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Theorem dalem23 33305
Description: Lemma for dath 33345. Show that auxiliary atom  G is an atom. (Contributed by NM, 2-Aug-2012.)
Hypotheses
Ref Expression
dalem.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 ) ) ) ) )
dalem.l  |-  .<_  =  ( le `  K )
dalem.j  |-  .\/  =  ( join `  K )
dalem.a  |-  A  =  ( Atoms `  K )
dalem.ps  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
dalem23.m  |-  ./\  =  ( meet `  K )
dalem23.o  |-  O  =  ( LPlanes `  K )
dalem23.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem23.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
dalem23.g  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
Assertion
Ref Expression
dalem23  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  A )

Proof of Theorem dalem23
StepHypRef Expression
1 dalem23.g . 2  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
2 dalem.ph . . . . . . . 8  |-  ( 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 ) ) ) ) )
32dalemkehl 33232 . . . . . . 7  |-  ( ph  ->  K  e.  HL )
43adantr 471 . . . . . 6  |-  ( (
ph  /\  ps )  ->  K  e.  HL )
5 dalem.ps . . . . . . . 8  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
65dalemccea 33292 . . . . . . 7  |-  ( ps 
->  c  e.  A
)
76adantl 472 . . . . . 6  |-  ( (
ph  /\  ps )  ->  c  e.  A )
82dalempea 33235 . . . . . . 7  |-  ( ph  ->  P  e.  A )
98adantr 471 . . . . . 6  |-  ( (
ph  /\  ps )  ->  P  e.  A )
105dalemddea 33293 . . . . . . 7  |-  ( ps 
->  d  e.  A
)
1110adantl 472 . . . . . 6  |-  ( (
ph  /\  ps )  ->  d  e.  A )
122dalemsea 33238 . . . . . . 7  |-  ( ph  ->  S  e.  A )
1312adantr 471 . . . . . 6  |-  ( (
ph  /\  ps )  ->  S  e.  A )
14 dalem.j . . . . . . 7  |-  .\/  =  ( join `  K )
15 dalem.a . . . . . . 7  |-  A  =  ( Atoms `  K )
1614, 15hlatj4 32983 . . . . . 6  |-  ( ( K  e.  HL  /\  ( c  e.  A  /\  P  e.  A
)  /\  ( d  e.  A  /\  S  e.  A ) )  -> 
( ( c  .\/  P )  .\/  ( d 
.\/  S ) )  =  ( ( c 
.\/  d )  .\/  ( P  .\/  S ) ) )
174, 7, 9, 11, 13, 16syl122anc 1285 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( ( c  .\/  P )  .\/  ( d 
.\/  S ) )  =  ( ( c 
.\/  d )  .\/  ( P  .\/  S ) ) )
18173adant2 1033 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  P )  .\/  ( d 
.\/  S ) )  =  ( ( c 
.\/  d )  .\/  ( P  .\/  S ) ) )
19 dalem.l . . . . 5  |-  .<_  =  ( le `  K )
20 dalem23.o . . . . 5  |-  O  =  ( LPlanes `  K )
21 dalem23.y . . . . 5  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
22 dalem23.z . . . . 5  |-  Z  =  ( ( S  .\/  T )  .\/  U )
232, 19, 14, 15, 5, 20, 21, 22dalem22 33304 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  d )  .\/  ( P  .\/  S ) )  e.  O )
2418, 23eqeltrd 2539 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  P )  .\/  ( d 
.\/  S ) )  e.  O )
2533ad2ant1 1035 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
262, 19, 14, 15, 20, 21dalemply 33263 . . . . . . . 8  |-  ( ph  ->  P  .<_  Y )
275dalem-ccly 33294 . . . . . . . 8  |-  ( ps 
->  -.  c  .<_  Y )
28 nbrne2 4434 . . . . . . . 8  |-  ( ( P  .<_  Y  /\  -.  c  .<_  Y )  ->  P  =/=  c
)
2926, 27, 28syl2an 484 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  P  =/=  c )
3029necomd 2690 . . . . . 6  |-  ( (
