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Theorem dalem10 34870
Description: Lemma for dath 34933. Atom  D belongs to the axis of perspectivity  X. (Contributed by NM, 19-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 )
dalem10.m  |-  ./\  =  ( meet `  K )
dalem10.o  |-  O  =  ( LPlanes `  K )
dalem10.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem10.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
dalem10.x  |-  X  =  ( Y  ./\  Z
)
dalem10.d  |-  D  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )
Assertion
Ref Expression
dalem10  |-  ( ph  ->  D  .<_  X )

Proof of Theorem dalem10
StepHypRef Expression
1 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 ) ) ) ) )
21dalemkelat 34821 . . . 4  |-  ( ph  ->  K  e.  Lat )
3 dalemc.j . . . . 5  |-  .\/  =  ( join `  K )
4 dalemc.a . . . . 5  |-  A  =  ( Atoms `  K )
51, 3, 4dalempjqeb 34842 . . . 4  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
61, 4dalemreb 34838 . . . 4  |-  ( ph  ->  R  e.  ( Base `  K ) )
7 eqid 2467 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
8 dalemc.l . . . . 5  |-  .<_  =  ( le `  K )
97, 8, 3latlej1 15564 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  R  e.  ( Base `  K )
)  ->  ( P  .\/  Q )  .<_  ( ( P  .\/  Q ) 
.\/  R ) )
102, 5, 6, 9syl3anc 1228 . . 3  |-  ( ph  ->  ( P  .\/  Q
)  .<_  ( ( P 
.\/  Q )  .\/  R ) )
111, 3, 4dalemsjteb 34843 . . . 4  |-  ( ph  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
121, 4dalemueb 34841 . . . 4  |-  ( ph  ->  U  e.  ( Base `  K ) )
137, 8, 3latlej1 15564 . . . 4  |-  ( ( K  e.  Lat  /\  ( S  .\/  T )  e.  ( Base `  K
)  /\  U  e.  ( Base `  K )
)  ->  ( S  .\/  T )  .<_  ( ( S  .\/  T ) 
.\/  U ) )
142, 11, 12, 13syl3anc 1228 . . 3  |-  ( ph  ->  ( S  .\/  T
)  .<_  ( ( S 
.\/  T )  .\/  U ) )
15 dalem10.y . . . . 5  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
16 dalem10.o . . . . . 6  |-  O  =  ( LPlanes `  K )
171, 16dalemyeb 34846 . . . . 5  |-  ( ph  ->  Y  e.  ( Base `  K ) )
1815, 17syl5eqelr 2560 . . . 4  |-  ( ph  ->  ( ( P  .\/  Q )  .\/  R )  e.  ( Base `  K
) )
19 dalem10.z . . . . 5  |-  Z  =  ( ( S  .\/  T )  .\/  U )
201dalemzeo 34830 . . . . . 6  |-  ( ph  ->  Z  e.  O )
217, 16lplnbase 34731 . . . . . 6  |-  ( Z  e.  O  ->  Z  e.  ( Base `  K
) )
2220, 21syl 16 . . . . 5  |-  ( ph  ->  Z  e.  ( Base `  K ) )
2319, 22syl5eqelr 2560 . . . 4  |-  ( ph  ->  ( ( S  .\/  T )  .\/  U )  e.  ( Base `  K
) )
24 dalem10.m . . . . 5  |-  ./\  =  ( meet `  K )
257, 8, 24latmlem12 15587 . . . 4  |-  ( ( K  e.  Lat  /\  ( ( P  .\/  Q )  e.  ( Base `  K )  /\  (
( P  .\/  Q
)  .\/  R )  e.  ( Base `  K
) )  /\  (
( S  .\/  T
)  e.  ( Base `  K )  /\  (
( S  .\/  T
)  .\/  U )  e.  ( Base `  K
) ) )  -> 
( ( ( P 
.\/  Q )  .<_  ( ( P  .\/  Q )  .\/  R )  /\  ( S  .\/  T )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  (
( P  .\/  Q
)  ./\  ( S  .\/  T ) )  .<_  ( ( ( P 
.\/  Q )  .\/  R )  ./\  ( ( S  .\/  T )  .\/  U ) ) ) )
262, 5, 18, 11, 23, 25syl122anc 1237 . . 3  |-  ( ph  ->  ( ( ( P 
.\/  Q )  .<_  ( ( P  .\/  Q )  .\/  R )  /\  ( S  .\/  T )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  (
( P  .\/  Q
)  ./\  ( S  .\/  T ) )  .<_  ( ( ( P 
.\/  Q )  .\/  R )  ./\  ( ( S  .\/  T )  .\/  U ) ) ) )
2710, 14, 26mp2and 679 . 2  |-  ( ph  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( ( ( P 
.\/  Q )  .\/  R )  ./\  ( ( S  .\/  T )  .\/  U ) ) )
28 dalem10.d . 2  |-  D  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )
29 dalem10.x . . 3  |-  X  =  ( Y  ./\  Z
)
3015, 19oveq12i 6307 . . 3  |-  ( Y 
./\  Z )  =  ( ( ( P 
.\/  Q )  .\/  R )  ./\  ( ( S  .\/  T )  .\/  U ) )
3129, 30eqtri 2496 . 2  |-  X  =  ( ( ( P 
.\/  Q )  .\/  R )  ./\  ( ( S  .\/  T )  .\/  U ) )
3227, 28, 313brtr4g 4485 1  |-  ( ph  ->  D  .<_  X )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   Basecbs 14507   lecple 14579   joincjn 15448   meetcmee 15449   Latclat 15549   Atomscatm 34461   HLchlt 34548   LPlanesclpl 34689
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-poset 15450  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-lat 15550  df-ats 34465  df-atl 34496  df-cvlat 34520  df-hlat 34549  df-lplanes 34696
This theorem is referenced by:  dalem11  34871  dalem16  34876  dalem54  34923
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