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Theorem snatpsubN 38707
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 = (Atomsβ€˜πΎ)
snpsub.s 𝑆 = (PSubSpβ€˜πΎ)
Assertion
Ref Expression
snatpsubN ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) β†’ {𝑃} ∈ 𝑆)

Proof of Theorem snatpsubN
Dummy variables π‘ž 𝑝 π‘Ÿ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snssi 4811 . . . . . 6 (𝑃 ∈ 𝐴 β†’ {𝑃} βŠ† 𝐴)
21adantl 482 . . . . 5 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) β†’ {𝑃} βŠ† 𝐴)
3 atllat 38256 . . . . . . . . . . . . . . 15 (𝐾 ∈ AtLat β†’ 𝐾 ∈ Lat)
4 eqid 2732 . . . . . . . . . . . . . . . 16 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
5 snpsub.a . . . . . . . . . . . . . . . 16 𝐴 = (Atomsβ€˜πΎ)
64, 5atbase 38245 . . . . . . . . . . . . . . 15 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ (Baseβ€˜πΎ))
7 eqid 2732 . . . . . . . . . . . . . . . 16 (joinβ€˜πΎ) = (joinβ€˜πΎ)
84, 7latjidm 18417 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Baseβ€˜πΎ)) β†’ (𝑃(joinβ€˜πΎ)𝑃) = 𝑃)
93, 6, 8syl2an 596 . . . . . . . . . . . . . 14 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) β†’ (𝑃(joinβ€˜πΎ)𝑃) = 𝑃)
109adantr 481 . . . . . . . . . . . . 13 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) ∧ π‘Ÿ ∈ 𝐴) β†’ (𝑃(joinβ€˜πΎ)𝑃) = 𝑃)
1110breq2d 5160 . . . . . . . . . . . 12 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) ∧ π‘Ÿ ∈ 𝐴) β†’ (π‘Ÿ(leβ€˜πΎ)(𝑃(joinβ€˜πΎ)𝑃) ↔ π‘Ÿ(leβ€˜πΎ)𝑃))
12 eqid 2732 . . . . . . . . . . . . . . . 16 (leβ€˜πΎ) = (leβ€˜πΎ)
1312, 5atcmp 38267 . . . . . . . . . . . . . . 15 ((𝐾 ∈ AtLat ∧ π‘Ÿ ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) β†’ (π‘Ÿ(leβ€˜πΎ)𝑃 ↔ π‘Ÿ = 𝑃))
14133com23 1126 . . . . . . . . . . . . . 14 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ π‘Ÿ ∈ 𝐴) β†’ (π‘Ÿ(leβ€˜πΎ)𝑃 ↔ π‘Ÿ = 𝑃))
15143expa 1118 . . . . . . . . . . . . 13 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) ∧ π‘Ÿ ∈ 𝐴) β†’ (π‘Ÿ(leβ€˜πΎ)𝑃 ↔ π‘Ÿ = 𝑃))
1615biimpd 228 . . . . . . . . . . . 12 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) ∧ π‘Ÿ ∈ 𝐴) β†’ (π‘Ÿ(leβ€˜πΎ)𝑃 β†’ π‘Ÿ = 𝑃))
1711, 16sylbid 239 . . . . . . . . . . 11 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) ∧ π‘Ÿ ∈ 𝐴) β†’ (π‘Ÿ(leβ€˜πΎ)(𝑃(joinβ€˜πΎ)𝑃) β†’ π‘Ÿ = 𝑃))
1817adantld 491 . . . . . . . . . 10 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) ∧ π‘Ÿ ∈ 𝐴) β†’ (((𝑝 = 𝑃 ∧ π‘ž = 𝑃) ∧ π‘Ÿ(leβ€˜πΎ)(𝑃(joinβ€˜πΎ)𝑃)) β†’ π‘Ÿ = 𝑃))
19 velsn 4644 . . . . . . . . . . . . 13 (𝑝 ∈ {𝑃} ↔ 𝑝 = 𝑃)
20 velsn 4644 . . . . . . . . . . . . 13 (π‘ž ∈ {𝑃} ↔ π‘ž = 𝑃)
2119, 20anbi12i 627 . . . . . . . . . . . 12 ((𝑝 ∈ {𝑃} ∧ π‘ž ∈ {𝑃}) ↔ (𝑝 = 𝑃 ∧ π‘ž = 𝑃))
2221anbi1i 624 . . . . . . . . . . 11 (((𝑝 ∈ {𝑃} ∧ π‘ž ∈ {𝑃}) ∧ π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž)) ↔ ((𝑝 = 𝑃 ∧ π‘ž = 𝑃) ∧ π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž)))
