Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lvolex3N Structured version   Unicode version

Theorem lvolex3N 30272
Description: There is an atom outside of a lattice plane i.e. a 3-dimensional lattice volume exists. (Contributed by NM, 28-Jul-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
lvolex3.l  |-  .<_  =  ( le `  K )
lvolex3.a  |-  A  =  ( Atoms `  K )
lvolex3.p  |-  P  =  ( LPlanes `  K )
Assertion
Ref Expression
lvolex3N  |-  ( ( K  e.  HL  /\  X  e.  P )  ->  E. q  e.  A  -.  q  .<_  X )
Distinct variable groups:    A, q    K, q    .<_ , q    X, q
Allowed substitution hint:    P( q)

Proof of Theorem lvolex3N
Dummy variables  r 
s  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2435 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
2 lvolex3.l . . . 4  |-  .<_  =  ( le `  K )
3 eqid 2435 . . . 4  |-  ( join `  K )  =  (
join `  K )
4 lvolex3.a . . . 4  |-  A  =  ( Atoms `  K )
5 lvolex3.p . . . 4  |-  P  =  ( LPlanes `  K )
61, 2, 3, 4, 5islpln2 30270 . . 3  |-  ( K  e.  HL  ->  ( X  e.  P  <->  ( X  e.  ( Base `  K
)  /\  E. r  e.  A  E. s  e.  A  E. t  e.  A  ( r  =/=  s  /\  -.  t  .<_  ( r ( join `  K ) s )  /\  X  =  ( ( r ( join `  K ) s ) ( join `  K
) t ) ) ) ) )
7 simp1l 981 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( r  e.  A  /\  s  e.  A
) )  /\  t  e.  A  /\  (
r  =/=  s  /\  -.  t  .<_  ( r ( join `  K
) s )  /\  X  =  ( (
r ( join `  K
) s ) (
join `  K )
t ) ) )  ->  K  e.  HL )
8 simp1rl 1022 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( r  e.  A  /\  s  e.  A
) )  /\  t  e.  A  /\  (
r  =/=  s  /\  -.  t  .<_  ( r ( join `  K
) s )  /\  X  =  ( (
r ( join `  K
) s ) (
join `  K )
t ) ) )  ->  r  e.  A
)
9 simp1rr 1023 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( r  e.  A  /\  s  e.  A
) )  /\  t  e.  A  /\  (
r  =/=  s  /\  -.  t  .<_  ( r ( join `  K
) s )  /\  X  =  ( (
r ( join `  K
) s ) (
join `  K )
t ) ) )  ->  s  e.  A
)
10 simp2 958 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( r  e.  A  /\  s  e.  A
) )  /\  t  e.  A  /\  (
r  =/=  s  /\  -.  t  .<_  ( r ( join `  K
) s )  /\  X  =  ( (
r ( join `  K
) s ) (
join `  K )
t ) ) )  ->  t  e.  A
)
113, 2, 43dim3 30203 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( r  e.  A  /\  s  e.  A  /\  t  e.  A
) )  ->  E. q  e.  A  -.  q  .<_  ( ( r (
join `  K )
s ) ( join `  K ) t ) )
127, 8, 9, 10, 11syl13anc 1186 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( r  e.  A  /\  s  e.  A
) )  /\  t  e.  A  /\  (
r  =/=  s  /\  -.  t  .<_  ( r ( join `  K
) s )  /\  X  =  ( (
r ( join `  K
) s ) (
join `  K )
t ) ) )  ->  E. q  e.  A  -.  q  .<_  ( ( r ( join `  K
) s ) (
join `  K )
t ) )
13 simp33 995 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( r  e.  A  /\  s  e.  A
) )  /\  t  e.  A  /\  (
r  =/=  s  /\  -.  t  .<_  ( r ( join `  K
) s )  /\  X  =  ( (
r ( join `  K
) s ) (
join `  K )
t ) ) )  ->  X  =  ( ( r ( join `  K ) s ) ( join `  K
) t ) )
14 breq2 4208 . . . . . . . . . 10  |-  ( X  =  ( ( r ( join `  K
) s ) (
join `  K )
t )  ->  (
q  .<_  X  <->  q  .<_  ( ( r ( join `  K ) s ) ( join `  K
) t ) ) )
1514notbid 286 . . . . . . . . 9  |-  ( X  =  ( ( r ( join `  K
) s ) (
join `  K )
t )  ->  ( -.  q  .<_  X  <->  -.  q  .<_  ( ( r (
join `  K )
s ) ( join `  K ) t ) ) )
1615rexbidv 2718 . . . . . . . 8  |-  ( X  =  ( ( r ( join `  K
) s ) (
join `  K )
t )  ->  ( E. q  e.  A  -.  q  .<_  X  <->  E. q  e.  A  -.  q  .<_  ( ( r (
join `  K )
s ) ( join `  K ) t ) ) )
1713, 16syl 16 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( r  e.  A  /\  s  e.  A
) )  /\  t  e.  A  /\  (
r  =/=  s  /\  -.  t  .<_  ( r ( join `  K
) s )  /\  X  =  ( (
r ( join `  K
) s ) (
join `  K )
t ) ) )  ->  ( E. q  e.  A  -.  q  .<_  X  <->  E. q  e.  A  -.  q  .<_  ( ( r ( join `  K
) s ) (
join `  K )
t ) ) )
1812, 17mpbird 224 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( r  e.  A  /\  s  e.  A
) )  /\  t  e.  A  /\  (
r  =/=  s  /\  -.  t  .<_  ( r ( join `  K
) s )  /\  X  =  ( (
r ( join `  K
) s ) (
join `  K )
t ) ) )  ->  E. q  e.  A  -.  q  .<_  X )
1918rexlimdv3a 2824 . . . . 5  |-  ( ( K  e.  HL  /\  ( r  e.  A  /\  s  e.  A
) )  ->  ( E. t  e.  A  ( r  =/=  s  /\  -.  t  .<_  ( r ( join `  K
) s )  /\  X  =  ( (
r ( join `  K
) s ) (
join `  K )
t ) )  ->  E. q  e.  A  -.  q  .<_  X ) )
2019rexlimdvva 2829 . . . 4  |-  ( K  e.  HL  ->  ( E. r  e.  A  E. s  e.  A  E. t  e.  A  ( r  =/=  s  /\  -.  t  .<_  ( r ( join `  K
) s )  /\  X  =  ( (
r ( join `  K
) s ) (
join `  K )
t ) )  ->  E. q  e.  A  -.  q  .<_  X ) )
2120adantld 454 . . 3  |-  ( K  e.  HL  ->  (
( X  e.  (
Base `  K )  /\  E. r  e.  A  E. s  e.  A  E. t  e.  A  ( r  =/=  s  /\  -.  t  .<_  ( r ( join `  K
) s )  /\  X  =  ( (
r ( join `  K
) s ) (
join `  K )
t ) ) )  ->  E. q  e.  A  -.  q  .<_  X ) )
226, 21sylbid 207 . 2  |-  ( K  e.  HL  ->  ( X  e.  P  ->  E. q  e.  A  -.  q  .<_  X ) )
2322imp 419 1  |-  ( ( K  e.  HL  /\  X  e.  P )  ->  E. q  e.  A  -.  q  .<_  X )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   lecple 13528   joincjn 14393   Atomscatm 29998   HLchlt 30085   LPlanesclpl 30226
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-lat 14467  df-clat 14529  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-llines 30232  df-lplanes 30233
  Copyright terms: Public domain W3C validator