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Theorem lvolex3N 29100
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 2283 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
2 lvolex3.l . . . 4  |-  .<_  =  ( le `  K )
3 eqid 2283 . . . 4  |-  ( join `  K )  =  (
join `  K )
4 lvolex3.a . . . 4  |-  A  =  ( Atoms `  K )
5 lvolex3.p . . . 4  |-  P  =  ( LPlanes `  K )
61, 2, 3, 4, 5islpln2 29098 . . 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 979 . . . . . . . 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 1020 . . . . . . . 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 1021 . . . . . . . 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 956 . . . . . . . 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 29031 . . . . . . . 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 1184 . . . . . . 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 993 . . . . . . . 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 4027 . . . . . . . . . 10  |-  ( X  =  ( ( r ( join `  K
) s ) (
join `  K )
t )  ->  (
q  .<_  X  <->  q  .<_  ( ( r ( join `  K ) s ) ( join `  K
) t ) ) )
1514notbid 285 . . . . . . . . 9  |-  ( X  =  ( ( r ( join `  K
) s ) (
join `  K )
t )  ->  ( -.  q  .<_  X  <->  -.  q  .<_  ( ( r (
join `  K )
s ) ( join `  K ) t ) ) )
1615rexbidv 2564 . . . . . . . 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 15 . . . . . . 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 223 . . . . . 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 2669 . . . . 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 2674 . . . 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 453 . . 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 206 . 2  |-  ( K  e.  HL  ->  ( X  e.  P  ->  E. q  e.  A  -.  q  .<_  X ) )
2322imp 418 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   Atomscatm 28826   HLchlt 28913   LPlanesclpl 29054
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 28739  df-ol 28741  df-oml 28742  df-covers 28829  df-ats 28830  df-atl 28861  df-cvlat 28885  df-hlat 28914  df-llines 29060  df-lplanes 29061
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