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Theorem snatpsubN 35531
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 4476 . . . . . 6 (𝑃𝐴 → {𝑃} ⊆ 𝐴)
21adantl 473 . . . . 5 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → {𝑃} ⊆ 𝐴)
3 atllat 35082 . . . . . . . . . . . . . . 15 (𝐾 ∈ AtLat → 𝐾 ∈ Lat)
4 eqid 2752 . . . . . . . . . . . . . . . 16 (Base‘𝐾) = (Base‘𝐾)
5 snpsub.a . . . . . . . . . . . . . . . 16 𝐴 = (Atoms‘𝐾)
64, 5atbase 35071 . . . . . . . . . . . . . . 15 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
7 eqid 2752 . . . . . . . . . . . . . . . 16 (join‘𝐾) = (join‘𝐾)
84, 7latjidm 17267 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑃(join‘𝐾)𝑃) = 𝑃)
93, 6, 8syl2an 495 . . . . . . . . . . . . . 14 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → (𝑃(join‘𝐾)𝑃) = 𝑃)
109adantr 472 . . . . . . . . . . . . 13 (((𝐾 ∈ AtLat ∧ 𝑃𝐴) ∧ 𝑟𝐴) → (𝑃(join‘𝐾)𝑃) = 𝑃)
1110breq2d 4808 . . . . . . . . . . . 12 (((𝐾 ∈ AtLat ∧ 𝑃𝐴) ∧ 𝑟𝐴) → (𝑟(le‘𝐾)(𝑃(join‘𝐾)𝑃) ↔ 𝑟(le‘𝐾)𝑃))
12 eqid 2752 . . . . . . . . . . . . . . . 16 (le‘𝐾) = (le‘𝐾)
1312, 5atcmp 35093 . . . . . . . . . . . . . . 15 ((𝐾 ∈ AtLat ∧ 𝑟𝐴𝑃𝐴) → (𝑟(le‘𝐾)𝑃𝑟 = 𝑃))
14133com23 1120 . . . . . . . . . . . . . 14 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑟𝐴) → (𝑟(le‘𝐾)𝑃𝑟 = 𝑃))
15143expa 1111 . . . . . . . . . . . . 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 484 . . . . . . . . . 10 (((𝐾 ∈ AtLat ∧ 𝑃𝐴) ∧ 𝑟𝐴) → (((𝑝 = 𝑃𝑞 = 𝑃) ∧ 𝑟(le‘𝐾)(𝑃(join‘𝐾)𝑃)) → 𝑟 = 𝑃))
19 velsn 4329 . . . . . . . . . . . . 13 (𝑝 ∈ {𝑃} ↔ 𝑝 = 𝑃)
20 velsn 4329 . . . . . . . . . . . . 13 (𝑞 ∈ {𝑃} ↔ 𝑞 = 𝑃)
2119, 20anbi12i 735 . . . . . . . . . . . 12 ((𝑝 ∈ {𝑃} ∧ 𝑞 ∈ {𝑃}) ↔ (𝑝 = 𝑃𝑞 = 𝑃))
2221anbi1i 733 . . . . . . . . . . 11 (((𝑝 ∈ {𝑃} ∧ 𝑞 ∈ {𝑃}) ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞)) ↔ ((𝑝 = 𝑃𝑞 = 𝑃) ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞)))
23 oveq12 6814 . . . . . . . . . . . . 13 ((𝑝 = 𝑃𝑞 = 𝑃) → (𝑝(join‘𝐾)𝑞) = (𝑃(join‘𝐾)𝑃))
2423breq2d 4808 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑞 = 𝑃) → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ↔ 𝑟(le‘𝐾)(𝑃(join‘𝐾)𝑃)))
2524pm5.32i 672 . . . . . . . . . . 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 4329 . . . . . . . . . 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 633 . . . . . . . 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 3098 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → ((𝑝 ∈ {𝑃} ∧ 𝑞 ∈ {𝑃}) → ∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑃})))
3231ralrimivv 3100 . . . . 5 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → ∀𝑝 ∈ {𝑃}∀𝑞 ∈ {𝑃}∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑃}))
332, 32jca 555 . . . 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 35526 . . 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 444 1 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → {𝑃} ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1624  wcel 2131  wral 3042  wss 3707  {csn 4313   class class class wbr 4796  cfv 6041  (class class class)co 6805  Basecbs 16051  lecple 16142  joincjn 17137  Latclat 17238  Atomscatm 35045  AtLatcal 35046  PSubSpcpsubsp 35277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-oprab 6809  df-preset 17121  df-poset 17139  df-plt 17151  df-lub 17167  df-glb 17168  df-join 17169  df-meet 17170  df-p0 17232  df-lat 17239  df-covers 35048  df-ats 35049  df-atl 35080  df-psubsp 35284
This theorem is referenced by:  pointpsubN  35532  pclfinN  35681  pclfinclN  35731
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