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Theorem dalem21 34891
Description: Lemma for dath 34933. 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 34820 . . 3  |-  ( ph  ->  K  e.  HL )
323ad2ant1 1017 . 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 34889 . . 3  |-  ( (
ph  /\  ps )  ->  ( c  .\/  d
)  e.  ( LLines `  K ) )
983adant2 1015 . 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 34850 . . 3  |-  ( ph  ->  ( P  .\/  S
)  e.  ( LLines `  K ) )
13123ad2ant1 1017 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  .\/  S
)  e.  ( LLines `  K ) )
141, 4, 5, 6, 10, 11dalemply 34851 . . . . . . 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 34852 . . . . . 6  |-  ( (
ph  /\  Y  =  Z )  ->  S  .<_  Y )
181dalemkelat 34821 . . . . . . . 8  |-  ( ph  ->  K  e.  Lat )
191, 6dalempeb 34836 . . . . . . . 8  |-  ( ph  ->  P  e.  ( Base `  K ) )
201, 6dalemseb 34839 . . . . . . . 8  |-  ( ph  ->  S  e.  ( Base `  K ) )
211, 10dalemyeb 34846 . . . . . . . 8  |-  ( ph  ->  Y  e.  ( Base `  K ) )
22 eqid 2467 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
2322, 4, 5latjle12 15566 . . . . . . . 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 1230 . . . . . . 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 919 . . . . 5  |-  ( (
ph  /\  Y  =  Z )  ->  ( P  .\/  S )  .<_  Y )
27263adant3 1016 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  .\/  S
)  .<_  Y )
287dalem-ccly 34882 . . . . . . 7  |-  ( ps 
->  -.  c  .<_  Y )
2928adantl 466 . . . . . 6  |-  ( (
ph  /\  ps )  ->  -.  c  .<_  Y )
3018adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ps )  ->  K  e.  Lat )
317, 6dalemcceb 34886 . . . . . . . . 9  |-  ( ps 
->  c  e.  ( Base `  K ) )
3231adantl 466 . . . . . . . 8  |-  ( (
ph  /\  ps )  ->  c  e.  ( Base `  K ) )
337dalemddea 34881 . . . . . . . . . 10  |-  ( ps 
->  d  e.  A
)
3422, 6atbase 34487 . . . . . . . . . 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 15564 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  c  e.  ( Base `  K )  /\  d  e.  ( Base `  K
) )  ->  c  .<_  ( c  .\/  d
) )
3830, 32, 36, 37syl3anc 1228 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  c  .<_  ( c  .\/  d ) )
39 eqid 2467 . . . . . . . . . 10  |-  ( LLines `  K )  =  (
LLines `  K )
4022, 39llnbase 34706 . . . . . . . . 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 15560 . . . . . . . 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 1230 . . . . . . 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 1015 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  ( c  .\/  d
)  .<_  Y )
48 nbrne2 4471 . . . 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 2738 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  d
)  =/=  ( P 
.\/  S ) )
51 hlatl 34558 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  AtLat )
522, 51syl 16 . . . . 5  |-  ( ph  ->  K  e.  AtLat )
5352adantr 465 . . . 4  |-  ( (
ph  /\  ps )  ->  K  e.  AtLat )
541dalempea 34823 . . . . . . 7  |-  ( ph  ->  P  e.  A )
551dalemsea 34826 . . . . . . 7  |-  ( ph  ->  S  e.  A )
5622, 5, 6hlatjcl 34564 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
572, 54, 55, 56syl3anc 1228 . . . . . 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 15556 . . . . 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 1228 . . . 4  |-  ( (
ph  /\  ps )  ->  ( ( c  .\/  d )  ./\  ( P  .\/  S ) )  e.  ( Base `  K
) )
621, 4, 5, 6, 10, 11dalemcea 34857 . . . . 5  |-  ( ph  ->  C  e.  A )
6362adantr 465 . . . 4  |-  ( (
ph  /\  ps )  ->  C  e.  A )
647dalemclccjdd 34885 . . . . . 6  |-  ( ps 
->  C  .<_  ( c 
.\/  d ) )
6564adantl 466 . . . . 5  |-  ( (
ph  /\  ps )  ->  C  .<_  ( c  .\/  d ) )
661dalemclpjs 34831 . . . . . 6  |-  ( ph  ->  C  .<_  ( P  .\/  S ) )
6766adantr 465 . . . . 5  |-  ( (
ph  /\  ps )  ->  C  .<_  ( P  .\/  S ) )
681, 6dalemceb 34835 . . . . . . 7  |-  ( ph  ->  C  e.  ( Base `  K ) )
6968adantr 465 . . . . . 6  |-  ( (
ph  /\  ps )  ->  C  e.  ( Base `  K ) )
7022, 4, 59latlem12 15582 . . . . . 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 1230 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( ( C  .<_  ( c  .\/  d )  /\  C  .<_  ( P 
.\/  S ) )  <-> 
C  .<_  ( ( c 
.\/  d )  ./\  ( P  .\/  S ) ) ) )
7265, 67, 71mpbi2and 919 . . . 4  |-  ( (
ph  /\  ps )  ->  C  .<_  ( (
c  .\/  d )  ./\  ( P  .\/  S
) ) )
73 eqid 2467 . . . . 5  |-  ( 0.
`  K )  =  ( 0. `  K
)
7422, 4, 73, 6atlen0 34508 . . . 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 1231 . . 3  |-  ( (
ph  /\  ps )  ->  ( ( c  .\/  d )  ./\  ( P  .\/  S ) )  =/=  ( 0. `  K ) )
76753adant2 1015 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  d )  ./\  ( P  .\/  S ) )  =/=  ( 0. `  K ) )
7759, 73, 6, 392llnmat 34721 . 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 1236 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   Basecbs 14507   lecple 14579   joincjn 15448   meetcmee 15449   0.cp0 15541   Latclat 15549   Atomscatm 34461   AtLatcal 34462   HLchlt 34548   LLinesclln 34688   LPlanesclpl 34689
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-poset 15450  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-lat 15550  df-clat 15612  df-oposet 34374  df-ol 34376  df-oml 34377  df-covers 34464  df-ats 34465  df-atl 34496  df-cvlat 34520  df-hlat 34549  df-llines 34695  df-lplanes 34696
This theorem is referenced by:  dalem22  34892
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