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Theorem dalem15 35142
Description: Lemma for dath 35200. 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 2443 . . . 4  |-  ( LVols `  K )  =  (
LVols `  K )
8 dalem15.y . . . 4  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
9 dalem15.z . . . 4  |-  Z  =  ( ( S  .\/  T )  .\/  U )
10 eqid 2443 . . . 4  |-  ( Y 
.\/  C )  =  ( Y  .\/  C
)
112, 3, 4, 5, 6, 7, 8, 9, 10dalem14 35141 . . 3  |-  ( (
ph  /\  Y  =/=  Z )  ->  ( Y  .\/  Z )  e.  (
LVols `  K ) )
122dalemkehl 35087 . . . . 5  |-  ( ph  ->  K  e.  HL )
132dalemyeo 35096 . . . . 5  |-  ( ph  ->  Y  e.  O )
142dalemzeo 35097 . . . . 5  |-  ( ph  ->  Z  e.  O )
15 dalem15.m . . . . . 6  |-  ./\  =  ( meet `  K )
16 dalem15.n . . . . . 6  |-  N  =  ( LLines `  K )
174, 15, 16, 6, 72lplnmj 35086 . . . . 5  |-  ( ( K  e.  HL  /\  Y  e.  O  /\  Z  e.  O )  ->  ( ( Y  ./\  Z )  e.  N  <->  ( Y  .\/  Z )  e.  (
LVols `  K ) ) )
1812, 13, 14, 17syl3anc 1229 . . . 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 2535 1  |-  ( (
ph  /\  Y  =/=  Z )  ->  X  e.  N )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804    =/= wne 2638   class class class wbr 4437   ` cfv 5578  (class class class)co 6281   Basecbs 14509   lecple 14581   joincjn 15447   meetcmee 15448   Atomscatm 34728   HLchlt 34815   LLinesclln 34955   LPlanesclpl 34956   LVolsclvol 34957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-preset 15431  df-poset 15449  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-lat 15550  df-clat 15612  df-oposet 34641  df-ol 34643  df-oml 34644  df-covers 34731  df-ats 34732  df-atl 34763  df-cvlat 34787  df-hlat 34816  df-llines 34962  df-lplanes 34963  df-lvols 34964
This theorem is referenced by:  dalem16  35143  dalem53  35189
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