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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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