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Theorem ispsubsp2 36762
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 36761 . 2 (𝐾𝐷 → (𝑋𝑆 ↔ (𝑋𝐴 ∧ ∀𝑞𝑋𝑟𝑋𝑝𝐴 (𝑝 (𝑞 𝑟) → 𝑝𝑋))))
6 ralcom 3351 . . . . . . 7 (∀𝑟𝑋𝑝𝐴 (𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ ∀𝑝𝐴𝑟𝑋 (𝑝 (𝑞 𝑟) → 𝑝𝑋))
7 r19.23v 3276 . . . . . . . 8 (∀𝑟𝑋 (𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ (∃𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))
87ralbii 3162 . . . . . . 7 (∀𝑝𝐴𝑟𝑋 (𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ ∀𝑝𝐴 (∃𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))
96, 8bitri 276 . . . . . 6 (∀𝑟𝑋𝑝𝐴 (𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ ∀𝑝𝐴 (∃𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))
109ralbii 3162 . . . . 5 (∀𝑞𝑋𝑟𝑋𝑝𝐴 (𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ ∀𝑞𝑋𝑝𝐴 (∃𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))
11 ralcom 3351 . . . . . 6 (∀𝑞𝑋𝑝𝐴 (∃𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ ∀𝑝𝐴𝑞𝑋 (∃𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))
12 r19.23v 3276 . . . . . . 7 (∀𝑞𝑋 (∃𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))
1312ralbii 3162 . . . . . 6 (∀𝑝𝐴𝑞𝑋 (∃𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ ∀𝑝𝐴 (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))
1411, 13bitri 276 . . . . 5 (∀𝑞𝑋𝑝𝐴 (∃𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ ∀𝑝𝐴 (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))
1510, 14bitri 276 . . . 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 628 . 2 (𝐾𝐷 → ((𝑋𝐴 ∧ ∀𝑞𝑋𝑟𝑋𝑝𝐴 (𝑝 (𝑞 𝑟) → 𝑝𝑋)) ↔ (𝑋𝐴 ∧ ∀𝑝𝐴 (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))))
185, 17bitrd 280 1 (𝐾𝐷 → (𝑋𝑆 ↔ (𝑋𝐴 ∧ ∀𝑝𝐴 (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  wrex 3136  wss 3933   class class class wbr 5057  cfv 6348  (class class class)co 7145  lecple 16560  joincjn 17542  Atomscatm 36279  PSubSpcpsubsp 36512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-psubsp 36519
This theorem is referenced by:  psubspi  36763  paddclN  36858
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