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Theorem atpsubN 34558
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 3609 . . 3 𝐴𝐴
2 ax-1 6 . . . . 5 (𝑟𝐴 → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟𝐴))
32rgen 2918 . . . 4 𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟𝐴)
43rgen2w 2921 . . 3 𝑝𝐴𝑞𝐴𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟𝐴)
51, 4pm3.2i 471 . 2 (𝐴𝐴 ∧ ∀𝑝𝐴𝑞𝐴𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟𝐴))
6 eqid 2621 . . 3 (le‘𝐾) = (le‘𝐾)
7 eqid 2621 . . 3 (join‘𝐾) = (join‘𝐾)
8 atpsub.a . . 3 𝐴 = (Atoms‘𝐾)
9 atpsub.s . . 3 𝑆 = (PSubSp‘𝐾)
106, 7, 8, 9ispsubsp 34550 . 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 384   = wceq 1480  wcel 1987  wral 2908  wss 3560   class class class wbr 4623  cfv 5857  (class class class)co 6615  lecple 15888  joincjn 16884  Atomscatm 34069  PSubSpcpsubsp 34301
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865  df-ov 6618  df-psubsp 34308
This theorem is referenced by:  pclvalN  34695  pclclN  34696
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