Theorem pclidN 37910

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 2738	. . . 4 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
2		pclid.s	. . . 4 ⊢ 𝑆 = (PSubSp‘𝐾)
3	1, 2	psubssat 37768	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
4		pclid.c	. . . 4 ⊢ 𝑈 = (PCl‘𝐾)
5	1, 2, 4	pclvalN 37904	. . 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 4899	. . 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 2778	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) → (𝑈‘𝑋) = 𝑋)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {crab 3068   ⊆ wss 3887  ∩ cint 4879  ‘cfv 6433  Atomscatm 37277  PSubSpcpsubsp 37510  PClcpclN 37901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-psubsp 37517  df-pclN 37902

This theorem is referenced by:  pclbtwnN  37911  pclunN  37912  pclun2N  37913  pclfinN  37914  pclss2polN  37935  pclfinclN  37964