Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  islpln Unicode version

Theorem islpln 29719
Description: The predicate "is a lattice plane". (Contributed by NM, 16-Jun-2012.)
Hypotheses
Ref Expression
lplnset.b  |-  B  =  ( Base `  K
)
lplnset.c  |-  C  =  (  <o  `  K )
lplnset.n  |-  N  =  ( LLines `  K )
lplnset.p  |-  P  =  ( LPlanes `  K )
Assertion
Ref Expression
islpln  |-  ( K  e.  A  ->  ( X  e.  P  <->  ( X  e.  B  /\  E. y  e.  N  y C X ) ) )
Distinct variable groups:    y, N    y, K    y, X
Allowed substitution hints:    A( y)    B( y)    C( y)    P( y)

Proof of Theorem islpln
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 lplnset.b . . . 4  |-  B  =  ( Base `  K
)
2 lplnset.c . . . 4  |-  C  =  (  <o  `  K )
3 lplnset.n . . . 4  |-  N  =  ( LLines `  K )
4 lplnset.p . . . 4  |-  P  =  ( LPlanes `  K )
51, 2, 3, 4lplnset 29718 . . 3  |-  ( K  e.  A  ->  P  =  { x  e.  B  |  E. y  e.  N  y C x } )
65eleq2d 2350 . 2  |-  ( K  e.  A  ->  ( X  e.  P  <->  X  e.  { x  e.  B  |  E. y  e.  N  y C x } ) )
7 breq2 4027 . . . 4  |-  ( x  =  X  ->  (
y C x  <->  y C X ) )
87rexbidv 2564 . . 3  |-  ( x  =  X  ->  ( E. y  e.  N  y C x  <->  E. y  e.  N  y C X ) )
98elrab 2923 . 2  |-  ( X  e.  { x  e.  B  |  E. y  e.  N  y C x }  <->  ( X  e.  B  /\  E. y  e.  N  y C X ) )
106, 9syl6bb 252 1  |-  ( K  e.  A  ->  ( X  e.  P  <->  ( X  e.  B  /\  E. y  e.  N  y C X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   {crab 2547   class class class wbr 4023   ` cfv 5255   Basecbs 13148    <o ccvr 29452   LLinesclln 29680   LPlanesclpl 29681
This theorem is referenced by:  islpln4  29720  lplni  29721  lplnbase  29723  lplnnle2at  29730
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  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
  Copyright terms: Public domain W3C validator