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Theorem dalem23 30567
Description: Lemma for dath 30607. 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 30494 . . . . . . 7  |-  ( ph  ->  K  e.  HL )
43adantr 453 . . . . . 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 30554 . . . . . . 7  |-  ( ps 
->  c  e.  A
)
76adantl 454 . . . . . 6  |-  ( (
ph  /\  ps )  ->  c  e.  A )
82dalempea 30497 . . . . . . 7  |-  ( ph  ->  P  e.  A )
98adantr 453 . . . . . 6  |-  ( (
ph  /\  ps )  ->  P  e.  A )
105dalemddea 30555 . . . . . . 7  |-  ( ps 
->  d  e.  A
)
1110adantl 454 . . . . . 6  |-  ( (
ph  /\  ps )  ->  d  e.  A )
122dalemsea 30500 . . . . . . 7  |-  ( ph  ->  S  e.  A )
1312adantr 453 . . . . . 6  |-  ( (
ph  /\  ps )  ->  S  e.  A )
14 dalem.j . . . . . . 7  |-  .\/  =  ( join `  K )
15 dalem.a . . . . . . 7  |-  A  =  ( Atoms `  K )
1614, 15hlatj4 30245 . . . . . 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 1194 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( ( c  .\/  P )  .\/  ( d 
.\/  S ) )  =  ( ( c 
.\/  d )  .\/  ( P  .\/  S ) ) )
18173adant2 977 . . . 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 30566 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  d )  .\/  ( P  .\/  S ) )  e.  O )
2418, 23eqeltrd 2512 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  P )  .\/  ( d 
.\/  S ) )  e.  O )
2533ad2ant1 979 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
262, 19, 14, 15, 20, 21dalemply 30525 . . . . . . . 8  |-  ( ph  ->  P  .<_  Y )
275dalem-ccly 30556 . . . . . . . 8  |-  ( ps 
->  -.  c  .<_  Y )
28 nbrne2 4233 . . . . . . . 8  |-  ( ( P  .<_  Y  /\  -.  c  .<_  Y )  ->  P  =/=  c
)
2926, 27, 28syl2an 465 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  P  =/=  c )
3029necomd 2689 . . . . . 6  |-  ( (
ph  /\  ps )  ->  c  =/=  P )
31 eqid 2438 . . . . . . 7  |-  ( LLines `  K )  =  (
LLines `  K )
3214, 15, 31llni2 30383 . . . . . 6  |-  ( ( ( K  e.  HL  /\  c  e.  A  /\  P  e.  A )  /\  c  =/=  P
)  ->  ( c  .\/  P )  e.  (
LLines `  K ) )
334, 7, 9, 30, 32syl31anc 1188 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( c  .\/  P
)  e.  ( LLines `  K ) )
34333adant2 977 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  P
)  e.  ( LLines `  K ) )
35103ad2ant3 981 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
d  e.  A )
36123ad2ant1 979 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  S  e.  A )
372, 19, 14, 15, 22dalemsly 30526 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z )  ->  S  .<_  Y )
38373adant3 978 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  S  .<_  Y )
395dalem-ddly 30557 . . . . . . . 8  |-  ( ps 
->  -.  d  .<_  Y )
40393ad2ant3 981 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  d  .<_  Y )
41 nbrne2 4233 . . . . . . 7  |-  ( ( S  .<_  Y  /\  -.  d  .<_  Y )  ->  S  =/=  d
)
4238, 40, 41syl2anc 644 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  S  =/=  d )
4342necomd 2689 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
d  =/=  S )
4414, 15, 31llni2 30383 . . . . 5  |-  ( ( ( K  e.  HL  /\  d  e.  A  /\  S  e.  A )  /\  d  =/=  S
)  ->  ( d  .\/  S )  e.  (
LLines `  K ) )
4525, 35, 36, 43, 44syl31anc 1188 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( d  .\/  S
)  e.  ( LLines `  K ) )
46 dalem23.m . . . . 5  |-  ./\  =  ( meet `  K )
4714, 46, 15, 31, 202llnmj 30431 . . . 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 1185 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( c 
.\/  P )  ./\  ( d  .\/  S
) )  e.  A  <->  ( ( c  .\/  P
)  .\/  ( d  .\/  S ) )  e.  O ) )
4924, 48mpbird 225 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  P )  ./\  ( d  .\/  S ) )  e.  A )
501, 49syl5eqel 2522 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   Basecbs 13474   lecple 13541   joincjn 14406   meetcmee 14407   Atomscatm 30135   HLchlt 30222   LLinesclln 30362   LPlanesclpl 30363
This theorem is referenced by:  dalem24  30568  dalem27  30570  dalem28  30571  dalem29  30572  dalem38  30581  dalem39  30582  dalem41  30584  dalem42  30585  dalem43  30586  dalem44  30587  dalem45  30588  dalem51  30594  dalem52  30595  dalem54  30597  dalem55  30598  dalem57  30600  dalem58  30601  dalem59  30602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-poset 14408  df-plt 14420  df-lub 14436  df-glb 14437  df-join 14438  df-meet 14439  df-p0 14473  df-lat 14480  df-clat 14542  df-oposet 30048  df-ol 30050  df-oml 30051  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223  df-llines 30369  df-lplanes 30370
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