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Theorem psubspi2N 35537
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 6820 . . . 4 (𝑞 = 𝑄 → (𝑞 𝑟) = (𝑄 𝑟))
21breq2d 4816 . . 3 (𝑞 = 𝑄 → (𝑃 (𝑞 𝑟) ↔ 𝑃 (𝑄 𝑟)))
3 oveq2 6821 . . . 4 (𝑟 = 𝑅 → (𝑄 𝑟) = (𝑄 𝑅))
43breq2d 4816 . . 3 (𝑟 = 𝑅 → (𝑃 (𝑄 𝑟) ↔ 𝑃 (𝑄 𝑅)))
52, 4rspc2ev 3463 . 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 35536 . 2 (((𝐾𝐷𝑋𝑆𝑃𝐴) ∧ ∃𝑞𝑋𝑟𝑋 𝑃 (𝑞 𝑟)) → 𝑃𝑋)
115, 10sylan2 492 1 (((𝐾𝐷𝑋𝑆𝑃𝐴) ∧ (𝑄𝑋𝑅𝑋𝑃 (𝑄 𝑅))) → 𝑃𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2139  wrex 3051   class class class wbr 4804  cfv 6049  (class class class)co 6813  lecple 16150  joincjn 17145  Atomscatm 35053  PSubSpcpsubsp 35285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-psubsp 35292
This theorem is referenced by:  pclclN  35680  pclfinN  35689  pclfinclN  35739
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