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Theorem 4at2 30413
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 30412 . 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 30163 . . . . . 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 2438 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
98, 2, 3hlatjcl 30166 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
1093ad2ant1 979 . . . . 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 991 . . . . . 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 30089 . . . . . 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 992 . . . . . 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 30089 . . . . . 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 14526 . . . . 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 1187 . . . 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 993 . . . . . 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 994 . . . . . 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 30166 . . . . . 6  |-  ( ( K  e.  HL  /\  T  e.  A  /\  U  e.  A )  ->  ( T  .\/  U
)  e.  ( Base `  K ) )
225, 19, 20, 21syl3anc 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
)  e.  ( Base `  K ) )
23 simp32 995 . . . . . 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 30089 . . . . . 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 996 . . . . . 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 30089 . . . . . 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 14526 . . . . 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 1187 . . . 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 4227 . . 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 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 ) ) ) )
3318, 30eqeq12d 2452 . . 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 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 ) ) ) )
354, 32, 343bitr4d 278 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 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   lecple 13538   joincjn 14403   Latclat 14476   Atomscatm 30063   HLchlt 30150
This theorem is referenced by:  lplncvrlvol2  30414
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
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 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-lat 14477  df-clat 14539  df-oposet 29976  df-ol 29978  df-oml 29979  df-covers 30066  df-ats 30067  df-atl 30098  df-cvlat 30122  df-hlat 30151  df-llines 30297  df-lplanes 30298  df-lvols 30299
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