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Theorem psubcli2N 34039
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 2609 . . 3 (Atoms‘𝐾) = (Atoms‘𝐾)
2 psubcli2.p . . 3 = (⊥𝑃𝐾)
3 psubcli2.c . . 3 𝐶 = (PSubCl‘𝐾)
41, 2, 3ispsubclN 34037 . 2 (𝐾𝐷 → (𝑋𝐶 ↔ (𝑋 ⊆ (Atoms‘𝐾) ∧ ( ‘( 𝑋)) = 𝑋)))
54simplbda 651 1 ((𝐾𝐷𝑋𝐶) → ( ‘( 𝑋)) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  wss 3539  cfv 5790  Atomscatm 33364  𝑃cpolN 34002  PSubClcpscN 34034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-iota 5754  df-fun 5792  df-fv 5798  df-psubclN 34035
This theorem is referenced by:  psubclsubN  34040  pmapidclN  34042  poml6N  34055  osumcllem3N  34058  osumclN  34067  pmapojoinN  34068  pexmidN  34069  pexmidlem6N  34075
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