Theorem psubcli2N 37090

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 2821	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
2		psubcli2.p	. . 3 ⊢ ⊥ = (⊥𝑃‘𝐾)
3		psubcli2.c	. . 3 ⊢ 𝐶 = (PSubCl‘𝐾)
4	1, 2, 3	ispsubclN 37088	. 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝐶 ↔ (𝑋 ⊆ (Atoms‘𝐾) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)))
5	4	simplbda 502	1 ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐶) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ⊆ wss 3936  ‘cfv 6355  Atomscatm 36414  ⊥_𝑃cpolN 37053  PSubClcpscN 37085

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-iota 6314  df-fun 6357  df-fv 6363  df-psubclN 37086

This theorem is referenced by:  psubclsubN  37091  pmapidclN  37093  poml6N  37106  osumcllem3N  37109  osumclN  37118  pmapojoinN  37119  pexmidN  37120  pexmidlem6N  37126