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Theorem dalem10 30471
Description: Lemma for dath 30534. 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 30422 . . . 4  |-  ( ph  ->  K  e.  Lat )
3 dalemc.j . . . . 5  |-  .\/  =  ( join `  K )
4 dalemc.a . . . . 5  |-  A  =  ( Atoms `  K )
51, 3, 4dalempjqeb 30443 . . . 4  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
61, 4dalemreb 30439 . . . 4  |-  ( ph  ->  R  e.  ( Base `  K ) )
7 eqid 2437 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
8 dalemc.l . . . . 5  |-  .<_  =  ( le `  K )
97, 8, 3latlej1 14490 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  R  e.  ( Base `  K )
)  ->  ( P  .\/  Q )  .<_  ( ( P  .\/  Q ) 
.\/  R ) )
102, 5, 6, 9syl3anc 1185 . . 3  |-  ( ph  ->  ( P  .\/  Q
)  .<_  ( ( P 
.\/  Q )  .\/  R ) )
111, 3, 4dalemsjteb 30444 . . . 4  |-  ( ph  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
121, 4dalemueb 30442 . . . 4  |-  ( ph  ->  U  e.  ( Base `  K ) )
137, 8, 3latlej1 14490 . . . 4  |-  ( ( K  e.  Lat  /\  ( S  .\/  T )  e.  ( Base `  K
)  /\  U  e.  ( Base `  K )
)  ->  ( S  .\/  T )  .<_  ( ( S  .\/  T ) 
.\/  U ) )
142, 11, 12, 13syl3anc 1185 . . 3  |-  ( ph  ->  ( S  .\/  T
)  .<_  ( ( S 
.\/  T )  .\/  U ) )
15 dalem10.y . . . . 5  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
16 dalem10.o . . . . . 6  |-  O  =  ( LPlanes `  K )
171, 16dalemyeb 30447 . . . . 5  |-  ( ph  ->  Y  e.  ( Base `  K ) )
1815, 17syl5eqelr 2522 . . . 4  |-  ( ph  ->  ( ( P  .\/  Q )  .\/  R )  e.  ( Base `  K
) )
19 dalem10.z . . . . 5  |-  Z  =  ( ( S  .\/  T )  .\/  U )
201dalemzeo 30431 . . . . . 6  |-  ( ph  ->  Z  e.  O )
217, 16lplnbase 30332 . . . . . 6  |-  ( Z  e.  O  ->  Z  e.  ( Base `  K
) )
2220, 21syl 16 . . . . 5  |-  ( ph  ->  Z  e.  ( Base `  K ) )
2319, 22syl5eqelr 2522 . . . 4  |-  ( ph  ->  ( ( S  .\/  T )  .\/  U )  e.  ( Base `  K
) )
24 dalem10.m . . . . 5  |-  ./\  =  ( meet `  K )
257, 8, 24latmlem12 14513 . . . 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 1194 . . 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 662 . 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 6094 . . 3  |-  ( Y 
./\  Z )  =  ( ( ( P 
.\/  Q )  .\/  R )  ./\  ( ( S  .\/  T )  .\/  U ) )
3129, 30eqtri 2457 . 2  |-  X  =  ( ( ( P 
.\/  Q )  .\/  R )  ./\  ( ( S  .\/  T )  .\/  U ) )
3227, 28, 313brtr4g 4245 1  |-  ( ph  ->  D  .<_  X )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4213   ` cfv 5455  (class class class)co 6082   Basecbs 13470   lecple 13537   joincjn 14402   meetcmee 14403   Latclat 14475   Atomscatm 30062   HLchlt 30149   LPlanesclpl 30290
This theorem is referenced by:  dalem11  30472  dalem16  30477  dalem54  30524
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-undef 6544  df-riota 6550  df-poset 14404  df-lub 14432  df-glb 14433  df-join 14434  df-meet 14435  df-lat 14476  df-ats 30066  df-atl 30097  df-cvlat 30121  df-hlat 30150  df-lplanes 30297
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