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Theorem 4at 30584
Description: Four atoms determine a lattice volume uniquely. Three-dimensional analog of ps-1 30448 and 3at 30461. (Contributed by NM, 11-Jul-2012.)
Hypotheses
Ref Expression
4at.l  |-  .<_  =  ( le `  K )
4at.j  |-  .\/  =  ( join `  K )
4at.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
4at  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  (
( ( P  .\/  Q )  .\/  ( R 
.\/  S ) ) 
.<_  ( ( T  .\/  U )  .\/  ( V 
.\/  W ) )  <-> 
( ( P  .\/  Q )  .\/  ( R 
.\/  S ) )  =  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) )

Proof of Theorem 4at
StepHypRef Expression
1 4at.l . . 3  |-  .<_  =  ( le `  K )
2 4at.j . . 3  |-  .\/  =  ( join `  K )
3 4at.a . . 3  |-  A  =  ( Atoms `  K )
41, 2, 34atlem12 30583 . 2  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  (
( ( P  .\/  Q )  .\/  ( R 
.\/  S ) ) 
.<_  ( ( T  .\/  U )  .\/  ( V 
.\/  W ) )  ->  ( ( P 
.\/  Q )  .\/  ( R  .\/  S ) )  =  ( ( T  .\/  U ) 
.\/  ( V  .\/  W ) ) ) )
5 simp11 988 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A ) )  ->  K  e.  HL )
6 hllat 30335 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
75, 6syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A ) )  ->  K  e.  Lat )
8 simp23 993 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A ) )  ->  T  e.  A )
9 simp31 994 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A ) )  ->  U  e.  A )
10 eqid 2443 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
1110, 2, 3hlatjcl 30338 . . . . . . 7  |-  ( ( K  e.  HL  /\  T  e.  A  /\  U  e.  A )  ->  ( T  .\/  U
)  e.  ( Base `  K ) )
125, 8, 9, 11syl3anc 1185 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A ) )  -> 
( T  .\/  U
)  e.  ( Base `  K ) )
13 simp32 995 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A ) )  ->  V  e.  A )
14 simp33 996 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A ) )  ->  W  e.  A )
1510, 2, 3hlatjcl 30338 . . . . . . 7  |-  ( ( K  e.  HL  /\  V  e.  A  /\  W  e.  A )  ->  ( V  .\/  W
)  e.  ( Base `  K ) )
165, 13, 14, 15syl3anc 1185 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A ) )  -> 
( V  .\/  W
)  e.  ( Base `  K ) )
1710, 2latjcl 14517 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( T  .\/  U )  e.  ( Base `  K
)  /\  ( V  .\/  W )  e.  (
Base `  K )
)  ->  ( ( T  .\/  U )  .\/  ( V  .\/  W ) )  e.  ( Base `  K ) )
187, 12, 16, 17syl3anc 1185 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A ) )  -> 
( ( T  .\/  U )  .\/  ( V 
.\/  W ) )  e.  ( Base `  K
) )
1910, 1latref 14520 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( T  .\/  U )  .\/  ( V 
.\/  W ) )  e.  ( Base `  K
) )  ->  (
( T  .\/  U
)  .\/  ( V  .\/  W ) )  .<_  ( ( T  .\/  U )  .\/  ( V 
.\/  W ) ) )
207, 18, 19syl2anc 644 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A ) )  -> 
( ( T  .\/  U )  .\/  ( V 
.\/  W ) ) 
.<_  ( ( T  .\/  U )  .\/  ( V 
.\/  W ) ) )
21 breq1 4246 . . . 4  |-  ( ( ( P  .\/  Q
)  .\/  ( R  .\/  S ) )  =  ( ( T  .\/  U )  .\/  ( V 
.\/  W ) )  ->  ( ( ( P  .\/  Q ) 
.\/  ( R  .\/  S ) )  .<_  ( ( T  .\/  U ) 
.\/  ( V  .\/  W ) )  <->  ( ( T  .\/  U )  .\/  ( V  .\/  W ) )  .<_  ( ( T  .\/  U )  .\/  ( V  .\/  W ) ) ) )
2220, 21syl5ibrcom 215 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A ) )  -> 
( ( ( P 
.\/  Q )  .\/  ( R  .\/  S ) )  =  ( ( T  .\/  U ) 
.\/  ( V  .\/  W ) )  ->  (
( P  .\/  Q
)  .\/  ( R  .\/  S ) )  .<_  ( ( T  .\/  U )  .\/  ( V 
.\/  W ) ) ) )
2322adantr 453 . 2  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  (
( ( P  .\/  Q )  .\/  ( R 
.\/  S ) )  =  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  ->  ( ( P  .\/  Q )  .\/  ( R  .\/  S ) )  .<_  ( ( T  .\/  U )  .\/  ( V  .\/  W ) ) ) )
244, 23impbid 185 1  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  (
( ( P  .\/  Q )  .\/  ( R 
.\/  S ) ) 
.<_  ( ( T  .\/  U )  .\/  ( V 
.\/  W ) )  <-> 
( ( P  .\/  Q )  .\/  ( R 
.\/  S ) )  =  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ 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   Latclat 14512   Atomscatm 30235   HLchlt 30322
This theorem is referenced by:  4at2  30585
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-3or 938  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  df-lplanes 30470  df-lvols 30471
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