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Theorem ispsubsp2 35554
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 35553 . 2 (𝐾𝐷 → (𝑋𝑆 ↔ (𝑋𝐴 ∧ ∀𝑞𝑋𝑟𝑋𝑝𝐴 (𝑝 (𝑞 𝑟) → 𝑝𝑋))))
6 ralcom 3246 . . . . . . 7 (∀𝑟𝑋𝑝𝐴 (𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ ∀𝑝𝐴𝑟𝑋 (𝑝 (𝑞 𝑟) → 𝑝𝑋))
7 r19.23v 3171 . . . . . . . 8 (∀𝑟𝑋 (𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ (∃𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))
87ralbii 3129 . . . . . . 7 (∀𝑝𝐴𝑟𝑋 (𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ ∀𝑝𝐴 (∃𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))
96, 8bitri 264 . . . . . 6 (∀𝑟𝑋𝑝𝐴 (𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ ∀𝑝𝐴 (∃𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))
109ralbii 3129 . . . . 5 (∀𝑞𝑋𝑟𝑋𝑝𝐴 (𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ ∀𝑞𝑋𝑝𝐴 (∃𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))
11 ralcom 3246 . . . . . 6 (∀𝑞𝑋𝑝𝐴 (∃𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ ∀𝑝𝐴𝑞𝑋 (∃𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))
12 r19.23v 3171 . . . . . . 7 (∀𝑞𝑋 (∃𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))
1312ralbii 3129 . . . . . 6 (∀𝑝𝐴𝑞𝑋 (∃𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ ∀𝑝𝐴 (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))
1411, 13bitri 264 . . . . 5 (∀𝑞𝑋𝑝𝐴 (∃𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ ∀𝑝𝐴 (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))
1510, 14bitri 264 . . . 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 614 . 2 (𝐾𝐷 → ((𝑋𝐴 ∧ ∀𝑞𝑋𝑟𝑋𝑝𝐴 (𝑝 (𝑞 𝑟) → 𝑝𝑋)) ↔ (𝑋𝐴 ∧ ∀𝑝𝐴 (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))))
185, 17bitrd 268 1 (𝐾𝐷 → (𝑋𝑆 ↔ (𝑋𝐴 ∧ ∀𝑝𝐴 (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  wrex 3062  wss 3723   class class class wbr 4786  cfv 6031  (class class class)co 6793  lecple 16156  joincjn 17152  Atomscatm 35072  PSubSpcpsubsp 35304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796  df-psubsp 35311
This theorem is referenced by:  psubspi  35555  paddclN  35650
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