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Theorem dalem21 33701
Description: Lemma for dath 33743. Show that lines  c d and  P S intersect at 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
) ) ) )
dalem21.m  |-  ./\  =  ( meet `  K )
dalem21.o  |-  O  =  ( LPlanes `  K )
dalem21.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem21.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
Assertion
Ref Expression
dalem21  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  d )  ./\  ( P  .\/  S ) )  e.  A )

Proof of Theorem dalem21
StepHypRef Expression
1 dalem.ph . . . 4  |-  ( 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 ) ) ) ) )
21dalemkehl 33630 . . 3  |-  ( ph  ->  K  e.  HL )
323ad2ant1 1009 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
4 dalem.l . . . 4  |-  .<_  =  ( le `  K )
5 dalem.j . . . 4  |-  .\/  =  ( join `  K )
6 dalem.a . . . 4  |-  A  =  ( Atoms `  K )
7 dalem.ps . . . 4  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
81, 4, 5, 6, 7dalemcjden 33699 . . 3  |-  ( (
ph  /\  ps )  ->  ( c  .\/  d
)  e.  ( LLines `  K ) )
983adant2 1007 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  d
)  e.  ( LLines `  K ) )
10 dalem21.o . . . 4  |-  O  =  ( LPlanes `  K )
11 dalem21.y . . . 4  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
121, 4, 5, 6, 10, 11dalempjsen 33660 . . 3  |-  ( ph  ->  ( P  .\/  S
)  e.  ( LLines `  K ) )
13123ad2ant1 1009 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  .\/  S
)  e.  ( LLines `  K ) )
141, 4, 5, 6, 10, 11dalemply 33661 . . . . . . 7  |-  ( ph  ->  P  .<_  Y )
1514adantr 465 . . . . . 6  |-  ( (
ph  /\  Y  =  Z )  ->  P  .<_  Y )
16 dalem21.z . . . . . . 7  |-  Z  =  ( ( S  .\/  T )  .\/  U )
171, 4, 5, 6, 16dalemsly 33662 . . . . . 6  |-  ( (
ph  /\  Y  =  Z )  ->  S  .<_  Y )
181dalemkelat 33631 . . . . . . . 8  |-  ( ph  ->  K  e.  Lat )
191, 6dalempeb 33646 . . . . . . . 8  |-  ( ph  ->  P  e.  ( Base `  K ) )
201, 6dalemseb 33649 . . . . . . . 8  |-  ( ph  ->  S  e.  ( Base `  K ) )
211, 10dalemyeb 33656 . . . . . . . 8  |-  ( ph  ->  Y  e.  ( Base `  K ) )
22 eqid 2454 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
2322, 4, 5latjle12 15355 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  S  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) ) )  -> 
( ( P  .<_  Y  /\  S  .<_  Y )  <-> 
( P  .\/  S
)  .<_  Y ) )
2418, 19, 20, 21, 23syl13anc 1221 . . . . . . 7  |-  ( ph  ->  ( ( P  .<_  Y  /\  S  .<_  Y )  <-> 
( P  .\/  S
)  .<_  Y ) )
2524adantr 465 . . . . . 6  |-  ( (
ph  /\  Y  =  Z )  ->  (
( P  .<_  Y  /\  S  .<_  Y )  <->  ( P  .\/  S )  .<_  Y ) )
2615, 17, 25mpbi2and 912 . . . . 5  |-  ( (
ph  /\  Y  =  Z )  ->  ( P  .\/  S )  .<_  Y )
27263adant3 1008 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  .\/  S
)  .<_  Y )
287dalem-ccly 33692 . . . . . . 7  |-  ( ps 
->  -.  c  .<_  Y )
2928adantl 466 . . . . . 6  |-  ( (
ph  /\  ps )  ->  -.  c  .<_  Y )
