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Theorem lvoli3 29766
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 959 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  X  e.  P
)
2 simpl3 960 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  Q  e.  A
)
3 simpr 447 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  -.  Q  .<_  X )
4 eqidd 2284 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  ( X  .\/  Q )  =  ( X 
.\/  Q ) )
5 breq2 4027 . . . . . 6  |-  ( y  =  X  ->  (
r  .<_  y  <->  r  .<_  X ) )
65notbid 285 . . . . 5  |-  ( y  =  X  ->  ( -.  r  .<_  y  <->  -.  r  .<_  X ) )
7 oveq1 5865 . . . . . 6  |-  ( y  =  X  ->  (
y  .\/  r )  =  ( X  .\/  r ) )
87eqeq2d 2294 . . . . 5  |-  ( y  =  X  ->  (
( X  .\/  Q
)  =  ( y 
.\/  r )  <->  ( X  .\/  Q )  =  ( X  .\/  r ) ) )
96, 8anbi12d 691 . . . 4  |-  ( y  =  X  ->  (
( -.  r  .<_  y  /\  ( X  .\/  Q )  =  ( y 
.\/  r ) )  <-> 
( -.  r  .<_  X  /\  ( X  .\/  Q )  =  ( X 
.\/  r ) ) ) )
10 breq1 4026 . . . . . 6  |-  ( r  =  Q  ->  (
r  .<_  X  <->  Q  .<_  X ) )
1110notbid 285 . . . . 5  |-  ( r  =  Q  ->  ( -.  r  .<_  X  <->  -.  Q  .<_  X ) )
12 oveq2 5866 . . . . . 6  |-  ( r  =  Q  ->  ( X  .\/  r )  =  ( X  .\/  Q
) )
1312eqeq2d 2294 . . . . 5  |-  ( r  =  Q  ->  (
( X  .\/  Q
)  =  ( X 
.\/  r )  <->  ( X  .\/  Q )  =  ( X  .\/  Q ) ) )
1411, 13anbi12d 691 . . . 4  |-  ( r  =  Q  ->  (
( -.  r  .<_  X  /\  ( X  .\/  Q )  =  ( X 
.\/  r ) )  <-> 
( -.  Q  .<_  X  /\  ( X  .\/  Q )  =  ( X 
.\/  Q ) ) ) )
159, 14rspc2ev 2892 . . 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 1186 . 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 958 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  K  e.  HL )
18 hllat 29553 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
1917, 18syl 15 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  K  e.  Lat )
20 eqid 2283 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
21 lvoli3.p . . . . . 6  |-  P  =  ( LPlanes `  K )
2220, 21lplnbase 29723 . . . . 5  |-  ( X  e.  P  ->  X  e.  ( Base `  K
) )
231, 22syl 15 . . . 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 29479 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
262, 25syl 15 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Q  e.  A )  /\  -.  Q  .<_  X )  ->  Q  e.  (
Base `  K )
)
27 lvoli3.j . . . . 5  |-  .\/  =  ( join `  K )
2820, 27latjcl 14156 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( X  .\/  Q )  e.  ( Base `  K
) )
2919, 23, 26, 28syl3anc 1182 . . 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 29765 . . 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 642 . 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 223 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   Latclat 14151   Atomscatm 29453   HLchlt 29540   LPlanesclpl 29681   LVolsclvol 29682
This theorem is referenced by:  dalem9  29861  dalem39  29900
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-lplanes 29688  df-lvols 29689
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