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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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