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Theorem dalemdea 30396
Description: Lemma for dath 30470. Frequently-used utility lemma. (Contributed by NM, 11-Aug-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 )
dalemdea.m  |-  ./\  =  ( meet `  K )
dalemdea.o  |-  O  =  ( LPlanes `  K )
dalemdea.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalemdea.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
dalemdea.d  |-  D  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )
Assertion
Ref Expression
dalemdea  |-  ( ph  ->  D  e.  A )

Proof of Theorem dalemdea
StepHypRef Expression
1 dalemdea.d . 2  |-  D  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )
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 dalemdea.o . . . 4  |-  O  =  ( LPlanes `  K )
7 dalemdea.y . . . 4  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
82, 3, 4, 5, 6, 7dalem2 30395 . . 3  |-  ( ph  ->  ( ( P  .\/  Q )  .\/  ( S 
.\/  T ) )  e.  O )
92dalemkehl 30357 . . . 4  |-  ( ph  ->  K  e.  HL )
102dalempea 30360 . . . . 5  |-  ( ph  ->  P  e.  A )
112dalemqea 30361 . . . . 5  |-  ( ph  ->  Q  e.  A )
122dalemrea 30362 . . . . . 6  |-  ( ph  ->  R  e.  A )
132dalemyeo 30366 . . . . . 6  |-  ( ph  ->  Y  e.  O )
144, 5, 6, 7lplnri1 30287 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  Y  e.  O )  ->  P  =/=  Q )
159, 10, 11, 12, 13, 14syl131anc 1197 . . . . 5  |-  ( ph  ->  P  =/=  Q )
16 eqid 2435 . . . . . 6  |-  ( LLines `  K )  =  (
LLines `  K )
174, 5, 16llni2 30246 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  Q )  e.  (
LLines `  K ) )
189, 10, 11, 15, 17syl31anc 1187 . . . 4  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( LLines `  K ) )
192dalemsea 30363 . . . . 5  |-  ( ph  ->  S  e.  A )
202dalemtea 30364 . . . . 5  |-  ( ph  ->  T  e.  A )
212dalemuea 30365 . . . . . 6  |-  ( ph  ->  U  e.  A )
222dalemzeo 30367 . . . . . 6  |-  ( ph  ->  Z  e.  O )
23 dalemdea.z . . . . . . 7  |-  Z  =  ( ( S  .\/  T )  .\/  U )
244, 5, 6, 23lplnri1 30287 . . . . . 6  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
)  /\  Z  e.  O )  ->  S  =/=  T )
259, 19, 20, 21, 22, 24syl131anc 1197 . . . . 5  |-  ( ph  ->  S  =/=  T )
264, 5, 16llni2 30246 . . . . 5  |-  ( ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  /\  S  =/=  T
)  ->  ( S  .\/  T )  e.  (
LLines `  K ) )
279, 19, 20, 25, 26syl31anc 1187 . . . 4  |-  ( ph  ->  ( S  .\/  T
)  e.  ( LLines `  K ) )
28 dalemdea.m . . . . 5  |-  ./\  =  ( meet `  K )
294, 28, 5, 16, 62llnmj 30294 . . . 4  |-  ( ( K  e.  HL  /\  ( P  .\/  Q )  e.  ( LLines `  K
)  /\  ( S  .\/  T )  e.  (
LLines `  K ) )  ->  ( ( ( P  .\/  Q ) 
./\  ( S  .\/  T ) )  e.  A  <->  ( ( P  .\/  Q
)  .\/  ( S  .\/  T ) )  e.  O ) )
309, 18, 27, 29syl3anc 1184 . . 3  |-  ( ph  ->  ( ( ( P 
.\/  Q )  ./\  ( S  .\/  T ) )  e.  A  <->  ( ( P  .\/  Q )  .\/  ( S  .\/  T ) )  e.  O ) )
318, 30mpbird 224 . 2  |-  ( ph  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  e.  A )
321, 31syl5eqel 2519 1  |-  ( ph  ->  D  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   lecple 13528   joincjn 14393   meetcmee 14394   Atomscatm 29998   HLchlt 30085   LLinesclln 30225   LPlanesclpl 30226
This theorem is referenced by:  dalemeea  30397  dalem3  30398  dalem16  30413  dalem52  30458  dalem57  30463  dalem60  30466
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-lat 14467  df-clat 14529  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-llines 30232  df-lplanes 30233
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