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Theorem psubspi2N 36415
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 7023 . . . 4 (𝑞 = 𝑄 → (𝑞 𝑟) = (𝑄 𝑟))
21breq2d 4974 . . 3 (𝑞 = 𝑄 → (𝑃 (𝑞 𝑟) ↔ 𝑃 (𝑄 𝑟)))
3 oveq2 7024 . . . 4 (𝑟 = 𝑅 → (𝑄 𝑟) = (𝑄 𝑅))
43breq2d 4974 . . 3 (𝑟 = 𝑅 → (𝑃 (𝑄 𝑟) ↔ 𝑃 (𝑄 𝑅)))
52, 4rspc2ev 3574 . 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 36414 . 2 (((𝐾𝐷𝑋𝑆𝑃𝐴) ∧ ∃𝑞𝑋𝑟𝑋 𝑃 (𝑞 𝑟)) → 𝑃𝑋)
115, 10sylan2 592 1 (((𝐾𝐷𝑋𝑆𝑃𝐴) ∧ (𝑄𝑋𝑅𝑋𝑃 (𝑄 𝑅))) → 𝑃𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1080   = wceq 1522  wcel 2081  wrex 3106   class class class wbr 4962  cfv 6225  (class class class)co 7016  lecple 16401  joincjn 17383  Atomscatm 35930  PSubSpcpsubsp 36163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-iota 6189  df-fun 6227  df-fv 6233  df-ov 7019  df-psubsp 36170
This theorem is referenced by:  pclclN  36558  pclfinN  36567  pclfinclN  36617
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