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Theorem llnbase 30320
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 3473 . . . 4  |-  ( X  e.  N  ->  -.  N  =  (/) )
2 llnbase.n . . . . 5  |-  N  =  ( LLines `  K )
32eqeq1i 2303 . . . 4  |-  ( N  =  (/)  <->  ( LLines `  K
)  =  (/) )
41, 3sylnib 295 . . 3  |-  ( X  e.  N  ->  -.  ( LLines `  K )  =  (/) )
5 fvprc 5535 . . 3  |-  ( -.  K  e.  _V  ->  (
LLines `  K )  =  (/) )
64, 5nsyl2 119 . 2  |-  ( X  e.  N  ->  K  e.  _V )
7 llnbase.b . . . 4  |-  B  =  ( Base `  K
)
8 eqid 2296 . . . 4  |-  (  <o  `  K )  =  ( 
<o  `  K )
9 eqid 2296 . . . 4  |-  ( Atoms `  K )  =  (
Atoms `  K )
107, 8, 9, 2islln 30317 . . 3  |-  ( K  e.  _V  ->  ( X  e.  N  <->  ( X  e.  B  /\  E. p  e.  ( Atoms `  K )
p (  <o  `  K
) X ) ) )
1110simprbda 606 . 2  |-  ( ( K  e.  _V  /\  X  e.  N )  ->  X  e.  B )
126, 11mpancom 650 1  |-  ( X  e.  N  ->  X  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   E.wrex 2557   _Vcvv 2801   (/)c0 3468   class class class wbr 4039   ` cfv 5271   Basecbs 13164    <o ccvr 30074   Atomscatm 30075   LLinesclln 30302
This theorem is referenced by:  islln2  30322  llnnleat  30324  llnneat  30325  atcvrlln2  30330  llnexatN  30332  llncmp  30333  2llnmat  30335  islpln3  30344  llnmlplnN  30350  lplnle  30351  lplnnle2at  30352  llncvrlpln2  30368  llncvrlpln  30369  2llnmj  30371  lplncmp  30373  lplnexatN  30374  lplnexllnN  30375  2llnm2N  30379  2llnm3N  30380  2llnm4  30381  2llnmeqat  30382  dalem21  30505  dalem54  30537  dalem55  30538  dalem57  30540  dalem60  30543  llnexchb2lem  30679  llnexchb2  30680  llnexch2N  30681
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-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
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-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  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-iota 5235  df-fun 5273  df-fv 5279  df-llines 30309
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