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

Theorem psubclssatN 35730
Description: A closed projective subspace is a set of atoms. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
psubclssat.a 𝐴 = (Atoms‘𝐾)
psubclssat.c 𝐶 = (PSubCl‘𝐾)
Assertion
Ref Expression
psubclssatN ((𝐾𝐷𝑋𝐶) → 𝑋𝐴)

Proof of Theorem psubclssatN
StepHypRef Expression
1 psubclssat.a . . 3 𝐴 = (Atoms‘𝐾)
2 eqid 2760 . . 3 (⊥𝑃𝐾) = (⊥𝑃𝐾)
3 psubclssat.c . . 3 𝐶 = (PSubCl‘𝐾)
41, 2, 3psubcliN 35727 . 2 ((𝐾𝐷𝑋𝐶) → (𝑋𝐴 ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋))
54simpld 477 1 ((𝐾𝐷𝑋𝐶) → 𝑋𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wss 3715  cfv 6049  Atomscatm 35053  𝑃cpolN 35691  PSubClcpscN 35723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-psubclN 35724
This theorem is referenced by:  pmapidclN  35731  psubclinN  35737  paddatclN  35738  pclfinclN  35739  poml6N  35744  osumcllem3N  35747  osumcllem9N  35753  osumcllem11N  35755  osumclN  35756
  Copyright terms: Public domain W3C validator