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Theorem snatpsubN 34513
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 4308 . . . . . 6 (𝑃𝐴 → {𝑃} ⊆ 𝐴)
21adantl 482 . . . . 5 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → {𝑃} ⊆ 𝐴)
3 atllat 34064 . . . . . . . . . . . . . . 15 (𝐾 ∈ AtLat → 𝐾 ∈ Lat)
4 eqid 2621 . . . . . . . . . . . . . . . 16 (Base‘𝐾) = (Base‘𝐾)
5 snpsub.a . . . . . . . . . . . . . . . 16 𝐴 = (Atoms‘𝐾)
64, 5atbase 34053 . . . . . . . . . . . . . . 15 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
7 eqid 2621 . . . . . . . . . . . . . . . 16 (join‘𝐾) = (join‘𝐾)
84, 7latjidm 16995 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑃(join‘𝐾)𝑃) = 𝑃)
93, 6, 8syl2an 494 . . . . . . . . . . . . . 14 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → (𝑃(join‘𝐾)𝑃) = 𝑃)
109adantr 481 . . . . . . . . . . . . 13 (((𝐾 ∈ AtLat ∧ 𝑃𝐴) ∧ 𝑟𝐴) → (𝑃(join‘𝐾)𝑃) = 𝑃)
1110breq2d 4625 . . . . . . . . . . . 12 (((𝐾 ∈ AtLat ∧ 𝑃𝐴) ∧ 𝑟𝐴) → (𝑟(le‘𝐾)(𝑃(join‘𝐾)𝑃) ↔ 𝑟(le‘𝐾)𝑃))
12 eqid 2621 . . . . . . . . . . . . . . . 16 (le‘𝐾) = (le‘𝐾)
1312, 5atcmp 34075 . . . . . . . . . . . . . . 15 ((𝐾 ∈ AtLat ∧ 𝑟𝐴𝑃𝐴) → (𝑟(le‘𝐾)𝑃𝑟 = 𝑃))
14133com23 1268 . . . . . . . . . . . . . 14 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑟𝐴) → (𝑟(le‘𝐾)𝑃𝑟 = 𝑃))
15143expa 1262 . . . . . . . . . . . . 13 (((𝐾 ∈ AtLat ∧ 𝑃𝐴) ∧ 𝑟𝐴) → (𝑟(le‘𝐾)𝑃𝑟 = 𝑃))
1615biimpd 219 . . . . . . . . . . . 12 (((𝐾 ∈ AtLat ∧ 𝑃𝐴) ∧ 𝑟𝐴) → (𝑟(le‘𝐾)𝑃𝑟 = 𝑃))
1711, 16sylbid 230 . . . . . . . . . . 11 (((𝐾 ∈ AtLat ∧ 𝑃𝐴) ∧ 𝑟𝐴) → (𝑟(le‘𝐾)(𝑃(join‘𝐾)𝑃) → 𝑟 = 𝑃))
1817adantld 483 . . . . . . . . . 10 (((𝐾 ∈ AtLat ∧ 𝑃𝐴) ∧ 𝑟𝐴) → (((𝑝 = 𝑃𝑞 = 𝑃) ∧ 𝑟(le‘𝐾)(𝑃(join‘𝐾)𝑃)) → 𝑟 = 𝑃))
19 velsn 4164 . . . . . . . . . . . . 13 (𝑝 ∈ {𝑃} ↔ 𝑝 = 𝑃)
20 velsn 4164 . . . . . . . . . . . . 13 (𝑞 ∈ {𝑃} ↔ 𝑞 = 𝑃)
2119, 20anbi12i 732 . . . . . . . . . . . 12 ((𝑝 ∈ {𝑃} ∧ 𝑞 ∈ {𝑃}) ↔ (𝑝 = 𝑃𝑞 = 𝑃))
2221anbi1i 730 . . . . . . . . . . 11 (((𝑝 ∈ {𝑃} ∧ 𝑞 ∈ {𝑃}) ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞)) ↔ ((𝑝 = 𝑃𝑞 = 𝑃) ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞)))
23 oveq12 6613 . . . . . . . . . . . . 13 ((𝑝 = 𝑃𝑞 = 𝑃) → (𝑝(join‘𝐾)𝑞) = (𝑃(join‘𝐾)𝑃))
2423breq2d 4625 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑞 = 𝑃) → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ↔ 𝑟(le‘𝐾)(𝑃(join‘𝐾)𝑃)))
2524pm5.32i 668 . . . . . . . . . . 11 (((𝑝 = 𝑃𝑞 = 𝑃) ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞)) ↔ ((𝑝 = 𝑃𝑞 = 𝑃) ∧ 𝑟(le‘𝐾)(𝑃(join‘𝐾)𝑃)))
2622, 25bitri 264 . . . . . . . . . 10 (((𝑝 ∈ {𝑃} ∧ 𝑞 ∈ {𝑃}) ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞)) ↔ ((𝑝 = 𝑃𝑞 = 𝑃) ∧ 𝑟(le‘𝐾)(𝑃(join‘𝐾)𝑃)))
27 velsn 4164 . . . . . . . . . 10 (𝑟 ∈ {𝑃} ↔ 𝑟 = 𝑃)
2818, 26, 273imtr4g 285 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃𝐴) ∧ 𝑟𝐴) → (((𝑝 ∈ {𝑃} ∧ 𝑞 ∈ {𝑃}) ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞)) → 𝑟 ∈ {𝑃}))
2928exp4b 631 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → (𝑟𝐴 → ((𝑝 ∈ {𝑃} ∧ 𝑞 ∈ {𝑃}) → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑃}))))
3029com23 86 . . . . . . 7 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → ((𝑝 ∈ {𝑃} ∧ 𝑞 ∈ {𝑃}) → (𝑟𝐴 → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑃}))))
3130ralrimdv 2962 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → ((𝑝 ∈ {𝑃} ∧ 𝑞 ∈ {𝑃}) → ∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑃})))
3231ralrimivv 2964 . . . . 5 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → ∀𝑝 ∈ {𝑃}∀𝑞 ∈ {𝑃}∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑃}))
332, 32jca 554 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → ({𝑃} ⊆ 𝐴 ∧ ∀𝑝 ∈ {𝑃}∀𝑞 ∈ {𝑃}∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑃})))
3433ex 450 . . 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 34508 . . 3 (𝐾 ∈ AtLat → ({𝑃} ∈ 𝑆 ↔ ({𝑃} ⊆ 𝐴 ∧ ∀𝑝 ∈ {𝑃}∀𝑞 ∈ {𝑃}∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑃}))))
3734, 36sylibrd 249 . 2 (𝐾 ∈ AtLat → (𝑃𝐴 → {𝑃} ∈ 𝑆))
3837imp 445 1 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → {𝑃} ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wss 3555  {csn 4148   class class class wbr 4613  cfv 5847  (class class class)co 6604  Basecbs 15781  lecple 15869  joincjn 16865  Latclat 16966  Atomscatm 34027  AtLatcal 34028  PSubSpcpsubsp 34259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-preset 16849  df-poset 16867  df-plt 16879  df-lub 16895  df-glb 16896  df-join 16897  df-meet 16898  df-p0 16960  df-lat 16967  df-covers 34030  df-ats 34031  df-atl 34062  df-psubsp 34266
This theorem is referenced by:  pointpsubN  34514  pclfinN  34663  pclfinclN  34713
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