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Theorem dalem10 33150
Description: Lemma for dath 33213. 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 33101 . . . 4  |-  ( ph  ->  K  e.  Lat )
3 dalemc.j . . . . 5  |-  .\/  =  ( join `  K )
4 dalemc.a . . . . 5  |-  A  =  ( Atoms `  K )
51, 3, 4dalempjqeb 33122 . . . 4  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
61, 4dalemreb 33118 . . . 4  |-  ( ph  ->  R  e.  ( Base `  K ) )
7 eqid 2428 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
8 dalemc.l . . . . 5  |-  .<_  =  ( le `  K )
97, 8, 3latlej1 16249 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  R  e.  ( Base `  K )
)  ->  ( P  .\/  Q )  .<_  ( ( P  .\/  Q ) 
.\/  R ) )
102, 5, 6, 9syl3anc 1264 . . 3  |-  ( ph  ->  ( P  .\/  Q
)  .<_  ( ( P 
.\/  Q )  .\/  R ) )
111, 3, 4dalemsjteb 33123 . . . 4  |-  ( ph  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
121, 4dalemueb 33121 . . . 4  |-  ( ph  ->  U  e.  ( Base `  K ) )
137, 8, 3latlej1 16249 . . . 4  |-  ( ( K  e.  Lat  /\  ( S  .\/  T )  e.  ( Base `  K
)  /\  U  e.  ( Base `  K )
)  ->  ( S  .\/  T )  .<_  ( ( S  .\/  T ) 
.\/  U ) )
142, 11, 12, 13syl3anc 1264 . . 3  |-  ( ph  ->  ( S  .\/  T
)  .<_  ( ( S 
.\/  T )  .\/  U ) )
15 dalem10.y . . . . 5  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
16 dalem10.o . . . . . 6  |-  O  =  ( LPlanes `  K )
171, 16dalemyeb 33126 . . . . 5  |-  ( ph  ->  Y  e.  ( Base `  K ) )
1815, 17syl5eqelr 2511 . . . 4  |-  ( ph  ->  ( ( P  .\/  Q )  .\/  R )  e.  ( Base `  K
) )
19 dalem10.z . . . . 5  |-  Z  =  ( ( S  .\/  T )  .\/  U )
201dalemzeo 33110 . . . . . 6  |-  ( ph  ->  Z  e.  O )
217, 16lplnbase 33011 . . . . . 6  |-  ( Z  e.  O  ->  Z  e.  ( Base `  K
) )
2220, 21syl 17 . . . . 5  |-  ( ph  ->  Z  e.  ( Base `  K ) )
2319, 22syl5eqelr 2511 . . . 4  |-  ( ph  ->  ( ( S  .\/  T )  .\/  U )  e.  ( Base `  K
) )
24 dalem10.m . . . . 5  |-  ./\  =  ( meet `  K )
257, 8, 24latmlem12 16272 . . . 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 1273 . . 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 683 . 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 6261 . . 3  |-  ( Y 
./\  Z )  =  ( ( ( P 
.\/  Q )  .\/  R )  ./\  ( ( S  .\/  T )  .\/  U ) )
3129, 30eqtri 2450 . 2  |-  X  =  ( ( ( P 
.\/  Q )  .\/  R )  ./\  ( ( S  .\/  T )  .\/  U ) )
3227, 28, 313brtr4g 4399 1  |-  ( ph  ->  D  .<_  X )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   class class class wbr 4366   ` cfv 5544  (class class class)co 6249   Basecbs 15064   lecple 15140   joincjn 16132   meetcmee 16133   Latclat 16234   Atomscatm 32741   HLchlt 32828   LPlanesclpl 32969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-poset 16134  df-lub 16163  df-glb 16164  df-join 16165  df-meet 16166  df-lat 16235  df-ats 32745  df-atl 32776  df-cvlat 32800  df-hlat 32829  df-lplanes 32976
This theorem is referenced by:  dalem11  33151  dalem16  33156  dalem54  33203
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