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Theorem psubspi2N 39731
Description: Property of a projective subspace. (Contributed by NM, 13-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
psubspset.l = (le‘𝐾)
psubspset.j = (join‘𝐾)
psubspset.a 𝐴 = (Atoms‘𝐾)
psubspset.s 𝑆 = (PSubSp‘𝐾)
Assertion
Ref Expression
psubspi2N (((𝐾𝐷𝑋𝑆𝑃𝐴) ∧ (𝑄𝑋𝑅𝑋𝑃 (𝑄 𝑅))) → 𝑃𝑋)

Proof of Theorem psubspi2N
Dummy variables 𝑟 𝑞 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 7438 . . . 4 (𝑞 = 𝑄 → (𝑞 𝑟) = (𝑄 𝑟))
21breq2d 5160 . . 3 (𝑞 = 𝑄 → (𝑃 (𝑞 𝑟) ↔ 𝑃 (𝑄 𝑟)))
3 oveq2 7439 . . . 4 (𝑟 = 𝑅 → (𝑄 𝑟) = (𝑄 𝑅))
43breq2d 5160 . . 3 (𝑟 = 𝑅 → (𝑃 (𝑄 𝑟) ↔ 𝑃 (𝑄 𝑅)))
52, 4rspc2ev 3635 . 2 ((𝑄𝑋𝑅𝑋𝑃 (𝑄 𝑅)) → ∃𝑞𝑋𝑟𝑋 𝑃 (𝑞 𝑟))
6 psubspset.l . . 3 = (le‘𝐾)
7 psubspset.j . . 3 = (join‘𝐾)
8 psubspset.a . . 3 𝐴 = (Atoms‘𝐾)
9 psubspset.s . . 3 𝑆 = (PSubSp‘𝐾)
106, 7, 8, 9psubspi 39730 . 2 (((𝐾𝐷𝑋𝑆𝑃𝐴) ∧ ∃𝑞𝑋𝑟𝑋 𝑃 (𝑞 𝑟)) → 𝑃𝑋)
115, 10sylan2 593 1 (((𝐾𝐷𝑋𝑆𝑃𝐴) ∧ (𝑄𝑋𝑅𝑋𝑃 (𝑄 𝑅))) → 𝑃𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wrex 3068   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:  pclclN  39874  pclfinN  39883  pclfinclN  39933
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