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Theorem psubcliN 35745
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 35744 . 2 (𝐾𝐷 → (𝑋𝐶 ↔ (𝑋𝐴 ∧ ( ‘( 𝑋)) = 𝑋)))
54biimpa 462 1 ((𝐾𝐷𝑋𝐶) → (𝑋𝐴 ∧ ( ‘( 𝑋)) = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wss 3723  cfv 6030  Atomscatm 35070  𝑃cpolN 35709  PSubClcpscN 35741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fv 6038  df-psubclN 35742
This theorem is referenced by:  psubclsubN  35747  psubclssatN  35748
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