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Theorem pclssidN 35916
Description: A set of atoms is included in its projective subspace closure. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
pclss.a 𝐴 = (Atoms‘𝐾)
pclss.c 𝑈 = (PCl‘𝐾)
Assertion
Ref Expression
pclssidN ((𝐾𝑉𝑋𝐴) → 𝑋 ⊆ (𝑈𝑋))

Proof of Theorem pclssidN
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ssintub 4685 . 2 𝑋 {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋𝑦}
2 pclss.a . . 3 𝐴 = (Atoms‘𝐾)
3 eqid 2799 . . 3 (PSubSp‘𝐾) = (PSubSp‘𝐾)
4 pclss.c . . 3 𝑈 = (PCl‘𝐾)
52, 3, 4pclvalN 35911 . 2 ((𝐾𝑉𝑋𝐴) → (𝑈𝑋) = {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋𝑦})
61, 5syl5sseqr 3850 1 ((𝐾𝑉𝑋𝐴) → 𝑋 ⊆ (𝑈𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  {crab 3093  wss 3769   cint 4667  cfv 6101  Atomscatm 35284  PSubSpcpsubsp 35517  PClcpclN 35908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-psubsp 35524  df-pclN 35909
This theorem is referenced by:  pclunN  35919  pcl0bN  35944  pclfinclN  35971
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