Theorem pclidN 39899

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 2736	. . . 4 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
2		pclid.s	. . . 4 ⊢ 𝑆 = (PSubSp‘𝐾)
3	1, 2	psubssat 39757	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
4		pclid.c	. . . 4 ⊢ 𝑈 = (PCl‘𝐾)
5	1, 2, 4	pclvalN 39893	. . 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 4967	. . 3 ⊢ (𝑋 ∈ 𝑆 → ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} = 𝑋)
8	7	adantl 481	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) → ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} = 𝑋)
9	6, 8	eqtrd 2776	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) → (𝑈‘𝑋) = 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  {crab 3435   ⊆ wss 3950  ∩ cint 4945  ‘cfv 6560  Atomscatm 39265  PSubSpcpsubsp 39499  PClcpclN 39890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-psubsp 39506  df-pclN 39891

This theorem is referenced by:  pclbtwnN  39900  pclunN  39901  pclun2N  39902  pclfinN  39903  pclss2polN  39924  pclfinclN  39953