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Theorem dalem23 33698
Description: Lemma for dath 33738. 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 33625 . . . . . . 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 33685 . . . . . . 7  |-  ( ps 
->  c  e.  A
)
76adantl 466 . . . . . 6  |-  ( (
ph  /\  ps )  ->  c  e.  A )
82dalempea 33628 . . . . . . 7  |-  ( ph  ->  P  e.  A )
98adantr 465 . . . . . 6  |-  ( (
ph  /\  ps )  ->  P  e.  A )
105dalemddea 33686 . . . . . . 7  |-  ( ps 
->  d  e.  A
)
1110adantl 466 . . . . . 6  |-  ( (
ph  /\  ps )  ->  d  e.  A )
122dalemsea 33631 . . . . . . 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 33376 . . . . . 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 1228 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( ( c  .\/  P )  .\/  ( d 
.\/  S ) )  =  ( ( c 
.\/  d )  .\/  ( P  .\/  S ) ) )
18173adant2 1007 . . . 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 33697 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  d )  .\/  ( P  .\/  S ) )  e.  O )
2418, 23eqeltrd 2542 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  P )  .\/  ( d 
.\/  S ) )  e.  O )
2533ad2ant1 1009 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
262, 19, 14, 15, 20, 21dalemply 33656 . . . . . . . 8  |-  ( ph  ->  P  .<_  Y )
275dalem-ccly 33687 . . . . . . . 8  |-  ( ps 
->  -.  c  .<_  Y )
28 nbrne2 4421 . . . . . . . 8  |-  ( ( P  .<_  Y  /\  -.  c  .<_  Y )  ->  P  =/=  c
)
2926, 27, 28syl2an 477 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  P  =/=  c )
3029necomd 2723 . . . . . 6  |-  ( (
ph  /\  ps )  ->  c  =/=  P )
31 eqid 2454 . . . . . . 7  |-  ( LLines `  K )  =  (
LLines `  K )
3214, 15, 31llni2 33514 . . . . . 6  |-  ( ( ( K  e.  HL  /\  c  e.  A  /\  P  e.  A )  /\  c  =/=  P
)  ->  ( c  .\/  P )  e.  (
LLines `  K ) )
334, 7, 9, 30, 32syl31anc 1222 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( c  .\/  P
)  e.  ( LLines `  K ) )
34333adant2 1007 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  P
)  e.  ( LLines `  K ) )
35103ad2ant3 1011 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
d  e.  A )
36123ad2ant1 1009 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  S  e.  A )
372, 19, 14, 15, 22dalemsly 33657 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z )  ->  S  .<_  Y )
38373adant3 1008 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  S  .<_  Y )
395dalem-ddly 33688 . . . . . . . 8  |-  ( ps 
->  -.  d  .<_  Y )
40393ad2ant3 1011 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  d  .<_  Y )
41 nbrne2 4421 . . . . . . 7  |-  ( ( S  .<_  Y  /\  -.  d  .<_  Y )  ->  S  =/=  d
)
4238, 40, 41syl2anc 661 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  S  =/=  d )
4342necomd 2723 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
d  =/=  S )
4414, 15, 31llni2 33514 . . . . 5  |-  ( ( ( K  e.  HL  /\  d  e.  A  /\  S  e.  A )  /\  d  =/=  S
)  ->  ( d  .\/  S )  e.  (
LLines `  K ) )
4525, 35, 36, 43, 44syl31anc 1222 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( d  .\/  S
)  e.  ( LLines `  K ) )
46 dalem23.m . . . . 5  |-  ./\  =  ( meet `  K )
4714, 46, 15, 31, 202llnmj 33562 . . . 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 1219 . . 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 2546 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 965    = wceq 1370    e. wcel 1758    =/= wne 2648   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   Basecbs 14295   lecple 14367   joincjn 15236   meetcmee 15237   Atomscatm 33266   HLchlt 33353   LLinesclln 33493   LPlanesclpl 33494
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 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
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 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-poset 15238  df-plt 15250  df-lub 15266  df-glb 15267  df-join 15268  df-meet 15269  df-p0 15331  df-lat 15338  df-clat 15400  df-oposet 33179  df-ol 33181  df-oml 33182  df-covers 33269  df-ats 33270  df-atl 33301  df-cvlat 33325  df-hlat 33354  df-llines 33500  df-lplanes 33501
This theorem is referenced by:  dalem24  33699  dalem27  33701  dalem28  33702  dalem29  33703  dalem38  33712  dalem39  33713  dalem41  33715  dalem42  33716  dalem43  33717  dalem44  33718  dalem45  33719  dalem51  33725  dalem52  33726  dalem54  33728  dalem55  33729  dalem57  33731  dalem58  33732  dalem59  33733
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