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Theorem islvol2aN 30390
Description: The predicate "is a lattice volume". (Contributed by NM, 16-Jul-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
islvol2a.l  |-  .<_  =  ( le `  K )
islvol2a.j  |-  .\/  =  ( join `  K )
islvol2a.a  |-  A  =  ( Atoms `  K )
islvol2a.v  |-  V  =  ( LVols `  K )
Assertion
Ref Expression
islvol2aN  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  (
( ( ( P 
.\/  Q )  .\/  R )  .\/  S )  e.  V  <->  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) ) )

Proof of Theorem islvol2aN
StepHypRef Expression
1 oveq1 6089 . . . . . . . . 9  |-  ( P  =  Q  ->  ( P  .\/  Q )  =  ( Q  .\/  Q
) )
2 simpl1 961 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  K  e.  HL )
3 simpl3 963 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  Q  e.  A )
4 islvol2a.j . . . . . . . . . . 11  |-  .\/  =  ( join `  K )
5 islvol2a.a . . . . . . . . . . 11  |-  A  =  ( Atoms `  K )
64, 5hlatjidm 30167 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( Q  .\/  Q
)  =  Q )
72, 3, 6syl2anc 644 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  ( Q  .\/  Q )  =  Q )
81, 7sylan9eqr 2491 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  /\  P  =  Q )  ->  ( P  .\/  Q )  =  Q )
98oveq1d 6097 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  /\  P  =  Q )  ->  (
( P  .\/  Q
)  .\/  R )  =  ( Q  .\/  R ) )
109oveq1d 6097 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  /\  P  =  Q )  ->  (
( ( P  .\/  Q )  .\/  R ) 
.\/  S )  =  ( ( Q  .\/  R )  .\/  S ) )
11 simprl 734 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  R  e.  A )
12 simprr 735 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  S  e.  A )
13 islvol2a.v . . . . . . . . 9  |-  V  =  ( LVols `  K )
144, 5, 133atnelvolN 30384 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  -.  ( ( Q  .\/  R )  .\/  S )  e.  V )
152, 3, 11, 12, 14syl13anc 1187 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  -.  ( ( Q  .\/  R )  .\/  S )  e.  V )
1615adantr 453 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  /\  P  =  Q )  ->  -.  ( ( Q  .\/  R )  .\/  S )  e.  V )
1710, 16eqneltrd 2530 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  /\  P  =  Q )  ->  -.  ( ( ( P 
.\/  Q )  .\/  R )  .\/  S )  e.  V )
1817ex 425 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  ( P  =  Q  ->  -.  ( ( ( P 
.\/  Q )  .\/  R )  .\/  S )  e.  V ) )
1918necon2ad 2653 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  (
( ( ( P 
.\/  Q )  .\/  R )  .\/  S )  e.  V  ->  P  =/=  Q ) )
20 hllat 30162 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
212, 20syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  K  e.  Lat )
22 eqid 2437 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2322, 5atbase 30088 . . . . . . 7  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
2423ad2antrl 710 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  R  e.  ( Base `  K
) )
2522, 4, 5hlatjcl 30165 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
2625adantr 453 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
27 islvol2a.l . . . . . . 7  |-  .<_  =  ( le `  K )
2822, 27, 4latleeqj2 14494 . . . . . 6  |-  ( ( K  e.  Lat  /\  R  e.  ( Base `  K )  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  ->  ( R  .<_  ( P  .\/  Q )  <->  ( ( P 
.\/  Q )  .\/  R )  =  ( P 
.\/  Q ) ) )
2921, 24, 26, 28syl3anc 1185 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  ( R  .<_  ( P  .\/  Q )  <->  ( ( P 
.\/  Q )  .\/  R )  =  ( P 
.\/  Q ) ) )
30 simpl2 962 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  P  e.  A )
314, 5, 133atnelvolN 30384 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  S  e.  A
) )  ->  -.  ( ( P  .\/  Q )  .\/  S )  e.  V )
322, 30, 3, 12, 31syl13anc 1187 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  -.  ( ( P  .\/  Q )  .\/  S )  e.  V )
33 oveq1 6089 . . . . . . . 8  |-  ( ( ( P  .\/  Q
)  .\/  R )  =  ( P  .\/  Q )  ->  ( (
( P  .\/  Q
)  .\/  R )  .\/  S )  =  ( ( P  .\/  Q
)  .\/  S )
)
3433eleq1d 2503 . . . . . . 7  |-  ( ( ( P  .\/  Q
)  .\/  R )  =  ( P  .\/  Q )  ->  ( (
