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Theorem dalem23 34709
Description: Lemma for dath 34749. 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 34636 . . . . . . 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 34696 . . . . . . 7  |-  ( ps 
->  c  e.  A
)
76adantl 466 . . . . . 6  |-  ( (
ph  /\  ps )  ->  c  e.  A )
82dalempea 34639 . . . . . . 7  |-  ( ph  ->  P  e.  A )
98adantr 465 . . . . . 6  |-  ( (
ph  /\  ps )  ->  P  e.  A )
105dalemddea 34697 . . . . . . 7  |-  ( ps 
->  d  e.  A
)
1110adantl 466 . . . . . 6  |-  ( (
ph  /\  ps )  ->  d  e.  A )
122dalemsea 34642 . . . . . . 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 34387 . . . . . 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 1237 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( ( c  .\/  P )  .\/  ( d 
.\/  S ) )  =  ( ( c 
.\/  d )  .\/  ( P  .\/  S ) ) )
18173adant2 1015 . . . 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 34708 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  d )  .\/  ( P  .\/  S ) )  e.  O )
2418, 23eqeltrd 2555 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  P )  .\/  ( d 
.\/  S ) )  e.  O )
2533ad2ant1 1017 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
262, 19, 14, 15, 20, 21dalemply 34667 . . . . . . . 8  |-  ( ph  ->  P  .<_  Y )
275dalem-ccly 34698 . . . . . . . 8  |-  ( ps 
->  -.  c  .<_  Y )
28 nbrne2 4465 . . . . . . . 8  |-  ( ( P  .<_  Y  /\  -.  c  .<_  Y )  ->  P  =/=  c
)
2926, 27, 28syl2an 477 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  P  =/=  c )
3029necomd 2738 . . . . . 6  |-  ( (
ph  /\  ps )  ->  c  =/=  P )
31 eqid 2467 . . . . . . 7  |-  ( LLines `  K )  =  (
LLines `  K )
3214, 15, 31llni2 34525 . . . . . 6  |-  ( ( ( K  e.  HL  /\  c  e.  A  /\  P  e.  A )  /\  c  =/=  P
)  ->  ( c  .\/  P )  e.  (
LLines `  K ) )
334, 7, 9, 30, 32syl31anc 1231 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( c  .\/  P
)  e.  ( LLines `  K ) )
34333adant2 1015 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  P
)  e.  ( LLines `  K ) )
35103ad2ant3 1019 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
d  e.  A )
36123ad2ant1 1017 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  S  e.  A )
372, 19, 14, 15, 22dalemsly 34668 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z )  ->  S  .<_  Y )
38373adant3 1016 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  S  .<_  Y )
395dalem-ddly 34699 . . . . . . . 8  |-  ( ps 
->  -.  d  .<_  Y )
40393ad2ant3 1019 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  d  .<_  Y )
41 nbrne2 4465 . . . . . . 7  |-  ( ( S  .<_  Y  /\  -.  d  .<_  Y )  ->  S  =/=  d
)
4238, 40, 41syl2anc 661 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  S  =/=  d )
4342necomd 2738 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
d  =/=  S )
4414, 15, 31llni2 34525 . . . . 5  |-  ( ( ( K  e.  HL  /\  d  e.  A  /\  S  e.  A )  /\  d  =/=  S
)  ->  ( d  .\/  S )  e.  (
LLines `  K ) )
4525, 35, 36, 43, 44syl31anc 1231 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( d  .\/  S
)  e.  ( LLines `  K ) )
46 dalem23.m . . . . 5  |-  ./\  =  ( meet `  K )
4714, 46, 15, 31, 202llnmj 34573 . . . 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 1228 . . 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 2559 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447   ` cfv 5588  (class class class)co 6285   Basecbs 14493   lecple 14565   joincjn 15434   meetcmee 15435   Atomscatm 34277   HLchlt 34364   LLinesclln 34504   LPlanesclpl 34505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-poset 15436  df-plt 15448  df-lub 15464  df-glb 15465  df-join 15466  df-meet 15467  df-p0 15529  df-lat 15536  df-clat 15598  df-oposet 34190  df-ol 34192  df-oml 34193  df-covers 34280  df-ats 34281  df-atl 34312  df-cvlat 34336  df-hlat 34365  df-llines 34511  df-lplanes 34512
This theorem is referenced by:  dalem24  34710  dalem27  34712  dalem28  34713  dalem29  34714  dalem38  34723  dalem39  34724  dalem41  34726  dalem42  34727  dalem43  34728  dalem44  34729  dalem45  34730  dalem51  34736  dalem52  34737  dalem54  34739  dalem55  34740  dalem57  34742  dalem58  34743  dalem59  34744
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