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Theorem pointpsubN 37046
 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 2801 . . . 4 (Atoms‘𝐾) = (Atoms‘𝐾)
2 pointpsub.p . . . 4 𝑃 = (Points‘𝐾)
31, 2ispointN 37037 . . 3 (𝐾 ∈ AtLat → (𝑋𝑃 ↔ ∃𝑞 ∈ (Atoms‘𝐾)𝑋 = {𝑞}))
4 pointpsub.s . . . . . . 7 𝑆 = (PSubSp‘𝐾)
51, 4snatpsubN 37045 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑞 ∈ (Atoms‘𝐾)) → {𝑞} ∈ 𝑆)
65ex 416 . . . . 5 (𝐾 ∈ AtLat → (𝑞 ∈ (Atoms‘𝐾) → {𝑞} ∈ 𝑆))
7 eleq1a 2888 . . . . 5 ({𝑞} ∈ 𝑆 → (𝑋 = {𝑞} → 𝑋𝑆))
86, 7syl6 35 . . . 4 (𝐾 ∈ AtLat → (𝑞 ∈ (Atoms‘𝐾) → (𝑋 = {𝑞} → 𝑋𝑆)))
98rexlimdv 3245 . . 3 (𝐾 ∈ AtLat → (∃𝑞 ∈ (Atoms‘𝐾)𝑋 = {𝑞} → 𝑋𝑆))
103, 9sylbid 243 . 2 (𝐾 ∈ AtLat → (𝑋𝑃𝑋𝑆))
1110imp 410 1 ((𝐾 ∈ AtLat ∧ 𝑋𝑃) → 𝑋𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2112  ∃wrex 3110  {csn 4528  ‘cfv 6328  Atomscatm 36558  AtLatcal 36559  PointscpointsN 36790  PSubSpcpsubsp 36791 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-proset 17534  df-poset 17552  df-plt 17564  df-lub 17580  df-glb 17581  df-join 17582  df-meet 17583  df-p0 17645  df-lat 17652  df-covers 36561  df-ats 36562  df-atl 36593  df-pointsN 36797  df-psubsp 36798 This theorem is referenced by: (None)
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