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Theorem psubspi 37498
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 37497 . . . . 5 (𝐾𝐷 → (𝑋𝑆 ↔ (𝑋𝐴 ∧ ∀𝑝𝐴 (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))))
65simplbda 503 . . . 4 ((𝐾𝐷𝑋𝑆) → ∀𝑝𝐴 (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))
76ex 416 . . 3 (𝐾𝐷 → (𝑋𝑆 → ∀𝑝𝐴 (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋)))
8 breq1 5056 . . . . . 6 (𝑝 = 𝑃 → (𝑝 (𝑞 𝑟) ↔ 𝑃 (𝑞 𝑟)))
982rexbidv 3219 . . . . 5 (𝑝 = 𝑃 → (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) ↔ ∃𝑞𝑋𝑟𝑋 𝑃 (𝑞 𝑟)))
10 eleq1 2825 . . . . 5 (𝑝 = 𝑃 → (𝑝𝑋𝑃𝑋))
119, 10imbi12d 348 . . . 4 (𝑝 = 𝑃 → ((∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ (∃𝑞𝑋𝑟𝑋 𝑃 (𝑞 𝑟) → 𝑃𝑋)))
1211rspccv 3534 . . 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 1349 1 (((𝐾𝐷𝑋𝑆𝑃𝐴) ∧ ∃𝑞𝑋𝑟𝑋 𝑃 (𝑞 𝑟)) → 𝑃𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1089   = wceq 1543  wcel 2110  wral 3061  wrex 3062  wss 3866   class class class wbr 5053  cfv 6380  (class class class)co 7213  lecple 16809  joincjn 17818  Atomscatm 37014  PSubSpcpsubsp 37247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-iota 6338  df-fun 6382  df-fv 6388  df-ov 7216  df-psubsp 37254
This theorem is referenced by:  psubspi2N  37499  paddidm  37592
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