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Theorem psubspi 39730
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 39729 . . . . 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 5151 . . . . . 6 (𝑝 = 𝑃 → (𝑝 (𝑞 𝑟) ↔ 𝑃 (𝑞 𝑟)))
982rexbidv 3220 . . . . 5 (𝑝 = 𝑃 → (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) ↔ ∃𝑞𝑋𝑟𝑋 𝑃 (𝑞 𝑟)))
10 eleq1 2827 . . . . 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 1346 1 (((𝐾𝐷𝑋𝑆𝑃𝐴) ∧ ∃𝑞𝑋𝑟𝑋 𝑃 (𝑞 𝑟)) → 𝑃𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wrex 3068  wss 3963   class class class wbr 5148  cfv 6563  (class class class)co 7431  lecple 17305  joincjn 18369  Atomscatm 39245  PSubSpcpsubsp 39479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-psubsp 39486
This theorem is referenced by:  psubspi2N  39731  paddidm  39824
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