Theorem psubcli2N 39928

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 2730	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
2		psubcli2.p	. . 3 ⊢ ⊥ = (⊥𝑃‘𝐾)
3		psubcli2.c	. . 3 ⊢ 𝐶 = (PSubCl‘𝐾)
4	1, 2, 3	ispsubclN 39926	. 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 2109   ⊆ wss 3916  ‘cfv 6513  Atomscatm 39251  ⊥_𝑃cpolN 39891  PSubClcpscN 39923

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389

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 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-iota 6466  df-fun 6515  df-fv 6521  df-psubclN 39924

This theorem is referenced by:  psubclsubN  39929  pmapidclN  39931  poml6N  39944  osumcllem3N  39947  osumclN  39956  pmapojoinN  39957  pexmidN  39958  pexmidlem6N  39964