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Theorem psubspi 39749
Description: Property of 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
psubspi (((𝐾𝐷𝑋𝑆𝑃𝐴) ∧ ∃𝑞𝑋𝑟𝑋 𝑃 (𝑞 𝑟)) → 𝑃𝑋)
Distinct variable groups:   𝐴,𝑟,𝑞   𝐾,𝑞,𝑟   𝑋,𝑞,𝑟   𝐴,𝑞   𝑃,𝑞,𝑟
Allowed substitution hints:   𝐷(𝑟,𝑞)   𝑆(𝑟,𝑞)   (𝑟,𝑞)   (𝑟,𝑞)

Proof of Theorem psubspi
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 psubspset.l . . . . . 6 = (le‘𝐾)
2 psubspset.j . . . . . 6 = (join‘𝐾)
3 psubspset.a . . . . . 6 𝐴 = (Atoms‘𝐾)
4 psubspset.s . . . . . 6 𝑆 = (PSubSp‘𝐾)
51, 2, 3, 4ispsubsp2 39748 . . . . 5 (𝐾𝐷 → (𝑋𝑆 ↔ (𝑋𝐴 ∧ ∀𝑝𝐴 (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))))
65simplbda 499 . . . 4 ((𝐾𝐷𝑋𝑆) → ∀𝑝𝐴 (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))
76ex 412 . . 3 (𝐾𝐷 → (𝑋𝑆 → ∀𝑝𝐴 (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋)))
8 breq1 5146 . . . . . 6 (𝑝 = 𝑃 → (𝑝 (𝑞 𝑟) ↔ 𝑃 (𝑞 𝑟)))
982rexbidv 3222 . . . . 5 (𝑝 = 𝑃 → (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) ↔ ∃𝑞𝑋𝑟𝑋 𝑃 (𝑞 𝑟)))
10 eleq1 2829 . . . . 5 (𝑝 = 𝑃 → (𝑝𝑋𝑃𝑋))
119, 10imbi12d 344 . . . 4 (𝑝 = 𝑃 → ((∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ (∃𝑞𝑋𝑟𝑋 𝑃 (𝑞 𝑟) → 𝑃𝑋)))
1211rspccv 3619 . . 3 (∀𝑝𝐴 (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋) → (𝑃𝐴 → (∃𝑞𝑋𝑟𝑋 𝑃 (𝑞 𝑟) → 𝑃𝑋)))
137, 12syl6 35 . 2 (𝐾𝐷 → (𝑋𝑆 → (𝑃𝐴 → (∃𝑞𝑋𝑟𝑋 𝑃 (𝑞 𝑟) → 𝑃𝑋))))
14133imp1 1348 1 (((𝐾𝐷𝑋𝑆𝑃𝐴) ∧ ∃𝑞𝑋𝑟𝑋 𝑃 (𝑞 𝑟)) → 𝑃𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  wss 3951   class class class wbr 5143  cfv 6561  (class class class)co 7431  lecple 17304  joincjn 18357  Atomscatm 39264  PSubSpcpsubsp 39498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-psubsp 39505
This theorem is referenced by:  psubspi2N  39750  paddidm  39843
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