Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  psubcli2N Structured version   Visualization version   GIF version

Theorem psubcli2N 37953
Description: Property of a closed projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
psubcli2.p = (⊥𝑃𝐾)
psubcli2.c 𝐶 = (PSubCl‘𝐾)
Assertion
Ref Expression
psubcli2N ((𝐾𝐷𝑋𝐶) → ( ‘( 𝑋)) = 𝑋)

Proof of Theorem psubcli2N
StepHypRef Expression
1 eqid 2738 . . 3 (Atoms‘𝐾) = (Atoms‘𝐾)
2 psubcli2.p . . 3 = (⊥𝑃𝐾)
3 psubcli2.c . . 3 𝐶 = (PSubCl‘𝐾)
41, 2, 3ispsubclN 37951 . 2 (𝐾𝐷 → (𝑋𝐶 ↔ (𝑋 ⊆ (Atoms‘𝐾) ∧ ( ‘( 𝑋)) = 𝑋)))
54simplbda 500 1 ((𝐾𝐷𝑋𝐶) → ( ‘( 𝑋)) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  wss 3887  cfv 6433  Atomscatm 37277  𝑃cpolN 37916  PSubClcpscN 37948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-psubclN 37949
This theorem is referenced by:  psubclsubN  37954  pmapidclN  37956  poml6N  37969  osumcllem3N  37972  osumclN  37981  pmapojoinN  37982  pexmidN  37983  pexmidlem6N  37989
  Copyright terms: Public domain W3C validator