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Theorem dalem21 30428
Description: Lemma for dath 30470. 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 30357 . . 3  |-  ( ph  ->  K  e.  HL )
323ad2ant1 978 . 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 30426 . . 3  |-  ( (
ph  /\  ps )  ->  ( c  .\/  d
)  e.  ( LLines `  K ) )
983adant2 976 . 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 30387 . . 3  |-  ( ph  ->  ( P  .\/  S
)  e.  ( LLines `  K ) )
13123ad2ant1 978 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  .\/  S
)  e.  ( LLines `  K ) )
141, 4, 5, 6, 10, 11dalemply 30388 . . . . . . 7  |-  ( ph  ->  P  .<_  Y )
1514adantr 452 . . . . . 6  |-  ( (
ph  /\  Y  =  Z )  ->  P  .<_  Y )
16 dalem21.z . . . . . . 7  |-  Z  =  ( ( S  .\/  T )  .\/  U )
171, 4, 5, 6, 16dalemsly 30389 . . . . . 6  |-  ( (
ph  /\  Y  =  Z )  ->  S  .<_  Y )
181dalemkelat 30358 . . . . . . . 8  |-  ( ph  ->  K  e.  Lat )
191, 6dalempeb 30373 . . . . . . . 8  |-  ( ph  ->  P  e.  ( Base `  K ) )
201, 6dalemseb 30376 . . . . . . . 8  |-  ( ph  ->  S  e.  ( Base `  K ) )
211, 10dalemyeb 30383 . . . . . . . 8  |-  ( ph  ->  Y  e.  ( Base `  K ) )
22 eqid 2435 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
2322, 4, 5latjle12 14483 . . . . . . . 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 1186 . . . . . . 7  |-  ( ph  ->  ( ( P  .<_  Y  /\  S  .<_  Y )  <-> 
( P  .\/  S
)  .<_  Y ) )
2524adantr 452 . . . . . 6  |-  ( (
ph  /\  Y  =  Z )  ->  (
( P  .<_  Y  /\  S  .<_  Y )  <->  ( P  .\/  S )  .<_  Y ) )
2615, 17, 25mpbi2and 888 . . . . 5  |-  ( (
ph  /\  Y  =  Z )  ->  ( P  .\/  S )  .<_  Y )
27263adant3 977 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  .\/  S
)  .<_  Y )
287dalem-ccly 30419 . . . . . . 7  |-  ( ps 
->  -.  c  .<_  Y )
2928adantl 453 . . . . . 6  |-  ( (
ph  /\  ps )  ->  -.  c  .<_  Y )
3018adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ps )  ->  K  e.  Lat )
317, 6dalemcceb 30423 . . . . . . . . 9  |-  ( ps 
->  c  e.  ( Base `  K ) )
3231adantl 453 . . . . . . . 8  |-  ( (
ph  /\  ps )  ->  c  e.  ( Base `  K ) )
337dalemddea 30418 . . . . . . . . . 10  |-  ( ps 
->  d  e.  A
)
3422, 6atbase 30024 . . . . . . . . . 10  |-  ( d  e.  A  ->  d  e.  ( Base `  K
) )
3533, 34syl 16 . . . . . . . . 9  |-  ( ps 
->  d  e.  ( Base `  K ) )
3635adantl 453 . . . . . . . 8  |-  ( (
ph  /\  ps )  ->  d  e.  ( Base `  K ) )
3722, 4, 5latlej1 14481 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  c  e.  ( Base `  K )  /\  d  e.  ( Base `  K
) )  ->  c  .<_  ( c  .\/  d
) )
3830, 32, 36, 37syl3anc 1184 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  c  .<_  ( c  .\/  d ) )
39 eqid 2435 . . . . . . . . . 10  |-  ( LLines `  K )  =  (
LLines `  K )
4022, 39llnbase 30243 . . . . . . . . 9  |-  ( ( c  .\/  d )  e.  ( LLines `  K
)  ->  ( c  .\/  d )  e.  (
Base `  K )
)
418, 40syl 16 . . . . . . . 8  |-  ( (
ph  /\  ps )  ->  ( c  .\/  d
)  e.  ( Base `  K ) )
4221adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ps )  ->  Y  e.  ( Base `  K ) )
4322, 4lattr 14477 . . . . . . . 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 1186 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  ( ( c  .<_  ( c  .\/  d
)  /\  ( c  .\/  d )  .<_  Y )  ->  c  .<_  Y ) )
4538, 44mpand 657 . . . . . 6  |-  ( (
