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Theorem dalemdea 29777
Description: Lemma for dath 29851. 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 29776 . . 3  |-  ( ph  ->  ( ( P  .\/  Q )  .\/  ( S 
.\/  T ) )  e.  O )
92dalemkehl 29738 . . . 4  |-  ( ph  ->  K  e.  HL )
102dalempea 29741 . . . . 5  |-  ( ph  ->  P  e.  A )
112dalemqea 29742 . . . . 5  |-  ( ph  ->  Q  e.  A )
122dalemrea 29743 . . . . . 6  |-  ( ph  ->  R  e.  A )
132dalemyeo 29747 . . . . . 6  |-  ( ph  ->  Y  e.  O )
144, 5, 6, 7lplnri1 29668 . . . . . 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 2388 . . . . . 6  |-  ( LLines `  K )  =  (
LLines `  K )
174, 5, 16llni2 29627 . . . . 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 29744 . . . . 5  |-  ( ph  ->  S  e.  A )
202dalemtea 29745 . . . . 5  |-  ( ph  ->  T  e.  A )
212dalemuea 29746 . . . . . 6  |-  ( ph  ->  U  e.  A )
222dalemzeo 29748 . . . . . 6  |-  ( ph  ->  Z  e.  O )
23 dalemdea.z . . . . . . 7  |-  Z  =  ( ( S  .\/  T )  .\/  U )
244, 5, 6, 23lplnri1 29668 . . . . . 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 29627 . . . . 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 29675 . . . 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 2472 1  |-  ( ph  ->  D  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2551   class class class wbr 4154   ` cfv 5395  (class class class)co 6021   Basecbs 13397   lecple 13464   joincjn 14329   meetcmee 14330   Atomscatm 29379   HLchlt 29466   LLinesclln 29606   LPlanesclpl 29607
This theorem is referenced by:  dalemeea  29778  dalem3  29779  dalem16  29794  dalem52  29839  dalem57  29844  dalem60  29847
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-undef 6480  df-riota 6486  df-poset 14331  df-plt 14343  df-lub 14359  df-glb 14360  df-join 14361  df-meet 14362  df-p0 14396  df-lat 14403  df-clat 14465  df-oposet 29292  df-ol 29294  df-oml 29295  df-covers 29382  df-ats 29383  df-atl 29414  df-cvlat 29438  df-hlat 29467  df-llines 29613  df-lplanes 29614
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