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Theorem dalem23 35122
Description: Lemma for dath 35162. 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 35049 . . . . . . 7  |-  ( ph  ->  K  e.  HL )
43adantr 465 . . . . . 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 35109 . . . . . . 7  |-  ( ps 
->  c  e.  A
)
76adantl 466 . . . . . 6  |-  ( (
ph  /\  ps )  ->  c  e.  A )
82dalempea 35052 . . . . . . 7  |-  ( ph  ->  P  e.  A )
98adantr 465 . . . . . 6  |-  ( (
ph  /\  ps )  ->  P  e.  A )
105dalemddea 35110 . . . . . . 7  |-  ( ps 
->  d  e.  A
)
1110adantl 466 . . . . . 6  |-  ( (
ph  /\  ps )  ->  d  e.  A )
122dalemsea 35055 . . . . . . 7  |-  ( ph  ->  S  e.  A )
1312adantr 465 . . . . . 6  |-  ( (
ph  /\  ps )  ->  S  e.  A )
14 dalem.j . . . . . . 7  |-  .\/  =  ( join `  K )
15 dalem.a . . . . . . 7  |-  A  =  ( Atoms `  K )
1614, 15hlatj4 34800 . . . . . 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 1236 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( ( c  .\/  P )  .\/  ( d 
.\/  S ) )  =  ( ( c 
.\/  d )  .\/  ( P  .\/  S ) ) )
18173adant2 1014 . . . 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 35121 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  d )  .\/  ( P  .\/  S ) )  e.  O )
2418, 23eqeltrd 2529 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  P )  .\/  ( d 
.\/  S ) )  e.  O )
2533ad2ant1 1016 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
262, 19, 14, 15, 20, 21dalemply 35080 . . . . . . . 8  |-  ( ph  ->  P  .<_  Y )
275dalem-ccly 35111 . . . . . . . 8  |-  ( ps 
->  -.  c  .<_  Y )
28 nbrne2 4451 . . . . . . . 8  |-  ( ( P  .<_  Y  /\  -.  c  .<_  Y )  ->  P  =/=  c
)
2926, 27, 28syl2an 477 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  P  =/=  c )
3029necomd 2712 . . . . . 6  |-  ( (
ph  /\  ps )  ->  c  =/=  P )
31 eqid 2441 . . . . . . 7  |-  ( LLines `  K )  =  (
LLines `  K )
3214, 15, 31llni2 34938 . . . . . 6  |-  ( ( ( K  e.  HL  /\  c  e.  A  /\  P  e.  A )  /\  c  =/=  P
)  ->  ( c  .\/  P )  e.  (
LLines `  K ) )
334, 7, 9, 30, 32syl31anc 1230 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( c  .\/  P
)  e.  ( LLines `  K ) )
34333adant2 1014 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  P
)  e.  ( LLines `  K ) )
35103ad2ant3 1018 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
d  e.  A )
36123ad2ant1 1016 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  S  e.  A )
372, 19, 14, 15, 22dalemsly 35081 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z )  ->  S  .<_  Y )
38373adant3 1015 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  S  .<_  Y )
395dalem-ddly 35112 . . . . . . . 8  |-  ( ps 
->  -.  d  .<_  Y )
40393ad2ant3 1018 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  d  .<_  Y )
41 nbrne2 4451 . . . . . . 7  |-  ( ( S  .<_  Y  /\  -.  d  .<_  Y )  ->  S  =/=  d
)
4238, 40, 41syl2anc 661 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  S  =/=  d )
4342necomd 2712 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
d  =/=  S )
4414, 15, 31llni2 34938 . . . . 5  |-  ( ( ( K  e.  HL  /\  d  e.  A  /\  S  e.  A )  /\  d  =/=  S
)  ->  ( d  .\/  S )  e.  (
LLines `  K ) )
4525, 35, 36, 43, 44syl31anc 1230 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( d  .\/  S
)  e.  ( LLines `  K ) )
46 dalem23.m . . . . 5  |-  ./\  =  ( meet `  K )
4714, 46, 15, 31, 202llnmj 34986 . . . 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 1227 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( c 
.\/  P )  ./\  ( d  .\/  S
) )  e.  A  <->  ( ( c  .\/  P
)  .\/  ( d  .\/  S ) )  e.  O ) )
4924, 48mpbird 232 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  P )  ./\  ( d  .\/  S ) )  e.  A )
501, 49syl5eqel 2533 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802    =/= wne 2636   class class class wbr 4433   ` cfv 5574  (class class class)co 6277   Basecbs 14504   lecple 14576   joincjn 15442   meetcmee 15443   Atomscatm 34690   HLchlt 34777   LLinesclln 34917   LPlanesclpl 34918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-preset 15426  df-poset 15444  df-plt 15457  df-lub 15473  df-glb 15474  df-join 15475  df-meet 15476  df-p0 15538  df-lat 15545  df-clat 15607  df-oposet 34603  df-ol 34605  df-oml 34606  df-covers 34693  df-ats 34694  df-atl 34725  df-cvlat 34749  df-hlat 34778  df-llines 34924  df-lplanes 34925
This theorem is referenced by:  dalem24  35123  dalem27  35125  dalem28  35126  dalem29  35127  dalem38  35136  dalem39  35137  dalem41  35139  dalem42  35140  dalem43  35141  dalem44  35142  dalem45  35143  dalem51  35149  dalem52  35150  dalem54  35152  dalem55  35153  dalem57  35155  dalem58  35156  dalem59  35157
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