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Theorem snatpsubN 34054
 Description: The singleton of an atom is a projective subspace. (Contributed by NM, 9-Sep-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
snpsub.a 𝐴 = (Atoms‘𝐾)
snpsub.s 𝑆 = (PSubSp‘𝐾)
Assertion
Ref Expression
snatpsubN ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → {𝑃} ∈ 𝑆)

Proof of Theorem snatpsubN
Dummy variables 𝑞 𝑝 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snssi 4280 . . . . . 6 (𝑃𝐴 → {𝑃} ⊆ 𝐴)
21adantl 481 . . . . 5 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → {𝑃} ⊆ 𝐴)
3 atllat 33605 . . . . . . . . . . . . . . 15 (𝐾 ∈ AtLat → 𝐾 ∈ Lat)
4 eqid 2610 . . . . . . . . . . . . . . . 16 (Base‘𝐾) = (Base‘𝐾)
5 snpsub.a . . . . . . . . . . . . . . . 16 𝐴 = (Atoms‘𝐾)
64, 5atbase 33594 . . . . . . . . . . . . . . 15 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
7 eqid 2610 . . . . . . . . . . . . . . . 16 (join‘𝐾) = (join‘𝐾)
84, 7latjidm 16897 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑃(join‘𝐾)𝑃) = 𝑃)
93, 6, 8syl2an 493 . . . . . . . . . . . . . 14 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → (𝑃(join‘𝐾)𝑃) = 𝑃)
109adantr 480 . . . . . . . . . . . . 13 (((𝐾 ∈ AtLat ∧ 𝑃𝐴) ∧ 𝑟𝐴) → (𝑃(join‘𝐾)𝑃) = 𝑃)
1110breq2d 4595 . . . . . . . . . . . 12 (((𝐾 ∈ AtLat ∧ 𝑃𝐴) ∧ 𝑟𝐴) → (𝑟(le‘𝐾)(𝑃(join‘𝐾)𝑃) ↔ 𝑟(le‘𝐾)𝑃))
12 eqid 2610 . . . . . . . . . . . . . . . 16 (le‘𝐾) = (le‘𝐾)
1312, 5atcmp 33616 . . . . . . . . . . . . . . 15 ((𝐾 ∈ AtLat ∧ 𝑟𝐴𝑃𝐴) → (𝑟(le‘𝐾)𝑃𝑟 = 𝑃))
14133com23 1263 . . . . . . . . . . . . . 14 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑟𝐴) → (𝑟(le‘𝐾)𝑃𝑟 = 𝑃))
15143expa 1257 . . . . . . . . . . . . 13 (((𝐾 ∈ AtLat ∧ 𝑃𝐴) ∧ 𝑟𝐴) → (𝑟(le‘𝐾)𝑃𝑟 = 𝑃))
1615biimpd 218 . . . . . . . . . . . 12 (((𝐾 ∈ AtLat ∧ 𝑃𝐴) ∧ 𝑟𝐴) → (𝑟(le‘𝐾)𝑃𝑟 = 𝑃))
1711, 16sylbid 229 . . . . . . . . . . 11 (((𝐾 ∈ AtLat ∧ 𝑃𝐴) ∧ 𝑟𝐴) → (𝑟(le‘𝐾)(𝑃(join‘𝐾)𝑃) → 𝑟 = 𝑃))
1817adantld 482 . . . . . . . . . 10 (((𝐾 ∈ AtLat ∧ 𝑃𝐴) ∧ 𝑟𝐴) → (((𝑝 = 𝑃𝑞 = 𝑃) ∧ 𝑟(le‘𝐾)(𝑃(join‘𝐾)𝑃)) → 𝑟 = 𝑃))
19 velsn 4141 . . . . . . . . . . . . 13 (𝑝 ∈ {𝑃} ↔ 𝑝 = 𝑃)
20 velsn 4141 . . . . . . . . . . . . 13 (𝑞 ∈ {𝑃} ↔ 𝑞 = 𝑃)
2119, 20anbi12i 729 . . . . . . . . . . . 12 ((𝑝 ∈ {𝑃} ∧ 𝑞 ∈ {𝑃}) ↔ (𝑝 = 𝑃𝑞 = 𝑃))
2221anbi1i 727 . . . . . . . . . . 11 (((𝑝 ∈ {𝑃} ∧ 𝑞 ∈ {𝑃}) ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞)) ↔ ((𝑝 = 𝑃𝑞 = 𝑃) ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞)))
