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Theorem pointpsubN 35536
Description: A point (singleton of an atom) is a projective subspace. Remark below Definition 15.1 of [MaedaMaeda] p. 61. (Contributed by NM, 13-Oct-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
pointpsub.p 𝑃 = (Points‘𝐾)
pointpsub.s 𝑆 = (PSubSp‘𝐾)
Assertion
Ref Expression
pointpsubN ((𝐾 ∈ AtLat ∧ 𝑋𝑃) → 𝑋𝑆)

Proof of Theorem pointpsubN
Dummy variable 𝑞 is distinct from all other variables.
StepHypRef Expression
1 eqid 2756 . . . 4 (Atoms‘𝐾) = (Atoms‘𝐾)
2 pointpsub.p . . . 4 𝑃 = (Points‘𝐾)
31, 2ispointN 35527 . . 3 (𝐾 ∈ AtLat → (𝑋𝑃 ↔ ∃𝑞 ∈ (Atoms‘𝐾)𝑋 = {𝑞}))
4 pointpsub.s . . . . . . 7 𝑆 = (PSubSp‘𝐾)
51, 4snatpsubN 35535 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑞 ∈ (Atoms‘𝐾)) → {𝑞} ∈ 𝑆)
65ex 449 . . . . 5 (𝐾 ∈ AtLat → (𝑞 ∈ (Atoms‘𝐾) → {𝑞} ∈ 𝑆))
7 eleq1a 2830 . . . . 5 ({𝑞} ∈ 𝑆 → (𝑋 = {𝑞} → 𝑋𝑆))
86, 7syl6 35 . . . 4 (𝐾 ∈ AtLat → (𝑞 ∈ (Atoms‘𝐾) → (𝑋 = {𝑞} → 𝑋𝑆)))
98rexlimdv 3164 . . 3 (𝐾 ∈ AtLat → (∃𝑞 ∈ (Atoms‘𝐾)𝑋 = {𝑞} → 𝑋𝑆))
103, 9sylbid 230 . 2 (𝐾 ∈ AtLat → (𝑋𝑃𝑋𝑆))
1110imp 444 1 ((𝐾 ∈ AtLat ∧ 𝑋𝑃) → 𝑋𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1628  wcel 2135  wrex 3047  {csn 4317  cfv 6045  Atomscatm 35049  AtLatcal 35050  PointscpointsN 35280  PSubSpcpsubsp 35281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-rep 4919  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-reu 3053  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-iun 4670  df-br 4801  df-opab 4861  df-mpt 4878  df-id 5170  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053  df-riota 6770  df-ov 6812  df-oprab 6813  df-preset 17125  df-poset 17143  df-plt 17155  df-lub 17171  df-glb 17172  df-join 17173  df-meet 17174  df-p0 17236  df-lat 17243  df-covers 35052  df-ats 35053  df-atl 35084  df-pointsN 35287  df-psubsp 35288
This theorem is referenced by: (None)
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