ph  /\  ps )  ->  c  =/=  P )
31 eqid 2461 . . . . . . 7  |-  ( LLines `  K )  =  (
LLines `  K )
3214, 15, 31llni2 33121 . . . . . 6  |-  ( ( ( K  e.  HL  /\  c  e.  A  /\  P  e.  A )  /\  c  =/=  P
)  ->  ( c  .\/  P )  e.  (
LLines `  K ) )
334, 7, 9, 30, 32syl31anc 1279 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( c  .\/  P
)  e.  ( LLines `  K ) )
34333adant2 1033 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  P
)  e.  ( LLines `  K ) )
35103ad2ant3 1037 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
d  e.  A )
36123ad2ant1 1035 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  S  e.  A )
372, 19, 14, 15, 22dalemsly 33264 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z )  ->  S  .<_  Y )
38373adant3 1034 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  S  .<_  Y )
395dalem-ddly 33295 . . . . . . . 8  |-  ( ps 
->  -.  d  .<_  Y )
40393ad2ant3 1037 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  d  .<_  Y )
41 nbrne2 4434 . . . . . . 7  |-  ( ( S  .<_  Y  /\  -.  d  .<_  Y )  ->  S  =/=  d
)
4238, 40, 41syl2anc 671 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  S  =/=  d )
4342necomd 2690 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
d  =/=  S )
4414, 15, 31llni2 33121 . . . . 5  |-  ( ( ( K  e.  HL  /\  d  e.  A  /\  S  e.  A )  /\  d  =/=  S
)  ->  ( d  .\/  S )  e.  (
LLines `  K ) )
4525, 35, 36, 43, 44syl31anc 1279 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( d  .\/  S
)  e.  ( LLines `  K ) )
46 dalem23.m . . . . 5  |-  ./\  =  ( meet `  K )
4714, 46, 15, 31, 202llnmj 33169 . . . 4  |-  ( ( K  e.  HL  /\  ( c  .\/  P
)  e.  ( LLines `  K )  /\  (
d  .\/  S )  e.  ( LLines `  K )
)  ->  ( (
( c  .\/  P
)  ./\  ( d  .\/  S ) )  e.  A  <->  ( ( c 
.\/  P )  .\/  ( d  .\/  S
) )  e.  O
) )
4825, 34, 45, 47syl3anc 1276 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( c 
.\/  P )  ./\  ( d  .\/  S
) )  e.  A  <->  ( ( c  .\/  P
)  .\/  ( d  .\/  S ) )  e.  O ) )
4924, 48mpbird 240 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  P )  ./\  ( d  .\/  S ) )  e.  A )
501, 49syl5eqel 2543 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 991    = wceq 1454    e. wcel 1897    =/= wne 2632   class class class wbr 4415   ` cfv 5600  (class class class)co 6314   Basecbs 15169   lecple 15245   joincjn 16237   meetcmee 16238   Atomscatm 32873   HLchlt 32960   LLinesclln 33100   LPlanesclpl 33101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-reu 2755  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6276  df-ov 6317  df-oprab 6318  df-preset 16221  df-poset 16239  df-plt 16252  df-lub 16268  df-glb 16269  df-join 16270  df-meet 16271  df-p0 16333  df-lat 16340  df-clat 16402  df-oposet 32786  df-ol 32788  df-oml 32789  df-covers 32876  df-ats 32877  df-atl 32908  df-cvlat 32932  df-hlat 32961  df-llines 33107  df-lplanes 33108
This theorem is referenced by:  dalem24  33306  dalem27  33308  dalem28  33309  dalem29  33310  dalem38  33319  dalem39  33320  dalem41  33322  dalem42  33323  dalem43  33324  dalem44  33325  dalem45  33326  dalem51  33332  dalem52  33333  dalem54  33335  dalem55  33336  dalem57  33338  dalem58  33339  dalem59  33340
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