Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  psubcli2N Structured version   Visualization version   GIF version

Theorem psubcli2N 37596
Description: Property of a closed projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
psubcli2.p = (⊥𝑃𝐾)
psubcli2.c 𝐶 = (PSubCl‘𝐾)
Assertion
Ref Expression
psubcli2N ((𝐾𝐷𝑋𝐶) → ( ‘( 𝑋)) = 𝑋)

Proof of Theorem psubcli2N
StepHypRef Expression
1 eqid 2738 . . 3 (Atoms‘𝐾) = (Atoms‘𝐾)
2 psubcli2.p . . 3 = (⊥𝑃𝐾)
3 psubcli2.c . . 3 𝐶 = (PSubCl‘𝐾)
41, 2, 3ispsubclN 37594 . 2 (𝐾𝐷 → (𝑋𝐶 ↔ (𝑋 ⊆ (Atoms‘𝐾) ∧ ( ‘( 𝑋)) = 𝑋)))
54simplbda 503 1 ((𝐾𝐷𝑋𝐶) → ( ‘( 𝑋)) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2114  wss 3843  cfv 6339  Atomscatm 36920  𝑃cpolN 37559  PSubClcpscN 37591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-iota 6297  df-fun 6341  df-fv 6347  df-psubclN 37592
This theorem is referenced by:  psubclsubN  37597  pmapidclN  37599  poml6N  37612  osumcllem3N  37615  osumclN  37624  pmapojoinN  37625  pexmidN  37626  pexmidlem6N  37632
  Copyright terms: Public domain W3C validator