3018adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ps )  ->  K  e.  Lat )
317, 6dalemcceb 33696 . . . . . . . . 9  |-  ( ps 
->  c  e.  ( Base `  K ) )
3231adantl 466 . . . . . . . 8  |-  ( (
ph  /\  ps )  ->  c  e.  ( Base `  K ) )
337dalemddea 33691 . . . . . . . . . 10  |-  ( ps 
->  d  e.  A
)
3422, 6atbase 33297 . . . . . . . . . 10  |-  ( d  e.  A  ->  d  e.  ( Base `  K
) )
3533, 34syl 16 . . . . . . . . 9  |-  ( ps 
->  d  e.  ( Base `  K ) )
3635adantl 466 . . . . . . . 8  |-  ( (
ph  /\  ps )  ->  d  e.  ( Base `  K ) )
3722, 4, 5latlej1 15353 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  c  e.  ( Base `  K )  /\  d  e.  ( Base `  K
) )  ->  c  .<_  ( c  .\/  d
) )
3830, 32, 36, 37syl3anc 1219 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  c  .<_  ( c  .\/  d ) )
39 eqid 2454 . . . . . . . . . 10  |-  ( LLines `  K )  =  (
LLines `  K )
4022, 39llnbase 33516 . . . . . . . . 9  |-  ( ( c  .\/  d )  e.  ( LLines `  K
)  ->  ( c  .\/  d )  e.  (
Base `  K )
)
418, 40syl 16 . . . . . . . 8  |-  ( (
ph  /\  ps )  ->  ( c  .\/  d
)  e.  ( Base `  K ) )
4221adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ps )  ->  Y  e.  ( Base `  K ) )
4322, 4lattr 15349 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( c  e.  (
Base `  K )  /\  ( c  .\/  d
)  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) ) )  -> 
( ( c  .<_  ( c  .\/  d
)  /\  ( c  .\/  d )  .<_  Y )  ->  c  .<_  Y ) )
4430, 32, 41, 42, 43syl13anc 1221 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  ( ( c  .<_  ( c  .\/  d
)  /\  ( c  .\/  d )  .<_  Y )  ->  c  .<_  Y ) )
4538, 44mpand 675 . . . . . 6  |-  ( (
ph  /\  ps )  ->  ( ( c  .\/  d )  .<_  Y  -> 
c  .<_  Y ) )
4629, 45mtod 177 . . . . 5  |-  ( (
ph  /\  ps )  ->  -.  ( c  .\/  d )  .<_  Y )
47463adant2 1007 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  ( c  .\/  d
)  .<_  Y )
48 nbrne2 4421 . . . 4  |-  ( ( ( P  .\/  S
)  .<_  Y  /\  -.  ( c  .\/  d
)  .<_  Y )  -> 
( P  .\/  S
)  =/=  ( c 
.\/  d ) )
4927, 47, 48syl2anc 661 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  .\/  S
)  =/=  ( c 
.\/  d ) )
5049necomd 2723 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  d
)  =/=  ( P 
.\/  S ) )
51 hlatl 33368 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  AtLat )
522, 51syl 16 . . . . 5  |-  ( ph  ->  K  e.  AtLat )
5352adantr 465 . . . 4  |-  ( (
ph  /\  ps )  ->  K  e.  AtLat )
541dalempea 33633 . . . . . . 7  |-  ( ph  ->  P  e.  A )
551dalemsea 33636 . . . . . . 7  |-  ( ph  ->  S  e.  A )
5622, 5, 6hlatjcl 33374 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
572, 54, 55, 56syl3anc 1219 . . . . . 6  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
5857adantr 465 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
59 dalem21.m . . . . . 6  |-  ./\  =  ( meet `  K )
6022, 59latmcl 15345 . . . . 5  |-  ( ( K  e.  Lat  /\  ( c  .\/  d
)  e.  ( Base `  K )  /\  ( P  .\/  S )  e.  ( Base `  K
) )  ->  (
( c  .\/  d
