Theorem psubcli2N 40396

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 2737	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
2		psubcli2.p	. . 3 ⊢ ⊥ = (⊥𝑃‘𝐾)
3		psubcli2.c	. . 3 ⊢ 𝐶 = (PSubCl‘𝐾)
4	1, 2, 3	ispsubclN 40394	. 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 1542   ∈ wcel 2114   ⊆ wss 3890  ‘cfv 6490  Atomscatm 39720  ⊥_𝑃cpolN 40359  PSubClcpscN 40391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-iota 6446  df-fun 6492  df-fv 6498  df-psubclN 40392

This theorem is referenced by:  psubclsubN  40397  pmapidclN  40399  poml6N  40412  osumcllem3N  40415  osumclN  40424  pmapojoinN  40425  pexmidN  40426  pexmidlem6N  40432