23 oveq12 6558 . . . . . . . . . . . . 13 ((𝑝 = 𝑃𝑞 = 𝑃) → (𝑝(join‘𝐾)𝑞) = (𝑃(join‘𝐾)𝑃))
2423breq2d 4595 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑞 = 𝑃) → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ↔ 𝑟(le‘𝐾)(𝑃(join‘𝐾)𝑃)))
2524pm5.32i 667 . . . . . . . . . . 11 (((𝑝 = 𝑃𝑞 = 𝑃) ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞)) ↔ ((𝑝 = 𝑃𝑞 = 𝑃) ∧ 𝑟(le‘𝐾)(𝑃(join‘𝐾)𝑃)))
2622, 25bitri 263 . . . . . . . . . 10 (((𝑝 ∈ {𝑃} ∧ 𝑞 ∈ {𝑃}) ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞)) ↔ ((𝑝 = 𝑃𝑞 = 𝑃) ∧ 𝑟(le‘𝐾)(𝑃(join‘𝐾)𝑃)))
27 velsn 4141 . . . . . . . . . 10 (𝑟 ∈ {𝑃} ↔ 𝑟 = 𝑃)
2818, 26, 273imtr4g 284 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃𝐴) ∧ 𝑟𝐴) → (((𝑝 ∈ {𝑃} ∧ 𝑞 ∈ {𝑃}) ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞)) → 𝑟 ∈ {𝑃}))
2928exp4b 630 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → (𝑟𝐴 → ((𝑝 ∈ {𝑃} ∧ 𝑞 ∈ {𝑃}) → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑃}))))
3029com23 84 . . . . . . 7 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → ((𝑝 ∈ {𝑃} ∧ 𝑞 ∈ {𝑃}) → (𝑟𝐴 → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑃}))))
3130ralrimdv 2951 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → ((𝑝 ∈ {𝑃} ∧ 𝑞 ∈ {𝑃}) → ∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑃})))
3231ralrimivv 2953 . . . . 5 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → ∀𝑝 ∈ {𝑃}∀𝑞 ∈ {𝑃}∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑃}))
332, 32jca 553 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → ({𝑃} ⊆ 𝐴 ∧ ∀𝑝 ∈ {𝑃}∀𝑞 ∈ {𝑃}∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑃})))
3433ex 449 . . 3 (𝐾 ∈ AtLat → (𝑃𝐴 → ({𝑃} ⊆ 𝐴 ∧ ∀𝑝 ∈ {𝑃}∀𝑞 ∈ {𝑃}∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑃}))))
35 snpsub.s . . . 4 𝑆 = (PSubSp‘𝐾)
3612, 7, 5, 35ispsubsp 34049 . . 3 (𝐾 ∈ AtLat → ({𝑃} ∈ 𝑆 ↔ ({𝑃} ⊆ 𝐴 ∧ ∀𝑝 ∈ {𝑃}∀𝑞 ∈ {𝑃}∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑃}))))
3734, 36sylibrd 248 . 2 (𝐾 ∈ AtLat → (𝑃𝐴 → {𝑃} ∈ 𝑆))
3837imp 444 1 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → {𝑃} ∈ 𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540  {csn 4125   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  Latclat 16868  Atomscatm 33568  AtLatcal 33569  PSubSpcpsubsp 33800 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-lat 16869  df-covers 33571  df-ats 33572  df-atl 33603  df-psubsp 33807 This theorem is referenced by:  pointpsubN  34055  pclfinN  34204  pclfinclN  34254
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