Theorem psubcli2N 39958

Description: Property of a closed projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
psubcli2.p	⊢ ⊥ = (⊥𝑃‘𝐾)
psubcli2.c	⊢ 𝐶 = (PSubCl‘𝐾)

Assertion

Ref	Expression
psubcli2N	⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐶) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)

Proof of Theorem psubcli2N

Step	Hyp	Ref	Expression
1		eqid 2735	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
2		psubcli2.p	. . 3 ⊢ ⊥ = (⊥𝑃‘𝐾)
3		psubcli2.c	. . 3 ⊢ 𝐶 = (PSubCl‘𝐾)
4	1, 2, 3	ispsubclN 39956	. 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝐶 ↔ (𝑋 ⊆ (Atoms‘𝐾) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)))
5	4	simplbda 499	1 ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐶) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ⊆ wss 3926  ‘cfv 6531  Atomscatm 39281  ⊥_𝑃cpolN 39921  PSubClcpscN 39953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-iota 6484  df-fun 6533  df-fv 6539  df-psubclN 39954

This theorem is referenced by:  psubclsubN  39959  pmapidclN  39961  poml6N  39974  osumcllem3N  39977  osumclN  39986  pmapojoinN  39987  pexmidN  39988  pexmidlem6N  39994