ph  /\  ps )  ->  ( ( c  .\/  d )  .<_  Y  -> 
c  .<_  Y ) )
4629, 45mtod 170 . . . . 5  |-  ( (
ph  /\  ps )  ->  -.  ( c  .\/  d )  .<_  Y )
47463adant2 976 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  ( c  .\/  d
)  .<_  Y )
48 nbrne2 4222 . . . 4  |-  ( ( ( P  .\/  S
)  .<_  Y  /\  -.  ( c  .\/  d
)  .<_  Y )  -> 
( P  .\/  S
)  =/=  ( c 
.\/  d ) )
4927, 47, 48syl2anc 643 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  .\/  S
)  =/=  ( c 
.\/  d ) )
5049necomd 2681 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  d
)  =/=  ( P 
.\/  S ) )
51 hlatl 30095 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  AtLat )
522, 51syl 16 . . . . 5  |-  ( ph  ->  K  e.  AtLat )
5352adantr 452 . . . 4  |-  ( (
ph  /\  ps )  ->  K  e.  AtLat )
541dalempea 30360 . . . . . . 7  |-  ( ph  ->  P  e.  A )
551dalemsea 30363 . . . . . . 7  |-  ( ph  ->  S  e.  A )
5622, 5, 6hlatjcl 30101 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
572, 54, 55, 56syl3anc 1184 . . . . . 6  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
5857adantr 452 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
59 dalem21.m . . . . . 6  |-  ./\  =  ( meet `  K )
6022, 59latmcl 14472 . . . . 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 1184 . . . 4  |-  ( (
ph  /\  ps )  ->  ( ( c  .\/  d )  ./\  ( P  .\/  S ) )  e.  ( Base `  K
) )
621, 4, 5, 6, 10, 11dalemcea 30394 . . . . 5  |-  ( ph  ->  C  e.  A )
6362adantr 452 . . . 4  |-  ( (
ph  /\  ps )  ->  C  e.  A )
647dalemclccjdd 30422 . . . . . 6  |-  ( ps 
->  C  .<_  ( c 
.\/  d ) )
6564adantl 453 . . . . 5  |-  ( (
ph  /\  ps )  ->  C  .<_  ( c  .\/  d ) )
661dalemclpjs 30368 . . . . . 6  |-  ( ph  ->  C  .<_  ( P  .\/  S ) )
6766adantr 452 . . . . 5  |-  ( (
ph  /\  ps )  ->  C  .<_  ( P  .\/  S ) )
681, 6dalemceb 30372 . . . . . . 7  |-  ( ph  ->  C  e.  ( Base `  K ) )
6968adantr 452 . . . . . 6  |-  ( (
ph  /\  ps )  ->  C  e.  ( Base `  K ) )
7022, 4, 59latlem12 14499 . . . . . 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 1186 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( ( C  .<_  ( c  .\/  d )  /\  C  .<_  ( P 
.\/  S ) )  <-> 
C  .<_  ( ( c 
.\/  d )  ./\  ( P  .\/  S ) ) ) )
7265, 67, 71mpbi2and 888 . . . 4  |-  ( (
ph  /\  ps )  ->  C  .<_  ( (
c  .\/  d )  ./\  ( P  .\/  S
) ) )
73 eqid 2435 . . . . 5  |-  ( 0.
`  K )  =  ( 0. `  K
)
7422, 4, 73, 6atlen0 30045 . . . 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 1187 . . 3  |-  ( (
ph  /\  ps )  ->  ( ( c  .\/  d )  ./\  ( P  .\/  S ) )  =/=  ( 0. `  K ) )
76753adant2 976 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  d )  ./\  ( P  .\/  S ) )  =/=  ( 0. `  K ) )
7759, 73, 6, 392llnmat 30258 . 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 1192 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  d )  ./\  ( P  .\/  S ) )  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   lecple 13528   joincjn 14393   meetcmee 14394   0.cp0 14458   Latclat 14466   Atomscatm 29998   AtLatcal 29999   HLchlt 30085   LLinesclln 30225   LPlanesclpl 30226
This theorem is referenced by:  dalem22  30429
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-lat 14467  df-clat 14529  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-llines 30232  df-lplanes 30233
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