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Theorem 4at 33249
Description: Four atoms determine a lattice volume uniquely. Three-dimensional analogue of ps-1 33113 and 3at 33126. (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 33248 . 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 1060 . . . . . 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 33000 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
75, 6syl 17 . . . . 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 1065 . . . . . . 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 1066 . . . . . . 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 2471 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
1110, 2, 3hlatjcl 33003 . . . . . . 7  |-  ( ( K  e.  HL  /\  T  e.  A  /\  U  e.  A )  ->  ( T  .\/  U
)  e.  ( Base `  K ) )
125, 8, 9, 11syl3anc 1292 . . . . . 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 1067 . . . . . . 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 1068 . . . . . . 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 33003 . . . . . . 7  |-  ( ( K  e.  HL  /\  V  e.  A  /\  W  e.  A )  ->  ( V  .\/  W
)  e.  ( Base `  K ) )
165, 13, 14, 15syl3anc 1292 . . . . . 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 16375 . . . . . 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 1292 . . . . 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 16377 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( T  .\/  U )  .\/  ( V 
.\/  W ) )  e.  ( Base `  K
) )  ->  (
( T  .\/  U
)  .\/  ( V  .\/  W ) )  .<_  ( ( T  .\/  U )  .\/  ( V 
.\/  W ) ) )
207, 18, 19syl2anc 673 . . . 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 4398 . . . 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 230 . . 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 472 . 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 195 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   Basecbs 15199   lecple 15275   joincjn 16267   Latclat 16369   Atomscatm 32900   HLchlt 32987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-preset 16251  df-poset 16269  df-plt 16282  df-lub 16298  df-glb 16299  df-join 16300  df-meet 16301  df-p0 16363  df-lat 16370  df-clat 16432  df-oposet 32813  df-ol 32815  df-oml 32816  df-covers 32903  df-ats 32904  df-atl 32935  df-cvlat 32959  df-hlat 32988  df-llines 33134  df-lplanes 33135  df-lvols 33136
This theorem is referenced by:  4at2  33250
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