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Theorem dalemdea 32679
Description: Lemma for dath 32753. 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 32678 . . 3  |-  ( ph  ->  ( ( P  .\/  Q )  .\/  ( S 
.\/  T ) )  e.  O )
92dalemkehl 32640 . . . 4  |-  ( ph  ->  K  e.  HL )
102dalempea 32643 . . . . 5  |-  ( ph  ->  P  e.  A )
112dalemqea 32644 . . . . 5  |-  ( ph  ->  Q  e.  A )
122dalemrea 32645 . . . . . 6  |-  ( ph  ->  R  e.  A )
132dalemyeo 32649 . . . . . 6  |-  ( ph  ->  Y  e.  O )
144, 5, 6, 7lplnri1 32570 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  Y  e.  O )  ->  P  =/=  Q )
159, 10, 11, 12, 13, 14syl131anc 1243 . . . . 5  |-  ( ph  ->  P  =/=  Q )
16 eqid 2402 . . . . . 6  |-  ( LLines `  K )  =  (
LLines `  K )
174, 5, 16llni2 32529 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  Q )  e.  (
LLines `  K ) )
189, 10, 11, 15, 17syl31anc 1233 . . . 4  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( LLines `  K ) )
192dalemsea 32646 . . . . 5  |-  ( ph  ->  S  e.  A )
202dalemtea 32647 . . . . 5  |-  ( ph  ->  T  e.  A )
212dalemuea 32648 . . . . . 6  |-  ( ph  ->  U  e.  A )
222dalemzeo 32650 . . . . . 6  |-  ( ph  ->  Z  e.  O )
23 dalemdea.z . . . . . . 7  |-  Z  =  ( ( S  .\/  T )  .\/  U )
244, 5, 6, 23lplnri1 32570 . . . . . 6  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
)  /\  Z  e.  O )  ->  S  =/=  T )
259, 19, 20, 21, 22, 24syl131anc 1243 . . . . 5  |-  ( ph  ->  S  =/=  T )
264, 5, 16llni2 32529 . . . . 5  |-  ( ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  /\  S  =/=  T
)  ->  ( S  .\/  T )  e.  (
LLines `  K ) )
279, 19, 20, 25, 26syl31anc 1233 . . . 4  |-  ( ph  ->  ( S  .\/  T
)  e.  ( LLines `  K ) )
28 dalemdea.m . . . . 5  |-  ./\  =  ( meet `  K )
294, 28, 5, 16, 62llnmj 32577 . . . 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 1230 . . 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 2494 1  |-  ( ph  ->  D  e.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   class class class wbr 4395   ` cfv 5569  (class class class)co 6278   Basecbs 14841   lecple 14916   joincjn 15897   meetcmee 15898   Atomscatm 32281   HLchlt 32368   LLinesclln 32508   LPlanesclpl 32509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-preset 15881  df-poset 15899  df-plt 15912  df-lub 15928  df-glb 15929  df-join 15930  df-meet 15931  df-p0 15993  df-lat 16000  df-clat 16062  df-oposet 32194  df-ol 32196  df-oml 32197  df-covers 32284  df-ats 32285  df-atl 32316  df-cvlat 32340  df-hlat 32369  df-llines 32515  df-lplanes 32516
This theorem is referenced by:  dalemeea  32680  dalem3  32681  dalem16  32696  dalem52  32741  dalem57  32746  dalem60  32749
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