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Theorem pclidN 38572
Description: The projective subspace closure of a projective subspace is itself. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
pclid.s 𝑆 = (PSubSp‘𝐾)
pclid.c 𝑈 = (PCl‘𝐾)
Assertion
Ref Expression
pclidN ((𝐾𝑉𝑋𝑆) → (𝑈𝑋) = 𝑋)

Proof of Theorem pclidN
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eqid 2731 . . . 4 (Atoms‘𝐾) = (Atoms‘𝐾)
2 pclid.s . . . 4 𝑆 = (PSubSp‘𝐾)
31, 2psubssat 38430 . . 3 ((𝐾𝑉𝑋𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
4 pclid.c . . . 4 𝑈 = (PCl‘𝐾)
51, 2, 4pclvalN 38566 . . 3 ((𝐾𝑉𝑋 ⊆ (Atoms‘𝐾)) → (𝑈𝑋) = {𝑦𝑆𝑋𝑦})
63, 5syldan 591 . 2 ((𝐾𝑉𝑋𝑆) → (𝑈𝑋) = {𝑦𝑆𝑋𝑦})
7 intmin 4965 . . 3 (𝑋𝑆 {𝑦𝑆𝑋𝑦} = 𝑋)
87adantl 482 . 2 ((𝐾𝑉𝑋𝑆) → {𝑦𝑆𝑋𝑦} = 𝑋)
96, 8eqtrd 2771 1 ((𝐾𝑉𝑋𝑆) → (𝑈𝑋) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  {crab 3431  wss 3944   cint 4943  cfv 6532  Atomscatm 37938  PSubSpcpsubsp 38172  PClcpclN 38563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  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-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-ov 7396  df-psubsp 38179  df-pclN 38564
This theorem is referenced by:  pclbtwnN  38573  pclunN  38574  pclun2N  38575  pclfinN  38576  pclss2polN  38597  pclfinclN  38626
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