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Theorem lvolex3N 30349
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 2296 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
2 lvolex3.l . . . 4  |-  .<_  =  ( le `  K )
3 eqid 2296 . . . 4  |-  ( join `  K )  =  (
join `  K )
4 lvolex3.a . . . 4  |-  A  =  ( Atoms `  K )
5 lvolex3.p . . . 4  |-  P  =  ( LPlanes `  K )
61, 2, 3, 4, 5islpln2 30347 . . 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 30280 . . . . . . . 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 4043 . . . . . . . . . 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 2577 . . . . . . . 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 2682 . . . . 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 2687 . . . 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 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   Atomscatm 30075   HLchlt 30162   LPlanesclpl 30303
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310
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