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Theorem lplnbase 29723
Description: A lattice plane is a lattice element. (Contributed by NM, 17-Jun-2012.)
Hypotheses
Ref Expression
lplnbase.b  |-  B  =  ( Base `  K
)
lplnbase.p  |-  P  =  ( LPlanes `  K )
Assertion
Ref Expression
lplnbase  |-  ( X  e.  P  ->  X  e.  B )

Proof of Theorem lplnbase
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 n0i 3460 . . . 4  |-  ( X  e.  P  ->  -.  P  =  (/) )
2 lplnbase.p . . . . 5  |-  P  =  ( LPlanes `  K )
32eqeq1i 2290 . . . 4  |-  ( P  =  (/)  <->  ( LPlanes `  K
)  =  (/) )
41, 3sylnib 295 . . 3  |-  ( X  e.  P  ->  -.  ( LPlanes `  K )  =  (/) )
5 fvprc 5519 . . 3  |-  ( -.  K  e.  _V  ->  (
LPlanes `  K )  =  (/) )
64, 5nsyl2 119 . 2  |-  ( X  e.  P  ->  K  e.  _V )
7 lplnbase.b . . . 4  |-  B  =  ( Base `  K
)
8 eqid 2283 . . . 4  |-  (  <o  `  K )  =  ( 
<o  `  K )
9 eqid 2283 . . . 4  |-  ( LLines `  K )  =  (
LLines `  K )
107, 8, 9, 2islpln 29719 . . 3  |-  ( K  e.  _V  ->  ( X  e.  P  <->  ( X  e.  B  /\  E. x  e.  ( LLines `  K )
x (  <o  `  K
) X ) ) )
1110simprbda 606 . 2  |-  ( ( K  e.  _V  /\  X  e.  P )  ->  X  e.  B )
126, 11mpancom 650 1  |-  ( X  e.  P  ->  X  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788   (/)c0 3455   class class class wbr 4023   ` cfv 5255   Basecbs 13148    <o ccvr 29452   LLinesclln 29680   LPlanesclpl 29681
This theorem is referenced by:  islpln2  29725  llnmlplnN  29728  lplnnle2at  29730  lplnneat  29734  lplnnelln  29735  llncvrlpln2  29746  2lplnmN  29748  lplncmp  29751  lplnexatN  29752  lplnexllnN  29753  2llnjaN  29755  islvol3  29765  lvoli3  29766  lvolnle3at  29771  lplncvrlvol2  29804  lplncvrlvol  29805  lvolcmp  29806  2lplnm2N  29810  2lplnmj  29811  dalemyeb  29838  dalem10  29862  dalem16  29868  dalem44  29905  dalem55  29916
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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
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-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  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-iota 5219  df-fun 5257  df-fv 5263  df-lplanes 29688
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