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Theorem lvoli3 30375
Description: Condition implying a 3-dim lattice volume. (Contributed by NM, 2-Aug-2012.)
Hypotheses
Ref Expression
lvoli3.l  |-  .<_  =  ( le `  K )
lvoli3.j  |-  .\/  =  ( join `  K )
lvoli3.a  |-  A  =  ( Atoms `  K )
lvoli3.p  |-  P  =  ( LPlanes `  K )
lvoli3.v  |-  V  =  ( LVols `  K )
Assertion
Ref Expression
lvoli3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  ( X  .\/  Q )  e.  V )

Proof of Theorem lvoli3
Dummy variables  y 
r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl2 962 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  X  e.  P
)
2 simpl3 963 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  Q  e.  A
)
3 simpr 449 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  -.  Q  .<_  X )
4 eqidd 2438 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  ( X  .\/  Q )  =  ( X 
.\/  Q ) )
5 breq2 4217 . . . . . 6  |-  ( y  =  X  ->  (
r  .<_  y  <->  r  .<_  X ) )
65notbid 287 . . . . 5  |-  ( y  =  X  ->  ( -.  r  .<_  y  <->  -.  r  .<_  X ) )
7 oveq1 6089 . . . . . 6  |-  ( y  =  X  ->  (
y  .\/  r )  =  ( X  .\/  r ) )
87eqeq2d 2448 . . . . 5  |-  ( y  =  X  ->  (
( X  .\/  Q
)  =  ( y 
.\/  r )  <->  ( X  .\/  Q )  =  ( X  .\/  r ) ) )
96, 8anbi12d 693 . . . 4  |-  ( y  =  X  ->  (
( -.  r  .<_  y  /\  ( X  .\/  Q )  =  ( y 
.\/  r ) )  <-> 
( -.  r  .<_  X  /\  ( X  .\/  Q )  =  ( X 
.\/  r ) ) ) )
10 breq1 4216 . . . . . 6  |-  ( r  =  Q  ->  (
r  .<_  X  <->  Q  .<_  X ) )
1110notbid 287 . . . . 5  |-  ( r  =  Q  ->  ( -.  r  .<_  X  <->  -.  Q  .<_  X ) )
12 oveq2 6090 . . . . . 6  |-  ( r  =  Q  ->  ( X  .\/  r )  =  ( X  .\/  Q
) )
1312eqeq2d 2448 . . . . 5  |-  ( r  =  Q  ->  (
( X  .\/  Q
)  =  ( X 
.\/  r )  <->  ( X  .\/  Q )  =  ( X  .\/  Q ) ) )
1411, 13anbi12d 693 . . . 4  |-  ( r  =  Q  ->  (
( -.  r  .<_  X  /\  ( X  .\/  Q )  =  ( X 
.\/  r ) )  <-> 
( -.  Q  .<_  X  /\  ( X  .\/  Q )  =  ( X 
.\/  Q ) ) ) )
159, 14rspc2ev 3061 . . 3  |-  ( ( X  e.  P  /\  Q  e.  A  /\  ( -.  Q  .<_  X  /\  ( X  .\/  Q )  =  ( X 
.\/  Q ) ) )  ->  E. y  e.  P  E. r  e.  A  ( -.  r  .<_  y  /\  ( X  .\/  Q )  =  ( y  .\/  r
) ) )
161, 2, 3, 4, 15syl112anc 1189 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  E. y  e.  P  E. r  e.  A  ( -.  r  .<_  y  /\  ( X  .\/  Q )  =  ( y 
.\/  r ) ) )
17 simpl1 961 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  K  e.  HL )
18 hllat 30162 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
1917, 18syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  K  e.  Lat )
20 eqid 2437 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
21 lvoli3.p . . . . . 6  |-  P  =  ( LPlanes `  K )
2220, 21lplnbase 30332 . . . . 5  |-  ( X  e.  P  ->  X  e.  ( Base `  K
) )
231, 22syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  X  e.  (
Base `  K )
)
24 lvoli3.a . . . . . 6  |-  A  =  ( Atoms `  K )
2520, 24atbase 30088 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
262, 25syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  Q  e.  (
Base `  K )
)
27 lvoli3.j . . . . 5  |-  .\/  =  ( join `  K )
2820, 27latjcl 14480 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( X  .\/  Q )  e.  ( Base `  K
) )
2919, 23, 26, 28syl3anc 1185 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  ( X  .\/  Q )  e.  ( Base `  K ) )
30 lvoli3.l . . . 4  |-  .<_  =  ( le `  K )
31 lvoli3.v . . . 4  |-  V  =  ( LVols `  K )
3220, 30, 27, 24, 21, 31islvol3 30374 . . 3  |-  ( ( K  e.  HL  /\  ( X  .\/  Q )  e.  ( Base `  K
) )  ->  (
( X  .\/  Q
)  e.  V  <->  E. y  e.  P  E. r  e.  A  ( -.  r  .<_  y  /\  ( X  .\/  Q )  =  ( y  .\/  r
) ) ) )
3317, 29, 32syl2anc 644 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  ( ( X 
.\/  Q )  e.  V  <->  E. y  e.  P  E. r  e.  A  ( -.  r  .<_  y  /\  ( X  .\/  Q )  =  ( y 
.\/  r ) ) ) )
3416, 33mpbird 225 1  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  ( X  .\/  Q )  e.  V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   E.wrex 2707   class class class wbr 4213   ` cfv 5455  (class class class)co 6082   Basecbs 13470   lecple 13537   joincjn 14402   Latclat 14475   Atomscatm 30062   HLchlt 30149   LPlanesclpl 30290   LVolsclvol 30291
This theorem is referenced by:  dalem9  30470  dalem39  30509
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-lplanes 30297  df-lvols 30298
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