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Theorem 3atnelvolN 30481
Description: The join of 3 atoms is not a lattice volume. (Contributed by NM, 17-Jul-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
3atnelvol.j  |-  .\/  =  ( join `  K )
3atnelvol.a  |-  A  =  ( Atoms `  K )
3atnelvol.v  |-  V  =  ( LVols `  K )
Assertion
Ref Expression
3atnelvolN  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  -.  ( ( P  .\/  Q )  .\/  R )  e.  V )

Proof of Theorem 3atnelvolN
StepHypRef Expression
1 hllat 30259 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
21adantr 453 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  K  e.  Lat )
3 eqid 2442 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
4 3atnelvol.j . . . . . 6  |-  .\/  =  ( join `  K )
5 3atnelvol.a . . . . . 6  |-  A  =  ( Atoms `  K )
63, 4, 5hlatjcl 30262 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
763adant3r3 1165 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
8 simpr3 966 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  R  e.  A )
93, 5atbase 30185 . . . . 5  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
108, 9syl 16 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  R  e.  ( Base `  K
) )
113, 4latjcl 14510 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  R  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  .\/  R )  e.  ( Base `  K ) )
122, 7, 10, 11syl3anc 1185 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  R )  e.  ( Base `  K
) )
13 eqid 2442 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
143, 13latref 14513 . . 3  |-  ( ( K  e.  Lat  /\  ( ( P  .\/  Q )  .\/  R )  e.  ( Base `  K
) )  ->  (
( P  .\/  Q
)  .\/  R )
( le `  K
) ( ( P 
.\/  Q )  .\/  R ) )
152, 12, 14syl2anc 644 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  R )
( le `  K
) ( ( P 
.\/  Q )  .\/  R ) )
16 3atnelvol.v . . . . 5  |-  V  =  ( LVols `  K )
1713, 4, 5, 16lvolnle3at 30477 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  Q )  .\/  R )  e.  V )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  -.  ( ( P  .\/  Q )  .\/  R ) ( le `  K
) ( ( P 
.\/  Q )  .\/  R ) )
1817an32s 781 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  (
( P  .\/  Q
)  .\/  R )  e.  V )  ->  -.  ( ( P  .\/  Q )  .\/  R ) ( le `  K
) ( ( P 
.\/  Q )  .\/  R ) )
1918ex 425 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( ( P  .\/  Q )  .\/  R )  e.  V  ->  -.  ( ( P  .\/  Q )  .\/  R ) ( le `  K
) ( ( P 
.\/  Q )  .\/  R ) ) )
2015, 19mt2d 112 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  -.  ( ( P  .\/  Q )  .\/  R )  e.  V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727   class class class wbr 4237   ` cfv 5483  (class class class)co 6110   Basecbs 13500   lecple 13567   joincjn 14432   Latclat 14505   Atomscatm 30159   HLchlt 30246   LVolsclvol 30388
This theorem is referenced by:  2atnelvolN  30482  islvol2aN  30487
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 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-undef 6572  df-riota 6578  df-poset 14434  df-plt 14446  df-lub 14462  df-glb 14463  df-join 14464  df-meet 14465  df-p0 14499  df-lat 14506  df-clat 14568  df-oposet 30072  df-ol 30074  df-oml 30075  df-covers 30162  df-ats 30163  df-atl 30194  df-cvlat 30218  df-hlat 30247  df-llines 30393  df-lplanes 30394  df-lvols 30395
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