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Theorem llnbase 30243
Description: A lattice line is a lattice element. (Contributed by NM, 16-Jun-2012.)
Hypotheses
Ref Expression
llnbase.b  |-  B  =  ( Base `  K
)
llnbase.n  |-  N  =  ( LLines `  K )
Assertion
Ref Expression
llnbase  |-  ( X  e.  N  ->  X  e.  B )

Proof of Theorem llnbase
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 n0i 3625 . . . 4  |-  ( X  e.  N  ->  -.  N  =  (/) )
2 llnbase.n . . . . 5  |-  N  =  ( LLines `  K )
32eqeq1i 2442 . . . 4  |-  ( N  =  (/)  <->  ( LLines `  K
)  =  (/) )
41, 3sylnib 296 . . 3  |-  ( X  e.  N  ->  -.  ( LLines `  K )  =  (/) )
5 fvprc 5714 . . 3  |-  ( -.  K  e.  _V  ->  (
LLines `  K )  =  (/) )
64, 5nsyl2 121 . 2  |-  ( X  e.  N  ->  K  e.  _V )
7 llnbase.b . . . 4  |-  B  =  ( Base `  K
)
8 eqid 2435 . . . 4  |-  (  <o  `  K )  =  ( 
<o  `  K )
9 eqid 2435 . . . 4  |-  ( Atoms `  K )  =  (
Atoms `  K )
107, 8, 9, 2islln 30240 . . 3  |-  ( K  e.  _V  ->  ( X  e.  N  <->  ( X  e.  B  /\  E. p  e.  ( Atoms `  K )
p (  <o  `  K
) X ) ) )
1110simprbda 607 . 2  |-  ( ( K  e.  _V  /\  X  e.  N )  ->  X  e.  B )
126, 11mpancom 651 1  |-  ( X  e.  N  ->  X  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   E.wrex 2698   _Vcvv 2948   (/)c0 3620   class class class wbr 4204   ` cfv 5446   Basecbs 13461    <o ccvr 29997   Atomscatm 29998   LLinesclln 30225
This theorem is referenced by:  islln2  30245  llnnleat  30247  llnneat  30248  atcvrlln2  30253  llnexatN  30255  llncmp  30256  2llnmat  30258  islpln3  30267  llnmlplnN  30273  lplnle  30274  lplnnle2at  30275  llncvrlpln2  30291  llncvrlpln  30292  2llnmj  30294  lplncmp  30296  lplnexatN  30297  lplnexllnN  30298  2llnm2N  30302  2llnm3N  30303  2llnm4  30304  2llnmeqat  30305  dalem21  30428  dalem54  30460  dalem55  30461  dalem57  30463  dalem60  30466  llnexchb2lem  30602  llnexchb2  30603  llnexch2N  30604
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-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
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-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  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-iota 5410  df-fun 5448  df-fv 5454  df-llines 30232
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