Theorem psubcli2N 39349

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 2727	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
2		psubcli2.p	. . 3 ⊢ ⊥ = (⊥𝑃‘𝐾)
3		psubcli2.c	. . 3 ⊢ 𝐶 = (PSubCl‘𝐾)
4	1, 2, 3	ispsubclN 39347	. 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 1534   ∈ wcel 2099   ⊆ wss 3944  ‘cfv 6542  Atomscatm 38672  ⊥_𝑃cpolN 39312  PSubClcpscN 39344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-psubclN 39345

This theorem is referenced by:  psubclsubN  39350  pmapidclN  39352  poml6N  39365  osumcllem3N  39368  osumclN  39377  pmapojoinN  39378  pexmidN  39379  pexmidlem6N  39385