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Theorem islvol2aN 29781
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 simpl1 958 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  K  e.  HL )
2 simpl3 960 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  Q  e.  A )
3 simprl 732 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  R  e.  A )
4 simprr 733 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  S  e.  A )
5 islvol2a.j . . . . . . . . 9  |-  .\/  =  ( join `  K )
6 islvol2a.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
7 islvol2a.v . . . . . . . . 9  |-  V  =  ( LVols `  K )
85, 6, 73atnelvolN 29775 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  -.  ( ( Q  .\/  R )  .\/  S )  e.  V )
91, 2, 3, 4, 8syl13anc 1184 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  -.  ( ( Q  .\/  R )  .\/  S )  e.  V )
109adantr 451 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  /\  P  =  Q )  ->  -.  ( ( Q  .\/  R )  .\/  S )  e.  V )
11 oveq1 5865 . . . . . . . . . 10  |-  ( P  =  Q  ->  ( P  .\/  Q )  =  ( Q  .\/  Q
) )
125, 6hlatjidm 29558 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( Q  .\/  Q
)  =  Q )
131, 2, 12syl2anc 642 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  ( Q  .\/  Q )  =  Q )
1411, 13sylan9eqr 2337 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  /\  P  =  Q )  ->  ( P  .\/  Q )  =  Q )
1514oveq1d 5873 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  /\  P  =  Q )  ->  (
( P  .\/  Q
)  .\/  R )  =  ( Q  .\/  R ) )
1615oveq1d 5873 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  /\  P  =  Q )  ->  (
( ( P  .\/  Q )  .\/  R ) 
.\/  S )  =  ( ( Q  .\/  R )  .\/  S ) )
1716eleq1d 2349 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  /\  P  =  Q )  ->  (
( ( ( P 
.\/  Q )  .\/  R )  .\/  S )  e.  V  <->  ( ( Q  .\/  R )  .\/  S )  e.  V ) )
1810, 17mtbird 292 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  /\  P  =  Q )  ->  -.  ( ( ( P 
.\/  Q )  .\/  R )  .\/  S )  e.  V )
1918ex 423 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  ( P  =  Q  ->  -.  ( ( ( P 
.\/  Q )  .\/  R )  .\/  S )  e.  V ) )
2019necon2ad 2494 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  (
( ( ( P 
.\/  Q )  .\/  R )  .\/  S )  e.  V  ->  P  =/=  Q ) )
21 hllat 29553 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
221, 21syl 15 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  K  e.  Lat )
23 eqid 2283 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2423, 6atbase 29479 . . . . . . 7  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
2524ad2antrl 708 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  R  e.  ( Base `  K
) )
2623, 5, 6hlatjcl 29556 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
2726adantr 451 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
28 islvol2a.l . . . . . . 7  |-  .<_  =  ( le `  K )
2923, 28, 5latleeqj2 14170 . . . . . 6  |-  ( ( K  e.  Lat  /\  R  e.  ( Base `  K )  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  ->  ( R  .<_  ( P  .\/  Q )  <->  ( ( P 
.\/  Q )  .\/  R )  =  ( P 
.\/  Q ) ) )
3022, 25, 27, 29syl3anc 1182 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  ( R  .<_  ( P  .\/  Q )  <->  ( ( P 
.\/  Q )  .\/  R )  =  ( P 
.\/  Q ) ) )
31 simpl2 959 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  P  e.  A )
325, 6, 73atnelvolN 29775 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  S  e.  A
) )  ->  -.  ( ( P  .\/  Q )  .\/  S )  e.  V )
331, 31, 2, 4, 32syl13anc 1184 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  -.  ( ( P  .\/  Q )  .\/  S )  e.  V )
34 oveq1 5865 . . . . . . . 8  |-  ( ( ( P  .\/  Q
)  .\/  R )  =  ( P  .\/  Q )  ->  ( (
( P  .\/  Q
)  .\/  R )  .\/  S )  =  ( ( P  .\/  Q
)  .\/  S )
)
3534eleq1d 2349 . . . . . . 7  |-  ( ( ( P  .\/  Q
)  .\/  R )  =  ( P  .\/  Q )  ->  ( (
( ( P  .\/  Q )  .\/  R ) 
.\/  S )  e.  V  <->  ( ( P 
.\/  Q )  .\/  S )  e.  V ) )
3635notbid 285 . . . . . 6  |-  ( ( ( P  .\/  Q
)  .\/  R )  =  ( P  .\/  Q )  ->  ( -.  ( ( ( P 
.\/  Q )  .\/  R )  .\/  S )  e.  V  <->  -.  (
( P  .\/  Q
)  .\/  S )  e.  V ) )
3733, 36syl5ibrcom 213 . . . . 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 ) )
3830, 37sylbid 206 . . . 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 ) )
3938con2d 107 . . 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 ) ) )
4023, 6atbase 29479 . . . . . . 7  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
4140ad2antll 709 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  S  e.  ( Base `  K
) )
4223, 5latjcl 14156 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  R  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  .\/  R )  e.  ( Base `  K ) )
4322, 27, 25, 42syl3anc 1182 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  R )  e.  ( Base `  K
) )
4423, 28, 5latleeqj2 14170 . . . . . 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 ) ) )
4522, 41, 43, 44syl3anc 1182 . . . . 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 ) ) )
465, 6, 73atnelvolN 29775 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  -.  ( ( P  .\/  Q )  .\/  R )  e.  V )
471, 31, 2, 3, 46syl13anc 1184 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  -.  ( ( P  .\/  Q )  .\/  R )  e.  V )
48 eleq1 2343 . . . . . . 7  |-  ( ( ( ( P  .\/  Q )  .\/  R ) 
.\/  S )  =  ( ( P  .\/  Q )  .\/  R )  ->  ( ( ( ( P  .\/  Q
)  .\/  R )  .\/  S )  e.  V  <->  ( ( P  .\/  Q
)  .\/  R )  e.  V ) )
4948notbid 285 . . . . . 6  |-  ( ( ( ( P  .\/  Q )  .\/  R ) 
.\/  S )  =  ( ( P  .\/  Q )  .\/  R )  ->  ( -.  (
( ( P  .\/  Q )  .\/  R ) 
.\/  S )  e.  V  <->  -.  ( ( P  .\/  Q )  .\/  R )  e.  V ) )
5047, 49syl5ibrcom 213 . . . . 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 ) )
5145, 50sylbid 206 . . . 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 ) )
5251con2d 107 . . 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 ) ) )
5320, 39, 523jcad 1133 . 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 ) ) ) )
5428, 5, 6, 7lvoli2 29770 . . 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 )
55543expia 1153 . 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 ) )
5653, 55impbid 183 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   Latclat 14151   Atomscatm 29453   HLchlt 29540   LVolsclvol 29682
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689
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