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Theorem ispsubsp2 39729
Description: The predicate "is a projective subspace". (Contributed by NM, 13-Jan-2012.)
Hypotheses
Ref Expression
psubspset.l = (le‘𝐾)
psubspset.j = (join‘𝐾)
psubspset.a 𝐴 = (Atoms‘𝐾)
psubspset.s 𝑆 = (PSubSp‘𝐾)
Assertion
Ref Expression
ispsubsp2 (𝐾𝐷 → (𝑋𝑆 ↔ (𝑋𝐴 ∧ ∀𝑝𝐴 (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))))
Distinct variable groups:   𝐴,𝑟   𝑞,𝑝,𝑟,𝐾   𝑋,𝑝,𝑞,𝑟   𝐴,𝑝,𝑞
Allowed substitution hints:   𝐷(𝑟,𝑞,𝑝)   𝑆(𝑟,𝑞,𝑝)   (𝑟,𝑞,𝑝)   (𝑟,𝑞,𝑝)

Proof of Theorem ispsubsp2
StepHypRef Expression
1 psubspset.l . . 3 = (le‘𝐾)
2 psubspset.j . . 3 = (join‘𝐾)
3 psubspset.a . . 3 𝐴 = (Atoms‘𝐾)
4 psubspset.s . . 3 𝑆 = (PSubSp‘𝐾)
51, 2, 3, 4ispsubsp 39728 . 2 (𝐾𝐷 → (𝑋𝑆 ↔ (𝑋𝐴 ∧ ∀𝑞𝑋𝑟𝑋𝑝𝐴 (𝑝 (𝑞 𝑟) → 𝑝𝑋))))
6 ralcom 3287 . . . . . . 7 (∀𝑟𝑋𝑝𝐴 (𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ ∀𝑝𝐴𝑟𝑋 (𝑝 (𝑞 𝑟) → 𝑝𝑋))
7 r19.23v 3181 . . . . . . . 8 (∀𝑟𝑋 (𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ (∃𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))
87ralbii 3091 . . . . . . 7 (∀𝑝𝐴𝑟𝑋 (𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ ∀𝑝𝐴 (∃𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))
96, 8bitri 275 . . . . . 6 (∀𝑟𝑋𝑝𝐴 (𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ ∀𝑝𝐴 (∃𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))
109ralbii 3091 . . . . 5 (∀𝑞𝑋𝑟𝑋𝑝𝐴 (𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ ∀𝑞𝑋𝑝𝐴 (∃𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))
11 ralcom 3287 . . . . . 6 (∀𝑞𝑋𝑝𝐴 (∃𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ ∀𝑝𝐴𝑞𝑋 (∃𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))
12 r19.23v 3181 . . . . . . 7 (∀𝑞𝑋 (∃𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))
1312ralbii 3091 . . . . . 6 (∀𝑝𝐴𝑞𝑋 (∃𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ ∀𝑝𝐴 (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))
1411, 13bitri 275 . . . . 5 (∀𝑞𝑋𝑝𝐴 (∃𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ ∀𝑝𝐴 (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))
1510, 14bitri 275 . . . 4 (∀𝑞𝑋𝑟𝑋𝑝𝐴 (𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ ∀𝑝𝐴 (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))
1615a1i 11 . . 3 (𝐾𝐷 → (∀𝑞𝑋𝑟𝑋𝑝𝐴 (𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ ∀𝑝𝐴 (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋)))
1716anbi2d 630 . 2 (𝐾𝐷 → ((𝑋𝐴 ∧ ∀𝑞𝑋𝑟𝑋𝑝𝐴 (𝑝 (𝑞 𝑟) → 𝑝𝑋)) ↔ (𝑋𝐴 ∧ ∀𝑝𝐴 (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))))
185, 17bitrd 279 1 (𝐾𝐷 → (𝑋𝑆 ↔ (𝑋𝐴 ∧ ∀𝑝𝐴 (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  wrex 3068  wss 3963   class class class wbr 5148  cfv 6563  (class class class)co 7431  lecple 17305  joincjn 18369  Atomscatm 39245  PSubSpcpsubsp 39479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-psubsp 39486
This theorem is referenced by:  psubspi  39730  paddclN  39825
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