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Theorem atpsubN 35357
 Description: The set of all atoms is a projective subspace. Remark below Definition 15.1 of [MaedaMaeda] p. 61. (Contributed by NM, 13-Oct-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
atpsub.a 𝐴 = (Atoms‘𝐾)
atpsub.s 𝑆 = (PSubSp‘𝐾)
Assertion
Ref Expression
atpsubN (𝐾𝑉𝐴𝑆)

Proof of Theorem atpsubN
Dummy variables 𝑞 𝑝 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssid 3657 . . 3 𝐴𝐴
2 ax-1 6 . . . . 5 (𝑟𝐴 → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟𝐴))
32rgen 2951 . . . 4 𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟𝐴)
43rgen2w 2954 . . 3 𝑝𝐴𝑞𝐴𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟𝐴)
51, 4pm3.2i 470 . 2 (𝐴𝐴 ∧ ∀𝑝𝐴𝑞𝐴𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟𝐴))
6 eqid 2651 . . 3 (le‘𝐾) = (le‘𝐾)
7 eqid 2651 . . 3 (join‘𝐾) = (join‘𝐾)
8 atpsub.a . . 3 𝐴 = (Atoms‘𝐾)
9 atpsub.s . . 3 𝑆 = (PSubSp‘𝐾)
106, 7, 8, 9ispsubsp 35349 . 2 (𝐾𝑉 → (𝐴𝑆 ↔ (𝐴𝐴 ∧ ∀𝑝𝐴𝑞𝐴𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟𝐴))))
115, 10mpbiri 248 1 (𝐾𝑉𝐴𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∀wral 2941   ⊆ wss 3607   class class class wbr 4685  ‘cfv 5926  (class class class)co 6690  lecple 15995  joincjn 16991  Atomscatm 34868  PSubSpcpsubsp 35100 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-psubsp 35107 This theorem is referenced by:  pclvalN  35494  pclclN  35495
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