Theorem snatpsubN 36890

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 ∧ 𝑃 ∈ 𝐴) → {𝑃} ∈ 𝑆)

Proof of Theorem snatpsubN

Dummy variables 𝑞 𝑝 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snssi 4744	. . . . . 6 ⊢ (𝑃 ∈ 𝐴 → {𝑃} ⊆ 𝐴)
2	1	adantl 484	. . . . 5 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → {𝑃} ⊆ 𝐴)
3		atllat 36440	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Lat)
4		eqid 2824	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐾) = (Base‘𝐾)
5		snpsub.a	. . . . . . . . . . . . . . . 16 ⊢ 𝐴 = (Atoms‘𝐾)
6	4, 5	atbase 36429	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
7		eqid 2824	. . . . . . . . . . . . . . . 16 ⊢ (join‘𝐾) = (join‘𝐾)
8	4, 7	latjidm 17687	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑃(join‘𝐾)𝑃) = 𝑃)
9	3, 6, 8	syl2an 597	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → (𝑃(join‘𝐾)𝑃) = 𝑃)
10	9	adantr 483	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) ∧ 𝑟 ∈ 𝐴) → (𝑃(join‘𝐾)𝑃) = 𝑃)
11	10	breq2d 5081	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) ∧ 𝑟 ∈ 𝐴) → (𝑟(le‘𝐾)(𝑃(join‘𝐾)𝑃) ↔ 𝑟(le‘𝐾)𝑃))
12		eqid 2824	. . . . . . . . . . . . . . . 16 ⊢ (le‘𝐾) = (le‘𝐾)
13	12, 5	atcmp 36451	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ AtLat ∧ 𝑟 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝑟(le‘𝐾)𝑃 ↔ 𝑟 = 𝑃))
14	13	3com23 1122	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) → (𝑟(le‘𝐾)𝑃 ↔ 𝑟 = 𝑃))
15	14	3expa 1114	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) ∧ 𝑟 ∈ 𝐴) → (𝑟(le‘𝐾)𝑃 ↔ 𝑟 = 𝑃))
16	15	biimpd 231	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) ∧ 𝑟 ∈ 𝐴) → (𝑟(le‘𝐾)𝑃 → 𝑟 = 𝑃))
17	11, 16	sylbid 242	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) ∧ 𝑟 ∈ 𝐴) → (𝑟(le‘𝐾)(𝑃(join‘𝐾)𝑃) → 𝑟 = 𝑃))
18	17	adantld 493	. . . . . . . . . 10 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) ∧ 𝑟 ∈ 𝐴) → (((𝑝 = 𝑃 ∧ 𝑞 = 𝑃) ∧ 𝑟(le‘𝐾)(𝑃(join‘𝐾)𝑃)) → 𝑟 = 𝑃))
19		velsn 4586	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ {𝑃} ↔ 𝑝 = 𝑃)
20		velsn 4586	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ {𝑃} ↔ 𝑞 = 𝑃)
21	19, 20	anbi12i 628	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ {𝑃} ∧ 𝑞 ∈ {𝑃}) ↔ (𝑝 = 𝑃 ∧ 𝑞 = 𝑃))
22	21	anbi1i 625	. . . . . . . . . . 11 ⊢ (((𝑝 ∈ {𝑃} ∧ 𝑞 ∈ {𝑃}) ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞)) ↔ ((𝑝 = 𝑃 ∧ 𝑞 = 𝑃) ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞)))
23		oveq12 7168	. . . . . . . . . . . . 13 ⊢ ((𝑝 = 𝑃 ∧ 𝑞 = 𝑃) → (𝑝(join‘𝐾)𝑞) = (𝑃(join‘𝐾)𝑃))
24	23	breq2d 5081	. . . . . . . . . . . 12 ⊢ ((𝑝 = 𝑃 ∧ 𝑞 = 𝑃) → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ↔ 𝑟(le‘𝐾)(𝑃(join‘𝐾)𝑃)))
