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Theorem ispsubsp 39083
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 39082 . . 3 (𝐾𝐷𝑆 = {𝑥 ∣ (𝑥𝐴 ∧ ∀𝑝𝑥𝑞𝑥𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑥))})
65eleq2d 2818 . 2 (𝐾𝐷 → (𝑋𝑆𝑋 ∈ {𝑥 ∣ (𝑥𝐴 ∧ ∀𝑝𝑥𝑞𝑥𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑥))}))
73fvexi 6905 . . . . 5 𝐴 ∈ V
87ssex 5321 . . . 4 (𝑋𝐴𝑋 ∈ V)
98adantr 480 . . 3 ((𝑋𝐴 ∧ ∀𝑝𝑋𝑞𝑋𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑋)) → 𝑋 ∈ V)
10 sseq1 4007 . . . 4 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
11 eleq2 2821 . . . . . . . 8 (𝑥 = 𝑋 → (𝑟𝑥𝑟𝑋))
1211imbi2d 340 . . . . . . 7 (𝑥 = 𝑋 → ((𝑟 (𝑝 𝑞) → 𝑟𝑥) ↔ (𝑟 (𝑝 𝑞) → 𝑟𝑋)))
1312ralbidv 3176 . . . . . 6 (𝑥 = 𝑋 → (∀𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑥) ↔ ∀𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑋)))
1413raleqbi1dv 3332 . . . . 5 (𝑥 = 𝑋 → (∀𝑞𝑥𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑥) ↔ ∀𝑞𝑋𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑋)))
1514raleqbi1dv 3332 . . . 4 (𝑥 = 𝑋 → (∀𝑝𝑥𝑞𝑥𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑥) ↔ ∀𝑝𝑋𝑞𝑋𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑋)))
1610, 15anbi12d 630 . . 3 (𝑥 = 𝑋 → ((𝑥𝐴 ∧ ∀𝑝𝑥𝑞𝑥𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑥)) ↔ (𝑋𝐴 ∧ ∀𝑝𝑋𝑞𝑋𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑋))))
179, 16elab3 3676 . 2 (𝑋 ∈ {𝑥 ∣ (𝑥𝐴 ∧ ∀𝑝𝑥𝑞𝑥𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑥))} ↔ (𝑋𝐴 ∧ ∀𝑝𝑋𝑞𝑋𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑋)))
186, 17bitrdi 287 1 (𝐾𝐷 → (𝑋𝑆 ↔ (𝑋𝐴 ∧ ∀𝑝𝑋𝑞𝑋𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1540  wcel 2105  {cab 2708  wral 3060  Vcvv 3473  wss 3948   class class class wbr 5148  cfv 6543  (class class class)co 7412  lecple 17211  joincjn 18274  Atomscatm 38600  PSubSpcpsubsp 38834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7415  df-psubsp 38841
This theorem is referenced by:  ispsubsp2  39084  0psubN  39087  snatpsubN  39088  linepsubN  39090  atpsubN  39091  psubssat  39092  pmapsub  39106  pclclN  39229  pclfinN  39238
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