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Theorem lvolex3N 29652
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 2387 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
2 lvolex3.l . . . 4  |-  .<_  =  ( le `  K )
3 eqid 2387 . . . 4  |-  ( join `  K )  =  (
join `  K )
4 lvolex3.a . . . 4  |-  A  =  ( Atoms `  K )
5 lvolex3.p . . . 4  |-  P  =  ( LPlanes `  K )
61, 2, 3, 4, 5islpln2 29650 . . 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 29583 . . . . . . . 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 4157 . . . . . . . . . 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 2670 . . . . . . . 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 2775 . . . . 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 2780 . . . 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 1649    e. wcel 1717    =/= wne 2550   E.wrex 2650   class class class wbr 4153   ` cfv 5394  (class class class)co 6020   Basecbs 13396   lecple 13463   joincjn 14328   Atomscatm 29378   HLchlt 29465   LPlanesclpl 29606
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-undef 6479  df-riota 6485  df-poset 14330  df-plt 14342  df-lub 14358  df-glb 14359  df-join 14360  df-meet 14361  df-p0 14395  df-lat 14402  df-clat 14464  df-oposet 29291  df-ol 29293  df-oml 29294  df-covers 29381  df-ats 29382  df-atl 29413  df-cvlat 29437  df-hlat 29466  df-llines 29612  df-lplanes 29613
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