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Theorem ispsubsp 34049
 Description: The predicate "is a projective subspace". (Contributed by NM, 2-Oct-2011.)
Hypotheses
Ref Expression
psubspset.l = (le‘𝐾)
psubspset.j = (join‘𝐾)
psubspset.a 𝐴 = (Atoms‘𝐾)
psubspset.s 𝑆 = (PSubSp‘𝐾)
Assertion
Ref Expression
ispsubsp (𝐾𝐷 → (𝑋𝑆 ↔ (𝑋𝐴 ∧ ∀𝑝𝑋𝑞𝑋𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑋))))
Distinct variable groups:   𝐴,𝑟   𝑞,𝑝,𝑟,𝐾   𝑋,𝑝,𝑞,𝑟
Allowed substitution hints:   𝐴(𝑞,𝑝)   𝐷(𝑟,𝑞,𝑝)   𝑆(𝑟,𝑞,𝑝)   (𝑟,𝑞,𝑝)   (𝑟,𝑞,𝑝)

Proof of Theorem ispsubsp
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 psubspset.l . . . 4 = (le‘𝐾)
2 psubspset.j . . . 4 = (join‘𝐾)
3 psubspset.a . . . 4 𝐴 = (Atoms‘𝐾)
4 psubspset.s . . . 4 𝑆 = (PSubSp‘𝐾)
51, 2, 3, 4psubspset 34048 . . 3 (𝐾𝐷𝑆 = {𝑥 ∣ (𝑥𝐴 ∧ ∀𝑝𝑥𝑞𝑥𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑥))})
65eleq2d 2673 . 2 (𝐾𝐷 → (𝑋𝑆𝑋 ∈ {𝑥 ∣ (𝑥𝐴 ∧ ∀𝑝𝑥𝑞𝑥𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑥))}))
7 fvex 6113 . . . . . 6 (Atoms‘𝐾) ∈ V
83, 7eqeltri 2684 . . . . 5 𝐴 ∈ V
98ssex 4730 . . . 4 (𝑋𝐴𝑋 ∈ V)
109adantr 480 . . 3 ((𝑋𝐴 ∧ ∀𝑝𝑋𝑞𝑋𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑋)) → 𝑋 ∈ V)
11 sseq1 3589 . . . 4 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
12 eleq2 2677 . . . . . . . 8 (𝑥 = 𝑋 → (𝑟𝑥𝑟𝑋))
1312imbi2d 329 . . . . . . 7 (𝑥 = 𝑋 → ((𝑟 (𝑝 𝑞) → 𝑟𝑥) ↔ (𝑟 (𝑝 𝑞) → 𝑟𝑋)))
1413ralbidv 2969 . . . . . 6 (𝑥 = 𝑋 → (∀𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑥) ↔ ∀𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑋)))
1514raleqbi1dv 3123 . . . . 5 (𝑥 = 𝑋 → (∀𝑞𝑥𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑥) ↔ ∀𝑞𝑋𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑋)))
1615raleqbi1dv 3123 . . . 4 (𝑥 = 𝑋 → (∀𝑝𝑥𝑞𝑥𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑥) ↔ ∀𝑝𝑋𝑞𝑋𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑋)))
1711, 16anbi12d 743 . . 3 (𝑥 = 𝑋 → ((𝑥𝐴 ∧ ∀𝑝𝑥𝑞𝑥𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑥)) ↔ (𝑋𝐴 ∧ ∀𝑝𝑋𝑞𝑋𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑋))))
1810, 17elab3 3327 . 2 (𝑋 ∈ {𝑥 ∣ (𝑥𝐴 ∧ ∀𝑝𝑥𝑞𝑥𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑥))} ↔ (𝑋𝐴 ∧ ∀𝑝𝑋𝑞𝑋𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑋)))
196, 18syl6bb 275 1 (𝐾𝐷 → (𝑋𝑆 ↔ (𝑋𝐴 ∧ ∀𝑝𝑋𝑞𝑋𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑋))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  Vcvv 3173   ⊆ wss 3540   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  lecple 15775  joincjn 16767  Atomscatm 33568  PSubSpcpsubsp 33800 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-psubsp 33807 This theorem is referenced by:  ispsubsp2  34050  0psubN  34053  snatpsubN  34054  linepsubN  34056  atpsubN  34057  psubssat  34058  pmapsub  34072  pclclN  34195  pclfinN  34204
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