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Theorem psubcli2N 39276
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 2731 . . 3 (Atoms‘𝐾) = (Atoms‘𝐾)
2 psubcli2.p . . 3 = (⊥𝑃𝐾)
3 psubcli2.c . . 3 𝐶 = (PSubCl‘𝐾)
41, 2, 3ispsubclN 39274 . 2 (𝐾𝐷 → (𝑋𝐶 ↔ (𝑋 ⊆ (Atoms‘𝐾) ∧ ( ‘( 𝑋)) = 𝑋)))
54simplbda 499 1 ((𝐾𝐷𝑋𝐶) → ( ‘( 𝑋)) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2105  wss 3948  cfv 6543  Atomscatm 38599  𝑃cpolN 39239  PSubClcpscN 39271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-psubclN 39272
This theorem is referenced by:  psubclsubN  39277  pmapidclN  39279  poml6N  39292  osumcllem3N  39295  osumclN  39304  pmapojoinN  39305  pexmidN  39306  pexmidlem6N  39312
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