( ( P  .\/  Q )  .\/  R ) 
.\/  S )  e.  V  <->  ( ( P 
.\/  Q )  .\/  S )  e.  V ) )
3534notbid 287 . . . . . 6  |-  ( ( ( P  .\/  Q
)  .\/  R )  =  ( P  .\/  Q )  ->  ( -.  ( ( ( P 
.\/  Q )  .\/  R )  .\/  S )  e.  V  <->  -.  (
( P  .\/  Q
)  .\/  S )  e.  V ) )
3632, 35syl5ibrcom 215 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  (
( ( P  .\/  Q )  .\/  R )  =  ( P  .\/  Q )  ->  -.  (
( ( P  .\/  Q )  .\/  R ) 
.\/  S )  e.  V ) )
3729, 36sylbid 208 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  ( R  .<_  ( P  .\/  Q )  ->  -.  (
( ( P  .\/  Q )  .\/  R ) 
.\/  S )  e.  V ) )
3837con2d 110 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  (
( ( ( P 
.\/  Q )  .\/  R )  .\/  S )  e.  V  ->  -.  R  .<_  ( P  .\/  Q ) ) )
3922, 5atbase 30088 . . . . . . 7  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
4039ad2antll 711 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  S  e.  ( Base `  K
) )
4122, 4latjcl 14480 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  R  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  .\/  R )  e.  ( Base `  K ) )
4221, 26, 24, 41syl3anc 1185 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  R )  e.  ( Base `  K
) )
4322, 27, 4latleeqj2 14494 . . . . . 6  |-  ( ( K  e.  Lat  /\  S  e.  ( Base `  K )  /\  (
( P  .\/  Q
)  .\/  R )  e.  ( Base `  K
) )  ->  ( S  .<_  ( ( P 
.\/  Q )  .\/  R )  <->  ( ( ( P  .\/  Q ) 
.\/  R )  .\/  S )  =  ( ( P  .\/  Q ) 
.\/  R ) ) )
4421, 40, 42, 43syl3anc 1185 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  ( S  .<_  ( ( P 
.\/  Q )  .\/  R )  <->  ( ( ( P  .\/  Q ) 
.\/  R )  .\/  S )  =  ( ( P  .\/  Q ) 
.\/  R ) ) )
454, 5, 133atnelvolN 30384 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  -.  ( ( P  .\/  Q )  .\/  R )  e.  V )
462, 30, 3, 11, 45syl13anc 1187 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  -.  ( ( P  .\/  Q )  .\/  R )  e.  V )
47 eleq1 2497 . . . . . . 7  |-  ( ( ( ( P  .\/  Q )  .\/  R ) 
.\/  S )  =  ( ( P  .\/  Q )  .\/  R )  ->  ( ( ( ( P  .\/  Q
)  .\/  R )  .\/  S )  e.  V  <->  ( ( P  .\/  Q
)  .\/  R )  e.  V ) )
4847notbid 287 . . . . . 6  |-  ( ( ( ( P  .\/  Q )  .\/  R ) 
.\/  S )  =  ( ( P  .\/  Q )  .\/  R )  ->  ( -.  (
( ( P  .\/  Q )  .\/  R ) 
.\/  S )  e.  V  <->  -.  ( ( P  .\/  Q )  .\/  R )  e.  V ) )
4946, 48syl5ibrcom 215 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  (
( ( ( P 
.\/  Q )  .\/  R )  .\/  S )  =  ( ( P 
.\/  Q )  .\/  R )  ->  -.  (
( ( P  .\/  Q )  .\/  R ) 
.\/  S )  e.  V ) )
5044, 49sylbid 208 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  ( S  .<_  ( ( P 
.\/  Q )  .\/  R )  ->  -.  (
( ( P  .\/  Q )  .\/  R ) 
.\/  S )  e.  V ) )
5150con2d 110 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  (
( ( ( P 
.\/  Q )  .\/  R )  .\/  S )  e.  V  ->  -.  S  .<_  ( ( P 
.\/  Q )  .\/  R ) ) )
5219, 38, 513jcad 1136 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  (
( ( ( P 
.\/  Q )  .\/  R )  .\/  S )  e.  V  ->  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) ) )
5327, 4, 5, 13lvoli2 30379 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  (
( ( P  .\/  Q )  .\/  R ) 
.\/  S )  e.  V )
54533expia 1156 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  ->  ( ( ( P  .\/  Q ) 
.\/  R )  .\/  S )  e.  V ) )
5552, 54impbid 185 1  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  (
( ( ( P 
.\/  Q )  .\/  R )  .\/  S )  e.  V  <->  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2600   class class class wbr 4213   ` cfv 5455  (class class class)co 6082   Basecbs 13470   lecple 13537   joincjn 14402   Latclat 14475   Atomscatm 30062   HLchlt 30149   LVolsclvol 30291
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 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-undef 6544  df-riota 6550  df-poset 14404  df-plt 14416  df-lub 14432  df-glb 14433  df-join 14434  df-meet 14435  df-p0 14469  df-lat 14476  df-clat 14538  df-oposet 29975  df-ol 29977  df-oml 29978  df-covers 30065  df-ats 30066  df-atl 30097  df-cvlat 30121  df-hlat 30150  df-llines 30296  df-lplanes 30297  df-lvols 30298
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