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

Theorem 2atm 30338
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 991 . . 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 992 . . 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 985 . . . . 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 30175 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
53, 4syl 15 . . . 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 990 . . . . 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 2296 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
8 2atm.a . . . . . 6  |-  A  =  ( Atoms `  K )
97, 8atbase 30101 . . . . 5  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
106, 9syl 15 . . . 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 986 . . . . . 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 30101 . . . . . 6  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
1311, 12syl 15 . . . . 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 987 . . . . . 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 30101 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
1614, 15syl 15 . . . . 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 14172 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
195, 13, 16, 18syl3anc 1182 . . . 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 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 ) ) )  ->  R  e.  A )
21 simp22 989 . . . . 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 30178 . . . . 5  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  ( Base `  K ) )
233, 20, 21, 22syl3anc 1182 . . . 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 14200 . . . 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 1184 . . 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 887 . 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 30172 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
303, 29syl 15 . . 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 14173 . . . . . . 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 1182 . . . . . 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 2296 . . . . . . 7  |-  ( 0.
`  K )  =  ( 0. `  K
)
347, 24, 33, 8atlen0 30122 . . . . . 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 1185 . . . . 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 2475 . . . 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 993 . . . . . 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 30337 . . . . . 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 1197 . . . . 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 366 . . . 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 117 . . 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 30123 . . 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 1182 . 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 201 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 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   meetcmee 14095   0.cp0 14159   Latclat 14167   Atomscatm 30075   AtLatcal 30076   HLchlt 30162
This theorem is referenced by:  cdlemk12  31661  cdlemk12u  31683  cdlemk47  31760
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309
  Copyright terms: Public domain W3C validator