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Theorem snatpsubN 35727
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 4495 . . . . . 6 (𝑃𝐴 → {𝑃} ⊆ 𝐴)
21adantl 473 . . . . 5 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → {𝑃} ⊆ 𝐴)
3 atllat 35277 . . . . . . . . . . . . . . 15 (𝐾 ∈ AtLat → 𝐾 ∈ Lat)
4 eqid 2765 . . . . . . . . . . . . . . . 16 (Base‘𝐾) = (Base‘𝐾)
5 snpsub.a . . . . . . . . . . . . . . . 16 𝐴 = (Atoms‘𝐾)
64, 5atbase 35266 . . . . . . . . . . . . . . 15 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
7 eqid 2765 . . . . . . . . . . . . . . . 16 (join‘𝐾) = (join‘𝐾)
84, 7latjidm 17354 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑃(join‘𝐾)𝑃) = 𝑃)
93, 6, 8syl2an 589 . . . . . . . . . . . . . 14 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → (𝑃(join‘𝐾)𝑃) = 𝑃)
109adantr 472 . . . . . . . . . . . . 13 (((𝐾 ∈ AtLat ∧ 𝑃𝐴) ∧ 𝑟𝐴) → (𝑃(join‘𝐾)𝑃) = 𝑃)
1110breq2d 4823 . . . . . . . . . . . 12 (((𝐾 ∈ AtLat ∧ 𝑃𝐴) ∧ 𝑟𝐴) → (𝑟(le‘𝐾)(𝑃(join‘𝐾)𝑃) ↔ 𝑟(le‘𝐾)𝑃))
12 eqid 2765 . . . . . . . . . . . . . . . 16 (le‘𝐾) = (le‘𝐾)
1312, 5atcmp 35288 . . . . . . . . . . . . . . 15 ((𝐾 ∈ AtLat ∧ 𝑟𝐴𝑃𝐴) → (𝑟(le‘𝐾)𝑃𝑟 = 𝑃))
14133com23 1156 . . . . . . . . . . . . . 14 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑟𝐴) → (𝑟(le‘𝐾)𝑃𝑟 = 𝑃))
15143expa 1147 . . . . . . . . . . . . 13 (((𝐾 ∈ AtLat ∧ 𝑃𝐴) ∧ 𝑟𝐴) → (𝑟(le‘𝐾)𝑃𝑟 = 𝑃))
1615biimpd 220 . . . . . . . . . . . 12 (((𝐾 ∈ AtLat ∧ 𝑃𝐴) ∧ 𝑟𝐴) → (𝑟(le‘𝐾)𝑃𝑟 = 𝑃))
1711, 16sylbid 231 . . . . . . . . . . 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 4352 . . . . . . . . . . . . 13 (𝑝 ∈ {𝑃} ↔ 𝑝 = 𝑃)
20 velsn 4352 . . . . . . . . . . . . 13 (𝑞 ∈ {𝑃} ↔ 𝑞 = 𝑃)
2119, 20anbi12i 620 . . . . . . . . . . . 12 ((𝑝 ∈ {𝑃} ∧ 𝑞 ∈ {𝑃}) ↔ (𝑝 = 𝑃𝑞 = 𝑃))
2221anbi1i 617 . . . . . . . . . . 11 (((𝑝 ∈ {𝑃} ∧ 𝑞 ∈ {𝑃}) ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞)) ↔ ((𝑝 = 𝑃𝑞 = 𝑃) ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞)))
23 oveq12 6855 . . . . . . . . . . . . 13 ((𝑝 = 𝑃𝑞 = 𝑃) → (𝑝(join‘𝐾)𝑞) = (𝑃(join‘𝐾)𝑃))
2423breq2d 4823 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑞 = 𝑃) → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ↔ 𝑟(le‘𝐾)(𝑃(join‘𝐾)𝑃)))
2524pm5.32i 570 . . . . . . . . . . 11 (((𝑝 = 𝑃𝑞 = 𝑃) ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞)) ↔ ((𝑝 = 𝑃𝑞 = 𝑃) ∧ 𝑟(le‘𝐾)(𝑃(join‘𝐾)𝑃)))
2622, 25bitri 266 . . . . . . . . . 10 (((𝑝 ∈ {𝑃} ∧ 𝑞 ∈ {𝑃}) ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞)) ↔ ((𝑝 = 𝑃𝑞 = 𝑃) ∧ 𝑟(le‘𝐾)(𝑃(join‘𝐾)𝑃)))
27 velsn 4352 . . . . . . . . . 10 (𝑟 ∈ {𝑃} ↔ 𝑟 = 𝑃)
2818, 26, 273imtr4g 287 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃𝐴) ∧ 𝑟𝐴) → (((𝑝 ∈ {𝑃} ∧ 𝑞 ∈ {𝑃}) ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞)) → 𝑟 ∈ {𝑃}))
2928exp4b 421 . . . . . . . 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 3115 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → ((𝑝 ∈ {𝑃} ∧ 𝑞 ∈ {𝑃}) → ∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑃})))
3231ralrimivv 3117 . . . . 5 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → ∀𝑝 ∈ {𝑃}∀𝑞 ∈ {𝑃}∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑃}))
332, 32jca 507 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → ({𝑃} ⊆ 𝐴 ∧ ∀𝑝 ∈ {𝑃}∀𝑞 ∈ {𝑃}∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑃})))
3433ex 401 . . 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 35722 . . 3 (𝐾 ∈ AtLat → ({𝑃} ∈ 𝑆 ↔ ({𝑃} ⊆ 𝐴 ∧ ∀𝑝 ∈ {𝑃}∀𝑞 ∈ {𝑃}∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑃}))))
3734, 36sylibrd 250 . 2 (𝐾 ∈ AtLat → (𝑃𝐴 → {𝑃} ∈ 𝑆))
3837imp 395 1 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → {𝑃} ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wral 3055  wss 3734  {csn 4336   class class class wbr 4811  cfv 6070  (class class class)co 6846  Basecbs 16144  lecple 16235  joincjn 17224  Latclat 17325  Atomscatm 35240  AtLatcal 35241  PSubSpcpsubsp 35473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-riota 6807  df-ov 6849  df-oprab 6850  df-proset 17208  df-poset 17226  df-plt 17238  df-lub 17254  df-glb 17255  df-join 17256  df-meet 17257  df-p0 17319  df-lat 17326  df-covers 35243  df-ats 35244  df-atl 35275  df-psubsp 35480
This theorem is referenced by:  pointpsubN  35728  pclfinN  35877  pclfinclN  35927
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