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Theorem pointpsubN 39754
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 2736 . . . 4 (Atoms‘𝐾) = (Atoms‘𝐾)
2 pointpsub.p . . . 4 𝑃 = (Points‘𝐾)
31, 2ispointN 39745 . . 3 (𝐾 ∈ AtLat → (𝑋𝑃 ↔ ∃𝑞 ∈ (Atoms‘𝐾)𝑋 = {𝑞}))
4 pointpsub.s . . . . . . 7 𝑆 = (PSubSp‘𝐾)
51, 4snatpsubN 39753 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑞 ∈ (Atoms‘𝐾)) → {𝑞} ∈ 𝑆)
65ex 412 . . . . 5 (𝐾 ∈ AtLat → (𝑞 ∈ (Atoms‘𝐾) → {𝑞} ∈ 𝑆))
7 eleq1a 2835 . . . . 5 ({𝑞} ∈ 𝑆 → (𝑋 = {𝑞} → 𝑋𝑆))
86, 7syl6 35 . . . 4 (𝐾 ∈ AtLat → (𝑞 ∈ (Atoms‘𝐾) → (𝑋 = {𝑞} → 𝑋𝑆)))
98rexlimdv 3152 . . 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 1539  wcel 2107  wrex 3069  {csn 4625  cfv 6560  Atomscatm 39265  AtLatcal 39266  PointscpointsN 39498  PSubSpcpsubsp 39499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-proset 18341  df-poset 18360  df-plt 18376  df-lub 18392  df-glb 18393  df-join 18394  df-meet 18395  df-p0 18471  df-lat 18478  df-covers 39268  df-ats 39269  df-atl 39300  df-pointsN 39505  df-psubsp 39506
This theorem is referenced by: (None)
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