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Theorem psubcli2N 39114
Description: Property of a closed projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
psubcli2.p βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
psubcli2.c 𝐢 = (PSubClβ€˜πΎ)
Assertion
Ref Expression
psubcli2N ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐢) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋)

Proof of Theorem psubcli2N
StepHypRef Expression
1 eqid 2731 . . 3 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
2 psubcli2.p . . 3 βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
3 psubcli2.c . . 3 𝐢 = (PSubClβ€˜πΎ)
41, 2, 3ispsubclN 39112 . 2 (𝐾 ∈ 𝐷 β†’ (𝑋 ∈ 𝐢 ↔ (𝑋 βŠ† (Atomsβ€˜πΎ) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋)))
54simplbda 499 1 ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐢) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105   βŠ† wss 3948  β€˜cfv 6543  Atomscatm 38437  βŠ₯𝑃cpolN 39077  PSubClcpscN 39109
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 39110
This theorem is referenced by:  psubclsubN  39115  pmapidclN  39117  poml6N  39130  osumcllem3N  39133  osumclN  39142  pmapojoinN  39143  pexmidN  39144  pexmidlem6N  39150
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