Theorem snatpsubN 40155

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 4766	. . . . . 6 ⊢ (𝑃 ∈ 𝐴 → {𝑃} ⊆ 𝐴)
2	1	adantl 481	. . . . 5 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → {𝑃} ⊆ 𝐴)
3		atllat 39705	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Lat)
4		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐾) = (Base‘𝐾)
5		snpsub.a	. . . . . . . . . . . . . . . 16 ⊢ 𝐴 = (Atoms‘𝐾)
6	4, 5	atbase 39694	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
7		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (join‘𝐾) = (join‘𝐾)
8	4, 7	latjidm 18399	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) ∧ 𝑟 ∈ 𝐴) → (𝑃(join‘𝐾)𝑃) = 𝑃)
11	10	breq2d 5112	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) ∧ 𝑟 ∈ 𝐴) → (𝑟(le‘𝐾)(𝑃(join‘𝐾)𝑃) ↔ 𝑟(le‘𝐾)𝑃))
12		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (le‘𝐾) = (le‘𝐾)
13	12, 5	atcmp 39716	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ AtLat ∧ 𝑟 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝑟(le‘𝐾)𝑃 ↔ 𝑟 = 𝑃))
14	13	3com23 1127	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) → (𝑟(le‘𝐾)𝑃 ↔ 𝑟 = 𝑃))
15	14	3expa 1119	. . . . . . . . . . . . 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 4598	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ {𝑃} ↔ 𝑝 = 𝑃)
20		velsn 4598	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ {𝑃} ↔ 𝑞 = 𝑃)
21	19, 20	anbi12i 629	. . . . . . . . . . . 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 7379	. . . . . . . . . . . . 13 ⊢ ((𝑝 = 𝑃 ∧ 𝑞 = 𝑃) → (𝑝(join‘𝐾)𝑞) = (𝑃(join‘𝐾)𝑃))
24	23	breq2d 5112	. . . . . . . . . . . 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 4598	. . . . . . . . . 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 3136	. . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → ((𝑝 ∈ {𝑃} ∧ 𝑞 ∈ {𝑃}) → ∀𝑟 ∈ 𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑃})))
32	31	ralrimivv 3179	. . . . 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 40150	. . 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 1542   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3903  {csn 4582   class class class wbr 5100  ‘cfv 6502  (class class class)co 7370  Basecbs 17150  lecple 17198  joincjn 18248  Latclat 18368  Atomscatm 39668  AtLatcal 39669  PSubSpcpsubsp 39901

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-riota 7327  df-ov 7373  df-oprab 7374  df-proset 18231  df-poset 18250  df-plt 18265  df-lub 18281  df-glb 18282  df-join 18283  df-meet 18284  df-p0 18360  df-lat 18369  df-covers 39671  df-ats 39672  df-atl 39703  df-psubsp 39908

This theorem is referenced by:  pointpsubN  40156  pclfinN  40305  pclfinclN  40355