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Theorem 4at 30141
Description: Four atoms determine a lattice volume uniquely. Three-dimensional analog of ps-1 30005 and 3at 30018. (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 30140 . 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 987 . . . . . 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 29892 . . . . . 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 992 . . . . . . 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 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 ) )  ->  U  e.  A )
10 eqid 2430 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
1110, 2, 3hlatjcl 29895 . . . . . . 7  |-  ( ( K  e.  HL  /\  T  e.  A  /\  U  e.  A )  ->  ( T  .\/  U
)  e.  ( Base `  K ) )
125, 8, 9, 11syl3anc 1184 . . . . . 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 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 ) )  ->  V  e.  A )
14 simp33 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 ) )  ->  W  e.  A )
1510, 2, 3hlatjcl 29895 . . . . . . 7  |-  ( ( K  e.  HL  /\  V  e.  A  /\  W  e.  A )  ->  ( V  .\/  W
)  e.  ( Base `  K ) )
165, 13, 14, 15syl3anc 1184 . . . . . 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 14462 . . . . . 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 1184 . . . . 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 14465 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( T  .\/  U )  .\/  ( V 
.\/  W ) )  e.  ( Base `  K
) )  ->  (
( T  .\/  U
)  .\/  ( V  .\/  W ) )  .<_  ( ( T  .\/  U )  .\/  ( V 
.\/  W ) ) )
207, 18, 19syl2anc 643 . . . 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 4202 . . . 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 214 . . 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 452 . 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 184 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2593   class class class wbr 4199   ` cfv 5440  (class class class)co 6067   Basecbs 13452   lecple 13519   joincjn 14384   Latclat 14457   Atomscatm 29792   HLchlt 29879
This theorem is referenced by:  4at2  30142
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 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-nel 2596  df-ral 2697  df-rex 2698  df-reu 2699  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-iun 4082  df-br 4200  df-opab 4254  df-mpt 4255  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-1st 6335  df-2nd 6336  df-undef 6529  df-riota 6535  df-poset 14386  df-plt 14398  df-lub 14414  df-glb 14415  df-join 14416  df-meet 14417  df-p0 14451  df-lat 14458  df-clat 14520  df-oposet 29705  df-ol 29707  df-oml 29708  df-covers 29795  df-ats 29796  df-atl 29827  df-cvlat 29851  df-hlat 29880  df-llines 30026  df-lplanes 30027  df-lvols 30028
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