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	⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) → (𝑈‘𝑋) = 𝑋)

Proof of Theorem pclidN

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2731	. . . 4 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
2		pclid.s	. . . 4 ⊢ 𝑆 = (PSubSp‘𝐾)
3	1, 2	psubssat 38430	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
4		pclid.c	. . . 4 ⊢ 𝑈 = (PCl‘𝐾)
5	1, 2, 4	pclvalN 38566	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ (Atoms‘𝐾)) → (𝑈‘𝑋) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦})
6	3, 5	syldan 591	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) → (𝑈‘𝑋) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦})
7		intmin 4965	. . 3 ⊢ (𝑋 ∈ 𝑆 → ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} = 𝑋)
8	7	adantl 482	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) → ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} = 𝑋)
9	6, 8	eqtrd 2771	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) → (𝑈‘𝑋) = 𝑋)

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