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Theorem pointpsubN 39734
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 2735 . . . 4 (Atoms‘𝐾) = (Atoms‘𝐾)
2 pointpsub.p . . . 4 𝑃 = (Points‘𝐾)
31, 2ispointN 39725 . . 3 (𝐾 ∈ AtLat → (𝑋𝑃 ↔ ∃𝑞 ∈ (Atoms‘𝐾)𝑋 = {𝑞}))
4 pointpsub.s . . . . . . 7 𝑆 = (PSubSp‘𝐾)
51, 4snatpsubN 39733 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑞 ∈ (Atoms‘𝐾)) → {𝑞} ∈ 𝑆)
65ex 412 . . . . 5 (𝐾 ∈ AtLat → (𝑞 ∈ (Atoms‘𝐾) → {𝑞} ∈ 𝑆))
7 eleq1a 2834 . . . . 5 ({𝑞} ∈ 𝑆 → (𝑋 = {𝑞} → 𝑋𝑆))
86, 7syl6 35 . . . 4 (𝐾 ∈ AtLat → (𝑞 ∈ (Atoms‘𝐾) → (𝑋 = {𝑞} → 𝑋𝑆)))
98rexlimdv 3151 . . 3 (𝐾 ∈ AtLat → (∃𝑞 ∈ (Atoms‘𝐾)𝑋 = {𝑞} → 𝑋𝑆))
103, 9sylbid 240 . 2 (𝐾 ∈ AtLat → (𝑋𝑃𝑋𝑆))
1110imp 406 1 ((𝐾 ∈ AtLat ∧ 𝑋𝑃) → 𝑋𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wrex 3068  {csn 4631  cfv 6563  Atomscatm 39245  AtLatcal 39246  PointscpointsN 39478  PSubSpcpsubsp 39479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-lat 18490  df-covers 39248  df-ats 39249  df-atl 39280  df-pointsN 39485  df-psubsp 39486
This theorem is referenced by: (None)
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