)  ./\  ( P  .\/  S ) )  e.  ( Base `  K
) )
6130, 41, 58, 60syl3anc 1219 . . . 4  |-  ( (
ph  /\  ps )  ->  ( ( c  .\/  d )  ./\  ( P  .\/  S ) )  e.  ( Base `  K
) )
621, 4, 5, 6, 10, 11dalemcea 33667 . . . . 5  |-  ( ph  ->  C  e.  A )
6362adantr 465 . . . 4  |-  ( (
ph  /\  ps )  ->  C  e.  A )
647dalemclccjdd 33695 . . . . . 6  |-  ( ps 
->  C  .<_  ( c 
.\/  d ) )
6564adantl 466 . . . . 5  |-  ( (
ph  /\  ps )  ->  C  .<_  ( c  .\/  d ) )
661dalemclpjs 33641 . . . . . 6  |-  ( ph  ->  C  .<_  ( P  .\/  S ) )
6766adantr 465 . . . . 5  |-  ( (
ph  /\  ps )  ->  C  .<_  ( P  .\/  S ) )
681, 6dalemceb 33645 . . . . . . 7  |-  ( ph  ->  C  e.  ( Base `  K ) )
6968adantr 465 . . . . . 6  |-  ( (
ph  /\  ps )  ->  C  e.  ( Base `  K ) )
7022, 4, 59latlem12 15371 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( C  e.  ( Base `  K )  /\  ( c  .\/  d
)  e.  ( Base `  K )  /\  ( P  .\/  S )  e.  ( Base `  K
) ) )  -> 
( ( C  .<_  ( c  .\/  d )  /\  C  .<_  ( P 
.\/  S ) )  <-> 
C  .<_  ( ( c 
.\/  d )  ./\  ( P  .\/  S ) ) ) )
7130, 69, 41, 58, 70syl13anc 1221 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( ( C  .<_  ( c  .\/  d )  /\  C  .<_  ( P 
.\/  S ) )  <-> 
C  .<_  ( ( c 
.\/  d )  ./\  ( P  .\/  S ) ) ) )
7265, 67, 71mpbi2and 912 . . . 4  |-  ( (
ph  /\  ps )  ->  C  .<_  ( (
c  .\/  d )  ./\  ( P  .\/  S
) ) )
73 eqid 2454 . . . . 5  |-  ( 0.
`  K )  =  ( 0. `  K
)
7422, 4, 73, 6atlen0 33318 . . . 4  |-  ( ( ( K  e.  AtLat  /\  ( ( c  .\/  d )  ./\  ( P  .\/  S ) )  e.  ( Base `  K
)  /\  C  e.  A )  /\  C  .<_  ( ( c  .\/  d )  ./\  ( P  .\/  S ) ) )  ->  ( (
c  .\/  d )  ./\  ( P  .\/  S
) )  =/=  ( 0. `  K ) )
7553, 61, 63, 72, 74syl31anc 1222 . . 3  |-  ( (
ph  /\  ps )  ->  ( ( c  .\/  d )  ./\  ( P  .\/  S ) )  =/=  ( 0. `  K ) )
76753adant2 1007 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  d )  ./\  ( P  .\/  S ) )  =/=  ( 0. `  K ) )
7759, 73, 6, 392llnmat 33531 . 2  |-  ( ( ( K  e.  HL  /\  ( c  .\/  d
)  e.  ( LLines `  K )  /\  ( P  .\/  S )  e.  ( LLines `  K )
)  /\  ( (
c  .\/  d )  =/=  ( P  .\/  S
)  /\  ( (
c  .\/  d )  ./\  ( P  .\/  S
) )  =/=  ( 0. `  K ) ) )  ->  ( (
c  .\/  d )  ./\  ( P  .\/  S
) )  e.  A
)
783, 9, 13, 50, 76, 77syl32anc 1227 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  d )  ./\  ( P  .\/  S ) )  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 14296   lecple 14368   joincjn 15237   meetcmee 15238   0.cp0 15330   Latclat 15338   Atomscatm 33271   AtLatcal 33272   HLchlt 33358   LLinesclln 33498   LPlanesclpl 33499
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 15239  df-plt 15251  df-lub 15267  df-glb 15268  df-join 15269  df-meet 15270  df-p0 15332  df-lat 15339  df-clat 15401  df-oposet 33184  df-ol 33186  df-oml 33187  df-covers 33274  df-ats 33275  df-atl 33306  df-cvlat 33330  df-hlat 33359  df-llines 33505  df-lplanes 33506
This theorem is referenced by:  dalem22  33702
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