23 oveq12 7420 . . . . . . . . . . . . 13 ((𝑝 = 𝑃 ∧ π‘ž = 𝑃) β†’ (𝑝(joinβ€˜πΎ)π‘ž) = (𝑃(joinβ€˜πΎ)𝑃))
2423breq2d 5160 . . . . . . . . . . . 12 ((𝑝 = 𝑃 ∧ π‘ž = 𝑃) β†’ (π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž) ↔ π‘Ÿ(leβ€˜πΎ)(𝑃(joinβ€˜πΎ)𝑃)))
2524pm5.32i 575 . . . . . . . . . . 11 (((𝑝 = 𝑃 ∧ π‘ž = 𝑃) ∧ π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž)) ↔ ((𝑝 = 𝑃 ∧ π‘ž = 𝑃) ∧ π‘Ÿ(leβ€˜πΎ)(𝑃(joinβ€˜πΎ)𝑃)))
2622, 25bitri 274 . . . . . . . . . 10 (((𝑝 ∈ {𝑃} ∧ π‘ž ∈ {𝑃}) ∧ π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž)) ↔ ((𝑝 = 𝑃 ∧ π‘ž = 𝑃) ∧ π‘Ÿ(leβ€˜πΎ)(𝑃(joinβ€˜πΎ)𝑃)))
27 velsn 4644 . . . . . . . . . 10 (π‘Ÿ ∈ {𝑃} ↔ π‘Ÿ = 𝑃)
2818, 26, 273imtr4g 295 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) ∧ π‘Ÿ ∈ 𝐴) β†’ (((𝑝 ∈ {𝑃} ∧ π‘ž ∈ {𝑃}) ∧ π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž)) β†’ π‘Ÿ ∈ {𝑃}))
2928exp4b 431 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) β†’ (π‘Ÿ ∈ 𝐴 β†’ ((𝑝 ∈ {𝑃} ∧ π‘ž ∈ {𝑃}) β†’ (π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž) β†’ π‘Ÿ ∈ {𝑃}))))
3029com23 86 . . . . . . 7 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) β†’ ((𝑝 ∈ {𝑃} ∧ π‘ž ∈ {𝑃}) β†’ (π‘Ÿ ∈ 𝐴 β†’ (π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž) β†’ π‘Ÿ ∈ {𝑃}))))
3130ralrimdv 3152 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) β†’ ((𝑝 ∈ {𝑃} ∧ π‘ž ∈ {𝑃}) β†’ βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž) β†’ π‘Ÿ ∈ {𝑃})))
3231ralrimivv 3198 . . . . 5 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) β†’ βˆ€π‘ ∈ {𝑃}βˆ€π‘ž ∈ {𝑃}βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž) β†’ π‘Ÿ ∈ {𝑃}))
332, 32jca 512 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) β†’ ({𝑃} βŠ† 𝐴 ∧ βˆ€π‘ ∈ {𝑃}βˆ€π‘ž ∈ {𝑃}βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž) β†’ π‘Ÿ ∈ {𝑃})))
3433ex 413 . . 3 (𝐾 ∈ AtLat β†’ (𝑃 ∈ 𝐴 β†’ ({𝑃} βŠ† 𝐴 ∧ βˆ€π‘ ∈ {𝑃}βˆ€π‘ž ∈ {𝑃}βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž) β†’ π‘Ÿ ∈ {𝑃}))))
35 snpsub.s . . . 4 𝑆 = (PSubSpβ€˜πΎ)
3612, 7, 5, 35ispsubsp 38702 . . 3 (𝐾 ∈ AtLat β†’ ({𝑃} ∈ 𝑆 ↔ ({𝑃} βŠ† 𝐴 ∧ βˆ€π‘ ∈ {𝑃}βˆ€π‘ž ∈ {𝑃}βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž) β†’ π‘Ÿ ∈ {𝑃}))))
3734, 36sylibrd 258 . 2 (𝐾 ∈ AtLat β†’ (𝑃 ∈ 𝐴 β†’ {𝑃} ∈ 𝑆))
3837imp 407 1 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) β†’ {𝑃} ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061   βŠ† wss 3948  {csn 4628   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7411  Basecbs 17146  lecple 17206  joincjn 18266  Latclat 18386  Atomscatm 38219  AtLatcal 38220  PSubSpcpsubsp 38453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-proset 18250  df-poset 18268  df-plt 18285  df-lub 18301  df-glb 18302  df-join 18303  df-meet 18304  df-p0 18380  df-lat 18387  df-covers 38222  df-ats 38223  df-atl 38254  df-psubsp 38460
This theorem is referenced by:  pointpsubN  38708  pclfinN  38857  pclfinclN  38907
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