Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  psubcliN Structured version   Visualization version   GIF version

Theorem psubcliN 39920
Description: Property of a closed projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
psubclset.a 𝐴 = (Atoms‘𝐾)
psubclset.p = (⊥𝑃𝐾)
psubclset.c 𝐶 = (PSubCl‘𝐾)
Assertion
Ref Expression
psubcliN ((𝐾𝐷𝑋𝐶) → (𝑋𝐴 ∧ ( ‘( 𝑋)) = 𝑋))

Proof of Theorem psubcliN
StepHypRef Expression
1 psubclset.a . . 3 𝐴 = (Atoms‘𝐾)
2 psubclset.p . . 3 = (⊥𝑃𝐾)
3 psubclset.c . . 3 𝐶 = (PSubCl‘𝐾)
41, 2, 3ispsubclN 39919 . 2 (𝐾𝐷 → (𝑋𝐶 ↔ (𝑋𝐴 ∧ ( ‘( 𝑋)) = 𝑋)))
54biimpa 476 1 ((𝐾𝐷𝑋𝐶) → (𝑋𝐴 ∧ ( ‘( 𝑋)) = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  wss 3962  cfv 6562  Atomscatm 39244  𝑃cpolN 39884  PSubClcpscN 39916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570  df-psubclN 39917
This theorem is referenced by:  psubclsubN  39922  psubclssatN  39923
  Copyright terms: Public domain W3C validator