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Theorem 3atnelvolN 30114
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 29892 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
21adantr 452 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  K  e.  Lat )
3 eqid 2430 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
4 3atnelvol.j . . . . . 6  |-  .\/  =  ( join `  K )
5 3atnelvol.a . . . . . 6  |-  A  =  ( Atoms `  K )
63, 4, 5hlatjcl 29895 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
763adant3r3 1164 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
8 simpr3 965 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  R  e.  A )
93, 5atbase 29818 . . . . 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 14462 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  R  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  .\/  R )  e.  ( Base `  K ) )
122, 7, 10, 11syl3anc 1184 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  R )  e.  ( Base `  K
) )
13 eqid 2430 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
143, 13latref 14465 . . 3  |-  ( ( K  e.  Lat  /\  ( ( P  .\/  Q )  .\/  R )  e.  ( Base `  K
) )  ->  (
( P  .\/  Q
)  .\/  R )
( le `  K
) ( ( P 
.\/  Q )  .\/  R ) )
152, 12, 14syl2anc 643 . 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 30110 . . . 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 780 . . 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 424 . 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 111 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4199   ` cfv 5440  (class class class)co 6067   Basecbs 13452   lecple 13519   joincjn 14384   Latclat 14457   Atomscatm 29792   HLchlt 29879   LVolsclvol 30021
This theorem is referenced by:  2atnelvolN  30115  islvol2aN  30120
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-nel 2596  df-ral 2697  df-rex 2698  df-reu 2699  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-iun 4082  df-br 4200  df-opab 4254  df-mpt 4255  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-1st 6335  df-2nd 6336  df-undef 6529  df-riota 6535  df-poset 14386  df-plt 14398  df-lub 14414  df-glb 14415  df-join 14416  df-meet 14417  df-p0 14451  df-lat 14458  df-clat 14520  df-oposet 29705  df-ol 29707  df-oml 29708  df-covers 29795  df-ats 29796  df-atl 29827  df-cvlat 29851  df-hlat 29880  df-llines 30026  df-lplanes 30027  df-lvols 30028
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