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Theorem islvol2aN 29599
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 29593 . . . . . . . 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 5907 . . . . . . . . . 10  |-  ( P  =  Q  ->  ( P  .\/  Q )  =  ( Q  .\/  Q
) )
125, 6hlatjidm 29376 . . . . . . . . . . 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 2370 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  /\  P  =  Q )  ->  ( P  .\/  Q )  =  Q )
1514oveq1d 5915 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  /\  P  =  Q )  ->  (
( P  .\/  Q
)  .\/  R )  =  ( Q  .\/  R ) )
1615oveq1d 5915 . . . . . . 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 2382 . . . . . 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 2527 . . 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 29371 . . . . . . 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 2316 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2423, 6atbase 29297 . . . . . . 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 29374 . . . . . . 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 14219 . . . . . 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 29593 . . . . . . 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 5907 . . . . . . . 8  |-  ( ( ( P  .\/  Q
)  .\/  R )  =  ( P  .\/  Q )  ->  ( (
( P  .\/  Q
)  .\/  R )  .\/  S )  =  ( ( P  .\/  Q
)  .\/  S )
)
3534eleq1d 2382 . . . . . . 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 29297 . . . . . . 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 14205 . . . . . . 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 14219 . . . . . 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 29593 . . . . . . 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 2376 . . . . . . 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 29588 . . 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 1633    e. wcel 1701    =/= wne 2479   class class class wbr 4060   ` cfv 5292  (class class class)co 5900   Basecbs 13195   lecple 13262   joincjn 14127   Latclat 14200   Atomscatm 29271   HLchlt 29358   LVolsclvol 29500
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-undef 6340  df-riota 6346  df-poset 14129  df-plt 14141  df-lub 14157  df-glb 14158  df-join 14159  df-meet 14160  df-p0 14194  df-lat 14201  df-clat 14263  df-oposet 29184  df-ol 29186  df-oml 29187  df-covers 29274  df-ats 29275  df-atl 29306  df-cvlat 29330  df-hlat 29359  df-llines 29505  df-lplanes 29506  df-lvols 29507
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