Theorem snatpsubN 39737

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 4768	. . . . . 6 ⊢ (𝑃 ∈ 𝐴 → {𝑃} ⊆ 𝐴)
2	1	adantl 481	. . . . 5 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → {𝑃} ⊆ 𝐴)
3		atllat 39286	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Lat)
4		eqid 2729	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐾) = (Base‘𝐾)
5		snpsub.a	. . . . . . . . . . . . . . . 16 ⊢ 𝐴 = (Atoms‘𝐾)
6	4, 5	atbase 39275	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
7		eqid 2729	. . . . . . . . . . . . . . . 16 ⊢ (join‘𝐾) = (join‘𝐾)
8	4, 7	latjidm 18403	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑃(join‘𝐾)𝑃) = 𝑃)
9	3, 6, 8	syl2an 596	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → (𝑃(join‘𝐾)𝑃) = 𝑃)
10	9	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) ∧ 𝑟 ∈ 𝐴) → (𝑃(join‘𝐾)𝑃) = 𝑃)
11	10	breq2d 5114	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) ∧ 𝑟 ∈ 𝐴) → (𝑟(le‘𝐾)(𝑃(join‘𝐾)𝑃) ↔ 𝑟(le‘𝐾)𝑃))
12		eqid 2729	. . . . . . . . . . . . . . . 16 ⊢ (le‘𝐾) = (le‘𝐾)
13	12, 5	atcmp 39297	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ AtLat ∧ 𝑟 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝑟(le‘𝐾)𝑃 ↔ 𝑟 = 𝑃))
14	13	3com23 1126	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) → (𝑟(le‘𝐾)𝑃 ↔ 𝑟 = 𝑃))
15	14	3expa 1118	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) ∧ 𝑟 ∈ 𝐴) → (𝑟(le‘𝐾)𝑃 ↔ 𝑟 = 𝑃))
16	15	biimpd 229	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) ∧ 𝑟 ∈ 𝐴) → (𝑟(le‘𝐾)𝑃 → 𝑟 = 𝑃))
17	11, 16	sylbid 240	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) ∧ 𝑟 ∈ 𝐴) → (𝑟(le‘𝐾)(𝑃(join‘𝐾)𝑃) → 𝑟 = 𝑃))
18	17	adantld 490	. . . . . . . . . 10 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) ∧ 𝑟 ∈ 𝐴) → (((𝑝 = 𝑃 ∧ 𝑞 = 𝑃) ∧ 𝑟(le‘𝐾)(𝑃(join‘𝐾)𝑃)) → 𝑟 = 𝑃))
19		velsn 4601	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ {𝑃} ↔ 𝑝 = 𝑃)
20		velsn 4601	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ {𝑃} ↔ 𝑞 = 𝑃)
21	19, 20	anbi12i 628	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ {𝑃} ∧ 𝑞 ∈ {𝑃}) ↔ (𝑝 = 𝑃 ∧ 𝑞 = 𝑃))
22	21	anbi1i 624	. . . . . . . . . . 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 7378	. . . . . . . . . . . . 13 ⊢ ((𝑝 = 𝑃 ∧ 𝑞 = 𝑃) → (𝑝(join‘𝐾)𝑞) = (𝑃(join‘𝐾)𝑃))
24	23	breq2d 5114	. . . . . . . . . . . 12 ⊢ ((𝑝 = 𝑃 ∧ 𝑞 = 𝑃) → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ↔ 𝑟(le‘𝐾)(𝑃(join‘𝐾)𝑃)))
25	24	pm5.32i 574	. . . . . . . . . . 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 275	. . . . . . . . . 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 4601	. . . . . . . . . 10 ⊢ (𝑟 ∈ {𝑃} ↔ 𝑟 = 𝑃)
28	18, 26, 27	3imtr4g 296	. . . . . . . . 9 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) ∧ 𝑟 ∈ 𝐴) → (((𝑝 ∈ {𝑃} ∧ 𝑞 ∈ {𝑃}) ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞)) → 𝑟 ∈ {𝑃}))
29	28	exp4b 430	. . . . . . . 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 3131	. . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → ((𝑝 ∈ {𝑃} ∧ 𝑞 ∈ {𝑃}) → ∀𝑟 ∈ 𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑃})))
32	31	ralrimivv 3176	. . . . 5 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → ∀𝑝 ∈ {𝑃}∀𝑞 ∈ {𝑃}∀𝑟 ∈ 𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑃}))
33	2, 32	jca 511	. . . 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 412	. . 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 39732	. . 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 259	. 2 ⊢ (𝐾 ∈ AtLat → (𝑃 ∈ 𝐴 → {𝑃} ∈ 𝑆))
38	37	imp 406	1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → {𝑃} ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ⊆ wss 3911  {csn 4585   class class class wbr 5102  ‘cfv 6499  (class class class)co 7369  Basecbs 17155  lecple 17203  joincjn 18252  Latclat 18372  Atomscatm 39249  AtLatcal 39250  PSubSpcpsubsp 39483

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-proset 18235  df-poset 18254  df-plt 18269  df-lub 18285  df-glb 18286  df-join 18287  df-meet 18288  df-p0 18364  df-lat 18373  df-covers 39252  df-ats 39253  df-atl 39284  df-psubsp 39490

This theorem is referenced by:  pointpsubN  39738  pclfinN  39887  pclfinclN  39937