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Theorem snatpsubN 35456
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 4447 . . . . . 6 (𝑃𝐴 → {𝑃} ⊆ 𝐴)
21adantl 473 . . . . 5 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → {𝑃} ⊆ 𝐴)
3 atllat 35007 . . . . . . . . . . . . . . 15 (𝐾 ∈ AtLat → 𝐾 ∈ Lat)
4 eqid 2724 . . . . . . . . . . . . . . . 16 (Base‘𝐾) = (Base‘𝐾)
5 snpsub.a . . . . . . . . . . . . . . . 16 𝐴 = (Atoms‘𝐾)
64, 5atbase 34996 . . . . . . . . . . . . . . 15 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
7 eqid 2724 . . . . . . . . . . . . . . . 16 (join‘𝐾) = (join‘𝐾)
84, 7latjidm 17196 . . . . . . . . . . . . . . 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 4772 . . . . . . . . . . . 12 (((𝐾 ∈ AtLat ∧ 𝑃𝐴) ∧ 𝑟𝐴) → (𝑟(le‘𝐾)(𝑃(join‘𝐾)𝑃) ↔ 𝑟(le‘𝐾)𝑃))
12 eqid 2724 . . . . . . . . . . . . . . . 16 (le‘𝐾) = (le‘𝐾)
1312, 5atcmp 35018 . . . . . . . . . . . . . . 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 4301 . . . . . . . . . . . . 13 (𝑝 ∈ {𝑃} ↔ 𝑝 = 𝑃)
20 velsn 4301 . . . . . . . . . . . . 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 6774 . . . . . . . . . . . . 13 ((𝑝 = 𝑃𝑞 = 𝑃) → (𝑝(join‘𝐾)𝑞) = (𝑃(join‘𝐾)𝑃))
2423breq2d 4772 . . . . . . . . . . . 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 4301 . . . . . . . . . 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 3070 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → ((𝑝 ∈ {𝑃} ∧ 𝑞 ∈ {𝑃}) → ∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑃})))
3231ralrimivv 3072 . . . . 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 35451 . . 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 1596  wcel 2103  wral 3014  wss 3680  {csn 4285   class class class wbr 4760  cfv 6001  (class class class)co 6765  Basecbs 15980  lecple 16071  joincjn 17066  Latclat 17167  Atomscatm 34970  AtLatcal 34971  PSubSpcpsubsp 35202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-riota 6726  df-ov 6768  df-oprab 6769  df-preset 17050  df-poset 17068  df-plt 17080  df-lub 17096  df-glb 17097  df-join 17098  df-meet 17099  df-p0 17161  df-lat 17168  df-covers 34973  df-ats 34974  df-atl 35005  df-psubsp 35209
This theorem is referenced by:  pointpsubN  35457  pclfinN  35606  pclfinclN  35656
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