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Theorem atpsubN 39220
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 4000 . . 3 𝐴𝐴
2 ax-1 6 . . . . 5 (𝑟𝐴 → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟𝐴))
32rgen 3059 . . . 4 𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟𝐴)
43rgen2w 3062 . . 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 2728 . . 3 (le‘𝐾) = (le‘𝐾)
7 eqid 2728 . . 3 (join‘𝐾) = (join‘𝐾)
8 atpsub.a . . 3 𝐴 = (Atoms‘𝐾)
9 atpsub.s . . 3 𝑆 = (PSubSp‘𝐾)
106, 7, 8, 9ispsubsp 39212 . 2 (𝐾𝑉 → (𝐴𝑆 ↔ (𝐴𝐴 ∧ ∀𝑝𝐴𝑞𝐴𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟𝐴))))
115, 10mpbiri 258 1 (𝐾𝑉𝐴𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  wral 3057  wss 3945   class class class wbr 5142  cfv 6542  (class class class)co 7414  lecple 17233  joincjn 18296  Atomscatm 38729  PSubSpcpsubsp 38963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7417  df-psubsp 38970
This theorem is referenced by:  pclvalN  39357  pclclN  39358
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