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Theorem psubspi2N 35549
 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 6799 . . . 4 (𝑞 = 𝑄 → (𝑞 𝑟) = (𝑄 𝑟))
21breq2d 4796 . . 3 (𝑞 = 𝑄 → (𝑃 (𝑞 𝑟) ↔ 𝑃 (𝑄 𝑟)))
3 oveq2 6800 . . . 4 (𝑟 = 𝑅 → (𝑄 𝑟) = (𝑄 𝑅))
43breq2d 4796 . . 3 (𝑟 = 𝑅 → (𝑃 (𝑄 𝑟) ↔ 𝑃 (𝑄 𝑅)))
52, 4rspc2ev 3472 . 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 35548 . 2 (((𝐾𝐷𝑋𝑆𝑃𝐴) ∧ ∃𝑞𝑋𝑟𝑋 𝑃 (𝑞 𝑟)) → 𝑃𝑋)
115, 10sylan2 572 1 (((𝐾𝐷𝑋𝑆𝑃𝐴) ∧ (𝑄𝑋𝑅𝑋𝑃 (𝑄 𝑅))) → 𝑃𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   ∧ w3a 1070   = wceq 1630   ∈ wcel 2144  ∃wrex 3061   class class class wbr 4784  ‘cfv 6031  (class class class)co 6792  lecple 16155  joincjn 17151  Atomscatm 35065  PSubSpcpsubsp 35297 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6795  df-psubsp 35304 This theorem is referenced by:  pclclN  35692  pclfinN  35701  pclfinclN  35751
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