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Theorem dalemdea 34458
Description: Lemma for dath 34532. 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 34457 . . 3  |-  ( ph  ->  ( ( P  .\/  Q )  .\/  ( S 
.\/  T ) )  e.  O )
92dalemkehl 34419 . . . 4  |-  ( ph  ->  K  e.  HL )
102dalempea 34422 . . . . 5  |-  ( ph  ->  P  e.  A )
112dalemqea 34423 . . . . 5  |-  ( ph  ->  Q  e.  A )
122dalemrea 34424 . . . . . 6  |-  ( ph  ->  R  e.  A )
132dalemyeo 34428 . . . . . 6  |-  ( ph  ->  Y  e.  O )
144, 5, 6, 7lplnri1 34349 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  Y  e.  O )  ->  P  =/=  Q )
159, 10, 11, 12, 13, 14syl131anc 1241 . . . . 5  |-  ( ph  ->  P  =/=  Q )
16 eqid 2467 . . . . . 6  |-  ( LLines `  K )  =  (
LLines `  K )
174, 5, 16llni2 34308 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  Q )  e.  (
LLines `  K ) )
189, 10, 11, 15, 17syl31anc 1231 . . . 4  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( LLines `  K ) )
192dalemsea 34425 . . . . 5  |-  ( ph  ->  S  e.  A )
202dalemtea 34426 . . . . 5  |-  ( ph  ->  T  e.  A )
212dalemuea 34427 . . . . . 6  |-  ( ph  ->  U  e.  A )
222dalemzeo 34429 . . . . . 6  |-  ( ph  ->  Z  e.  O )
23 dalemdea.z . . . . . . 7  |-  Z  =  ( ( S  .\/  T )  .\/  U )
244, 5, 6, 23lplnri1 34349 . . . . . 6  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
)  /\  Z  e.  O )  ->  S  =/=  T )
259, 19, 20, 21, 22, 24syl131anc 1241 . . . . 5  |-  ( ph  ->  S  =/=  T )
264, 5, 16llni2 34308 . . . . 5  |-  ( ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  /\  S  =/=  T
)  ->  ( S  .\/  T )  e.  (
LLines `  K ) )
279, 19, 20, 25, 26syl31anc 1231 . . . 4  |-  ( ph  ->  ( S  .\/  T
)  e.  ( LLines `  K ) )
28 dalemdea.m . . . . 5  |-  ./\  =  ( meet `  K )
294, 28, 5, 16, 62llnmj 34356 . . . 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 1228 . . 3  |-  ( ph  ->  ( ( ( P 
.\/  Q )  ./\  ( S  .\/  T ) )  e.  A  <->  ( ( P  .\/  Q )  .\/  ( S  .\/  T ) )  e.  O ) )
318, 30mpbird 232 . 2  |-  ( ph  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  e.  A )
321, 31syl5eqel 2559 1  |-  ( ph  ->  D  e.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   Basecbs 14483   lecple 14555   joincjn 15424   meetcmee 15425   Atomscatm 34060   HLchlt 34147   LLinesclln 34287   LPlanesclpl 34288
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
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 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-poset 15426  df-plt 15438  df-lub 15454  df-glb 15455  df-join 15456  df-meet 15457  df-p0 15519  df-lat 15526  df-clat 15588  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-llines 34294  df-lplanes 34295
This theorem is referenced by:  dalemeea  34459  dalem3  34460  dalem16  34475  dalem52  34520  dalem57  34525  dalem60  34528
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