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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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