25	24	pm5.32i 577	. . . . . . . . . . 11 ⊢ (((𝑝 = 𝑃 ∧ 𝑞 = 𝑃) ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞)) ↔ ((𝑝 = 𝑃 ∧ 𝑞 = 𝑃) ∧ 𝑟(le‘𝐾)(𝑃(join‘𝐾)𝑃)))
26	22, 25	bitri 277	. . . . . . . . . 10 ⊢ (((𝑝 ∈ {𝑃} ∧ 𝑞 ∈ {𝑃}) ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞)) ↔ ((𝑝 = 𝑃 ∧ 𝑞 = 𝑃) ∧ 𝑟(le‘𝐾)(𝑃(join‘𝐾)𝑃)))
27		velsn 4586	. . . . . . . . . 10 ⊢ (𝑟 ∈ {𝑃} ↔ 𝑟 = 𝑃)
28	18, 26, 27	3imtr4g 298	. . . . . . . . 9 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) ∧ 𝑟 ∈ 𝐴) → (((𝑝 ∈ {𝑃} ∧ 𝑞 ∈ {𝑃}) ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞)) → 𝑟 ∈ {𝑃}))
29	28	exp4b 433	. . . . . . . 8 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → (𝑟 ∈ 𝐴 → ((𝑝 ∈ {𝑃} ∧ 𝑞 ∈ {𝑃}) → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑃}))))
30	29	com23 86	. . . . . . 7 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → ((𝑝 ∈ {𝑃} ∧ 𝑞 ∈ {𝑃}) → (𝑟 ∈ 𝐴 → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑃}))))
31	30	ralrimdv 3191	. . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → ((𝑝 ∈ {𝑃} ∧ 𝑞 ∈ {𝑃}) → ∀𝑟 ∈ 𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑃})))
32	31	ralrimivv 3193	. . . . 5 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → ∀𝑝 ∈ {𝑃}∀𝑞 ∈ {𝑃}∀𝑟 ∈ 𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑃}))
33	2, 32	jca 514	. . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → ({𝑃} ⊆ 𝐴 ∧ ∀𝑝 ∈ {𝑃}∀𝑞 ∈ {𝑃}∀𝑟 ∈ 𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑃})))
34	33	ex 415	. . 3 ⊢ (𝐾 ∈ AtLat → (𝑃 ∈ 𝐴 → ({𝑃} ⊆ 𝐴 ∧ ∀𝑝 ∈ {𝑃}∀𝑞 ∈ {𝑃}∀𝑟 ∈ 𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑃}))))
35		snpsub.s	. . . 4 ⊢ 𝑆 = (PSubSp‘𝐾)
36	12, 7, 5, 35	ispsubsp 36885	. . 3 ⊢ (𝐾 ∈ AtLat → ({𝑃} ∈ 𝑆 ↔ ({𝑃} ⊆ 𝐴 ∧ ∀𝑝 ∈ {𝑃}∀𝑞 ∈ {𝑃}∀𝑟 ∈ 𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑃}))))
37	34, 36	sylibrd 261	. 2 ⊢ (𝐾 ∈ AtLat → (𝑃 ∈ 𝐴 → {𝑃} ∈ 𝑆))
38	37	imp 409	1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → {𝑃} ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1536   ∈ wcel 2113  ∀wral 3141   ⊆ wss 3939  {csn 4570   class class class wbr 5069  ‘cfv 6358  (class class class)co 7159  Basecbs 16486  lecple 16575  joincjn 17557  Latclat 17658  Atomscatm 36403  AtLatcal 36404  PSubSpcpsubsp 36636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7117  df-ov 7162  df-oprab 7163  df-proset 17541  df-poset 17559  df-plt 17571  df-lub 17587  df-glb 17588  df-join 17589  df-meet 17590  df-p0 17652  df-lat 17659  df-covers 36406  df-ats 36407  df-atl 36438  df-psubsp 36643

This theorem is referenced by:  pointpsubN  36891  pclfinN  37040  pclfinclN  37090