Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  2atm Structured version   Unicode version

Theorem 2atm 30498
Description: An atom majorized by two different atom joins (which could be atoms or lines) is equal to their intersection. (Contributed by NM, 30-Jun-2013.)
Hypotheses
Ref Expression
2atm.l  |-  .<_  =  ( le `  K )
2atm.j  |-  .\/  =  ( join `  K )
2atm.m  |-  ./\  =  ( meet `  K )
2atm.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
2atm  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  T  =  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) ) )

Proof of Theorem 2atm
StepHypRef Expression
1 simp31 994 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  T  .<_  ( P  .\/  Q ) )
2 simp32 995 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  T  .<_  ( R  .\/  S ) )
3 simp11 988 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  K  e.  HL )
4 hllat 30335 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
53, 4syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  K  e.  Lat )
6 simp23 993 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  T  e.  A )
7 eqid 2443 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
8 2atm.a . . . . . 6  |-  A  =  ( Atoms `  K )
97, 8atbase 30261 . . . . 5  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
106, 9syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  T  e.  ( Base `  K )
)
11 simp12 989 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  P  e.  A )
127, 8atbase 30261 . . . . . 6  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
1311, 12syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  P  e.  ( Base `  K )
)
14 simp13 990 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  Q  e.  A )
157, 8atbase 30261 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
1614, 15syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  Q  e.  ( Base `  K )
)
17 2atm.j . . . . . 6  |-  .\/  =  ( join `  K )
187, 17latjcl 14517 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
195, 13, 16, 18syl3anc 1185 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  ( P  .\/  Q )  e.  (
Base `  K )
)
20 simp21 991 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  R  e.  A )
21 simp22 992 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  S  e.  A )
227, 17, 8hlatjcl 30338 . . . . 5  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  ( Base `  K ) )
233, 20, 21, 22syl3anc 1185 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  ( R  .\/  S )  e.  (
Base `  K )
)
24 2atm.l . . . . 5  |-  .<_  =  ( le `  K )
25 2atm.m . . . . 5  |-  ./\  =  ( meet `  K )
267, 24, 25latlem12 14545 . . . 4  |-  ( ( K  e.  Lat  /\  ( T  e.  ( Base `  K )  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  ( R  .\/  S )  e.  (
Base `  K )
) )  ->  (
( T  .<_  ( P 
.\/  Q )  /\  T  .<_  ( R  .\/  S ) )  <->  T  .<_  ( ( P  .\/  Q
)  ./\  ( R  .\/  S ) ) ) )
275, 10, 19, 23, 26syl13anc 1187 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  ( ( T  .<_  ( P  .\/  Q )  /\  T  .<_  ( R  .\/  S ) )  <->  T  .<_  ( ( P  .\/  Q ) 
./\  ( R  .\/  S ) ) ) )
281, 2, 27mpbi2and 889 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  T  .<_  ( ( P  .\/  Q
)  ./\  ( R  .\/  S ) ) )
29 hlatl 30332 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
303, 29syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  K  e.  AtLat
)
317, 25latmcl 14518 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  ( R  .\/  S )  e.  (
Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  e.  ( Base `  K ) )
325, 19, 23, 31syl3anc 1185 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  e.  ( Base `  K ) )
33 eqid 2443 . . . . . . 7  |-  ( 0.
`  K )  =  ( 0. `  K
)
347, 24, 33, 8atlen0 30282 . . . . . 6  |-  ( ( ( K  e.  AtLat  /\  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  e.  ( Base `  K
)  /\  T  e.  A )  /\  T  .<_  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) ) )  ->  ( ( P 
.\/  Q )  ./\  ( R  .\/  S ) )  =/=  ( 0.
`  K ) )
3530, 32, 6, 28, 34syl31anc 1188 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  =/=  ( 0.
`  K ) )
3635neneqd 2624 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  -.  (
( P  .\/  Q
)  ./\  ( R  .\/  S ) )  =  ( 0. `  K
) )
37 simp33 996 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  ( P  .\/  Q )  =/=  ( R  .\/  S ) )
3817, 25, 33, 82atmat0 30497 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q
)  =/=  ( R 
.\/  S ) ) )  ->  ( (
( P  .\/  Q
)  ./\  ( R  .\/  S ) )  e.  A  \/  ( ( P  .\/  Q ) 
./\  ( R  .\/  S ) )  =  ( 0. `  K ) ) )
393, 11, 14, 20, 21, 37, 38syl33anc 1200 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  ( (
( P  .\/  Q
)  ./\  ( R  .\/  S ) )  e.  A  \/  ( ( P  .\/  Q ) 
./\  ( R  .\/  S ) )  =  ( 0. `  K ) ) )
4039ord 368 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  ( -.  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  e.  A  ->  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  =  ( 0.
`  K ) ) )
4136, 40mt3d 120 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  e.  A )
4224, 8atcmp 30283 . . 3  |-  ( ( K  e.  AtLat  /\  T  e.  A  /\  (
( P  .\/  Q
)  ./\  ( R  .\/  S ) )  e.  A )  ->  ( T  .<_  ( ( P 
.\/  Q )  ./\  ( R  .\/  S ) )  <->  T  =  (
( P  .\/  Q
)  ./\  ( R  .\/  S ) ) ) )
4330, 6, 41, 42syl3anc 1185 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  ( T  .<_  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  <->  T  =  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) ) ) )
4428, 43mpbid 203 1  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  T  =  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1654    e. wcel 1728    =/= wne 2606   class class class wbr 4243   ` cfv 5489  (class class class)co 6117   Basecbs 13507   lecple 13574   joincjn 14439   meetcmee 14440   0.cp0 14504   Latclat 14512   Atomscatm 30235   AtLatcal 30236   HLchlt 30322
This theorem is referenced by:  cdlemk12  31821  cdlemk12u  31843  cdlemk47  31920
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 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-rep 4351  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736
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 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2717  df-rex 2718  df-reu 2719  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-iun 4124  df-br 4244  df-opab 4298  df-mpt 4299  df-id 4533  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-1st 6385  df-2nd 6386  df-undef 6579  df-riota 6585  df-poset 14441  df-plt 14453  df-lub 14469  df-glb 14470  df-join 14471  df-meet 14472  df-p0 14506  df-lat 14513  df-clat 14575  df-oposet 30148  df-ol 30150  df-oml 30151  df-covers 30238  df-ats 30239  df-atl 30270  df-cvlat 30294  df-hlat 30323  df-llines 30469
  Copyright terms: Public domain W3C validator