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Theorem pointpsubN 36767
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 2818 . . . 4 (Atoms‘𝐾) = (Atoms‘𝐾)
2 pointpsub.p . . . 4 𝑃 = (Points‘𝐾)
31, 2ispointN 36758 . . 3 (𝐾 ∈ AtLat → (𝑋𝑃 ↔ ∃𝑞 ∈ (Atoms‘𝐾)𝑋 = {𝑞}))
4 pointpsub.s . . . . . . 7 𝑆 = (PSubSp‘𝐾)
51, 4snatpsubN 36766 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑞 ∈ (Atoms‘𝐾)) → {𝑞} ∈ 𝑆)
65ex 413 . . . . 5 (𝐾 ∈ AtLat → (𝑞 ∈ (Atoms‘𝐾) → {𝑞} ∈ 𝑆))
7 eleq1a 2905 . . . . 5 ({𝑞} ∈ 𝑆 → (𝑋 = {𝑞} → 𝑋𝑆))
86, 7syl6 35 . . . 4 (𝐾 ∈ AtLat → (𝑞 ∈ (Atoms‘𝐾) → (𝑋 = {𝑞} → 𝑋𝑆)))
98rexlimdv 3280 . . 3 (𝐾 ∈ AtLat → (∃𝑞 ∈ (Atoms‘𝐾)𝑋 = {𝑞} → 𝑋𝑆))
103, 9sylbid 241 . 2 (𝐾 ∈ AtLat → (𝑋𝑃𝑋𝑆))
1110imp 407 1 ((𝐾 ∈ AtLat ∧ 𝑋𝑃) → 𝑋𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wrex 3136  {csn 4557  cfv 6348  Atomscatm 36279  AtLatcal 36280  PointscpointsN 36511  PSubSpcpsubsp 36512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-proset 17526  df-poset 17544  df-plt 17556  df-lub 17572  df-glb 17573  df-join 17574  df-meet 17575  df-p0 17637  df-lat 17644  df-covers 36282  df-ats 36283  df-atl 36314  df-pointsN 36518  df-psubsp 36519
This theorem is referenced by: (None)
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