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Theorem atpsubN 36893
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 3992 . . 3 𝐴𝐴
2 ax-1 6 . . . . 5 (𝑟𝐴 → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟𝐴))
32rgen 3151 . . . 4 𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟𝐴)
43rgen2w 3154 . . 3 𝑝𝐴𝑞𝐴𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟𝐴)
51, 4pm3.2i 473 . 2 (𝐴𝐴 ∧ ∀𝑝𝐴𝑞𝐴𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟𝐴))
6 eqid 2824 . . 3 (le‘𝐾) = (le‘𝐾)
7 eqid 2824 . . 3 (join‘𝐾) = (join‘𝐾)
8 atpsub.a . . 3 𝐴 = (Atoms‘𝐾)
9 atpsub.s . . 3 𝑆 = (PSubSp‘𝐾)
106, 7, 8, 9ispsubsp 36885 . 2 (𝐾𝑉 → (𝐴𝑆 ↔ (𝐴𝐴 ∧ ∀𝑝𝐴𝑞𝐴𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟𝐴))))
115, 10mpbiri 260 1 (𝐾𝑉𝐴𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1536  wcel 2113  wral 3141  wss 3939   class class class wbr 5069  cfv 6358  (class class class)co 7159  lecple 16575  joincjn 17557  Atomscatm 36403  PSubSpcpsubsp 36636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-iota 6317  df-fun 6360  df-fv 6366  df-ov 7162  df-psubsp 36643
This theorem is referenced by:  pclvalN  37030  pclclN  37031
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