  <->  ( Y  .\/  Z )  e.  ( LVols `  K ) ) )
2011, 19mpbird 232 . 2  |-  ( (
ph  /\  Y  =/=  Z )  ->  ( Y  ./\ 
Z )  e.  N
)
211, 20syl5eqel 2552 1  |-  ( (
ph  /\  Y  =/=  Z )  ->  X  e.  N )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   Basecbs 14479   lecple 14551   joincjn 15420   meetcmee 15421   Atomscatm 33935   HLchlt 34022   LLinesclln 34162   LPlanesclpl 34163   LVolsclvol 34164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-poset 15422  df-plt 15434  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-p0 15515  df-lat 15522  df-clat 15584  df-oposet 33848  df-ol 33850  df-oml 33851  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023  df-llines 34169  df-lplanes 34170  df-lvols 34171
This theorem is referenced by:  dalem16  34350  dalem53  34396
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