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Theorem dalemdea 33588
Description: Lemma for dath 33662. 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 33587 . . 3  |-  ( ph  ->  ( ( P  .\/  Q )  .\/  ( S 
.\/  T ) )  e.  O )
92dalemkehl 33549 . . . 4  |-  ( ph  ->  K  e.  HL )
102dalempea 33552 . . . . 5  |-  ( ph  ->  P  e.  A )
112dalemqea 33553 . . . . 5  |-  ( ph  ->  Q  e.  A )
122dalemrea 33554 . . . . . 6  |-  ( ph  ->  R  e.  A )
132dalemyeo 33558 . . . . . 6  |-  ( ph  ->  Y  e.  O )
144, 5, 6, 7lplnri1 33479 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  Y  e.  O )  ->  P  =/=  Q )
159, 10, 11, 12, 13, 14syl131anc 1232 . . . . 5  |-  ( ph  ->  P  =/=  Q )
16 eqid 2450 . . . . . 6  |-  ( LLines `  K )  =  (
LLines `  K )
174, 5, 16llni2 33438 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  Q )  e.  (
LLines `  K ) )
189, 10, 11, 15, 17syl31anc 1222 . . . 4  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( LLines `  K ) )
192dalemsea 33555 . . . . 5  |-  ( ph  ->  S  e.  A )
202dalemtea 33556 . . . . 5  |-  ( ph  ->  T  e.  A )
212dalemuea 33557 . . . . . 6  |-  ( ph  ->  U  e.  A )
222dalemzeo 33559 . . . . . 6  |-  ( ph  ->  Z  e.  O )
23 dalemdea.z . . . . . . 7  |-  Z  =  ( ( S  .\/  T )  .\/  U )
244, 5, 6, 23lplnri1 33479 . . . . . 6  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
)  /\  Z  e.  O )  ->  S  =/=  T )
259, 19, 20, 21, 22, 24syl131anc 1232 . . . . 5  |-  ( ph  ->  S  =/=  T )
264, 5, 16llni2 33438 . . . . 5  |-  ( ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  /\  S  =/=  T
)  ->  ( S  .\/  T )  e.  (
LLines `  K ) )
279, 19, 20, 25, 26syl31anc 1222 . . . 4  |-  ( ph  ->  ( S  .\/  T
)  e.  ( LLines `  K ) )
28 dalemdea.m . . . . 5  |-  ./\  =  ( meet `  K )
294, 28, 5, 16, 62llnmj 33486 . . . 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 1219 . . 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 2540 1  |-  ( ph  ->  D  e.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1757    =/= wne 2641   class class class wbr 4376   ` cfv 5502  (class class class)co 6176   Basecbs 14262   lecple 14333   joincjn 15202   meetcmee 15203   Atomscatm 33190   HLchlt 33277   LLinesclln 33417   LPlanesclpl 33418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-reu 2799  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-poset 15204  df-plt 15216  df-lub 15232  df-glb 15233  df-join 15234  df-meet 15235  df-p0 15297  df-lat 15304  df-clat 15366  df-oposet 33103  df-ol 33105  df-oml 33106  df-covers 33193  df-ats 33194  df-atl 33225  df-cvlat 33249  df-hlat 33278  df-llines 33424  df-lplanes 33425
This theorem is referenced by:  dalemeea  33589  dalem3  33590  dalem16  33605  dalem52  33650  dalem57  33655  dalem60  33658
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