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Theorem ispsubsp2 36988
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 36987 . 2 (𝐾𝐷 → (𝑋𝑆 ↔ (𝑋𝐴 ∧ ∀𝑞𝑋𝑟𝑋𝑝𝐴 (𝑝 (𝑞 𝑟) → 𝑝𝑋))))
6 ralcom 3346 . . . . . . 7 (∀𝑟𝑋𝑝𝐴 (𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ ∀𝑝𝐴𝑟𝑋 (𝑝 (𝑞 𝑟) → 𝑝𝑋))
7 r19.23v 3272 . . . . . . . 8 (∀𝑟𝑋 (𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ (∃𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))
87ralbii 3160 . . . . . . 7 (∀𝑝𝐴𝑟𝑋 (𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ ∀𝑝𝐴 (∃𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))
96, 8bitri 278 . . . . . 6 (∀𝑟𝑋𝑝𝐴 (𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ ∀𝑝𝐴 (∃𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))
109ralbii 3160 . . . . 5 (∀𝑞𝑋𝑟𝑋𝑝𝐴 (𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ ∀𝑞𝑋𝑝𝐴 (∃𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))
11 ralcom 3346 . . . . . 6 (∀𝑞𝑋𝑝𝐴 (∃𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ ∀𝑝𝐴𝑞𝑋 (∃𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))
12 r19.23v 3272 . . . . . . 7 (∀𝑞𝑋 (∃𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))
1312ralbii 3160 . . . . . 6 (∀𝑝𝐴𝑞𝑋 (∃𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ ∀𝑝𝐴 (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))
1411, 13bitri 278 . . . . 5 (∀𝑞𝑋𝑝𝐴 (∃𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ ∀𝑝𝐴 (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))
1510, 14bitri 278 . . . 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 631 . 2 (𝐾𝐷 → ((𝑋𝐴 ∧ ∀𝑞𝑋𝑟𝑋𝑝𝐴 (𝑝 (𝑞 𝑟) → 𝑝𝑋)) ↔ (𝑋𝐴 ∧ ∀𝑝𝐴 (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))))
185, 17bitrd 282 1 (𝐾𝐷 → (𝑋𝑆 ↔ (𝑋𝐴 ∧ ∀𝑝𝐴 (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3133  wrex 3134  wss 3920   class class class wbr 5053  cfv 6344  (class class class)co 7150  lecple 16575  joincjn 17557  Atomscatm 36505  PSubSpcpsubsp 36738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3483  df-sbc 3760  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-if 4452  df-pw 4525  df-sn 4552  df-pr 4554  df-op 4558  df-uni 4826  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-iota 6303  df-fun 6346  df-fv 6352  df-ov 7153  df-psubsp 36745
This theorem is referenced by:  psubspi  36989  paddclN  37084
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