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Theorem 4at2 33561
Description: Four atoms determine a lattice volume uniquely. (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
4at2  |-  ( ( ( ( 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 4at2
StepHypRef Expression
1 4at.l . . 3  |-  .<_  =  ( le `  K )
2 4at.j . . 3  |-  .\/  =  ( join `  K )
3 4at.a . . 3  |-  A  =  ( Atoms `  K )
41, 2, 34at 33560 . 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 1018 . . . . . 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 33311 . . . . . 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 eqid 2451 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
98, 2, 3hlatjcl 33314 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
1093ad2ant1 1009 . . . . 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 ) )  -> 
( P  .\/  Q
)  e.  ( Base `  K ) )
11 simp21 1021 . . . . . 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 ) )  ->  R  e.  A )
128, 3atbase 33237 . . . . . 6  |-  ( R  e.  A  ->  R  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
)  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A ) )  ->  R  e.  ( Base `  K ) )
14 simp22 1022 . . . . . 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 ) )  ->  S  e.  A )
158, 3atbase 33237 . . . . . 6  |-  ( S  e.  A  ->  S  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
)  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A ) )  ->  S  e.  ( Base `  K ) )
178, 2latjass 15364 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( P  .\/  Q )  e.  ( Base `  K )  /\  R  e.  ( Base `  K
)  /\  S  e.  ( Base `  K )
) )  ->  (
( ( P  .\/  Q )  .\/  R ) 
.\/  S )  =  ( ( P  .\/  Q )  .\/  ( R 
.\/  S ) ) )
187, 10, 13, 16, 17syl13anc 1221 . . . 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 ) )  -> 
( ( ( P 
.\/  Q )  .\/  R )  .\/  S )  =  ( ( P 
.\/  Q )  .\/  ( R  .\/  S ) ) )
19 simp23 1023 . . . . . 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  e.  A )
20 simp31 1024 . . . . . 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 ) )  ->  U  e.  A )
218, 2, 3hlatjcl 33314 . . . . . 6  |-  ( ( K  e.  HL  /\  T  e.  A  /\  U  e.  A )  ->  ( T  .\/  U
)  e.  ( Base `  K ) )
225, 19, 20, 21syl3anc 1219 . . . . 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
)  e.  ( Base `  K ) )
23 simp32 1025 . . . . . 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  e.  A )
248, 3atbase 33237 . . . . . 6  |-  ( V  e.  A  ->  V  e.  ( Base `  K
) )
2523, 24syl 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 ) )  ->  V  e.  ( Base `  K ) )
26 simp33 1026 . . . . . 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 ) )  ->  W  e.  A )
278, 3atbase 33237 . . . . . 6  |-  ( W  e.  A  ->  W  e.  ( Base `  K
) )
2826, 27syl 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 ) )  ->  W  e.  ( Base `  K ) )
298, 2latjass 15364 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( T  .\/  U )  e.  ( Base `  K )  /\  V  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
) )  ->  (
( ( T  .\/  U )  .\/  V ) 
.\/  W )  =  ( ( T  .\/  U )  .\/  ( V 
.\/  W ) ) )
307, 22, 25, 28, 29syl13anc 1221 . . . 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 ) ) )
3118, 30breq12d 4400 . . 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 ) ) ) )
3231adantr 465 . 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 ) ) ) )
3318, 30eqeq12d 2472 . . 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 ) ) ) )
3433adantr 465 . 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 ) ) ) )
354, 32, 343bitr4d 285 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 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2642   class class class wbr 4387   ` cfv 5513  (class class class)co 6187   Basecbs 14273   lecple 14344   joincjn 15213   Latclat 15314   Atomscatm 33211   HLchlt 33298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-poset 15215  df-plt 15227  df-lub 15243  df-glb 15244  df-join 15245  df-meet 15246  df-p0 15308  df-lat 15315  df-clat 15377  df-oposet 33124  df-ol 33126  df-oml 33127  df-covers 33214  df-ats 33215  df-atl 33246  df-cvlat 33270  df-hlat 33299  df-llines 33445  df-lplanes 33446  df-lvols 33447
This theorem is referenced by:  lplncvrlvol2  33562
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