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Theorem 3atnelvolN 30557
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 30335 . . . 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 2443 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
4 3atnelvol.j . . . . . 6  |-  .\/  =  ( join `  K )
5 3atnelvol.a . . . . . 6  |-  A  =  ( Atoms `  K )
63, 4, 5hlatjcl 30338 . . . . 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 30261 . . . . 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 14517 . . . 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 2443 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
143, 13latref 14520 . . 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 30553 . . . 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 1654    e. wcel 1728   class class class wbr 4243   ` cfv 5489  (class class class)co 6117   Basecbs 13507   lecple 13574   joincjn 14439   Latclat 14512   Atomscatm 30235   HLchlt 30322   LVolsclvol 30464
This theorem is referenced by:  2atnelvolN  30558  islvol2aN  30563
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 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-rep 4351  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736
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 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2717  df-rex 2718  df-reu 2719  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-iun 4124  df-br 4244  df-opab 4298  df-mpt 4299  df-id 4533  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-1st 6385  df-2nd 6386  df-undef 6579  df-riota 6585  df-poset 14441  df-plt 14453  df-lub 14469  df-glb 14470  df-join 14471  df-meet 14472  df-p0 14506  df-lat 14513  df-clat 14575  df-oposet 30148  df-ol 30150  df-oml 30151  df-covers 30238  df-ats 30239  df-atl 30270  df-cvlat 30294  df-hlat 30323  df-llines 30469  df-lplanes 30470  df-lvols 30471
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