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Theorem psubcliN 39940
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 39939 . 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 1540  wcel 2108  wss 3951  cfv 6561  Atomscatm 39264  𝑃cpolN 39904  PSubClcpscN 39936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-psubclN 39937
This theorem is referenced by:  psubclsubN  39942  psubclssatN  39943
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