Theorem snatpsubN 37046

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 4701	. . . . . 6 ⊢ (𝑃 ∈ 𝐴 → {𝑃} ⊆ 𝐴)
2	1	adantl 485	. . . . 5 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → {𝑃} ⊆ 𝐴)
3		atllat 36596	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Lat)
4		eqid 2798	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐾) = (Base‘𝐾)
5		snpsub.a	. . . . . . . . . . . . . . . 16 ⊢ 𝐴 = (Atoms‘𝐾)
6	4, 5	atbase 36585	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
7		eqid 2798	. . . . . . . . . . . . . . . 16 ⊢ (join‘𝐾) = (join‘𝐾)
8	4, 7	latjidm 17676	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑃(join‘𝐾)𝑃) = 𝑃)
9	3, 6, 8	syl2an 598	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → (𝑃(join‘𝐾)𝑃) = 𝑃)
10	9	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) ∧ 𝑟 ∈ 𝐴) → (𝑃(join‘𝐾)𝑃) = 𝑃)
11	10	breq2d 5042	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) ∧ 𝑟 ∈ 𝐴) → (𝑟(le‘𝐾)(𝑃(join‘𝐾)𝑃) ↔ 𝑟(le‘𝐾)𝑃))
12		eqid 2798	. . . . . . . . . . . . . . . 16 ⊢ (le‘𝐾) = (le‘𝐾)
13	12, 5	atcmp 36607	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ AtLat ∧ 𝑟 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝑟(le‘𝐾)𝑃 ↔ 𝑟 = 𝑃))
14	13	3com23 1123	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) → (𝑟(le‘𝐾)𝑃 ↔ 𝑟 = 𝑃))
15	14	3expa 1115	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) ∧ 𝑟 ∈ 𝐴) → (𝑟(le‘𝐾)𝑃 ↔ 𝑟 = 𝑃))
16	15	biimpd 232	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) ∧ 𝑟 ∈ 𝐴) → (𝑟(le‘𝐾)𝑃 → 𝑟 = 𝑃))
17	11, 16	sylbid 243	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) ∧ 𝑟 ∈ 𝐴) → (𝑟(le‘𝐾)(𝑃(join‘𝐾)𝑃) → 𝑟 = 𝑃))
18	17	adantld 494	. . . . . . . . . 10 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) ∧ 𝑟 ∈ 𝐴) → (((𝑝 = 𝑃 ∧ 𝑞 = 𝑃) ∧ 𝑟(le‘𝐾)(𝑃(join‘𝐾)𝑃)) → 𝑟 = 𝑃))
19		velsn 4541	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ {𝑃} ↔ 𝑝 = 𝑃)
20		velsn 4541	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ {𝑃} ↔ 𝑞 = 𝑃)
21	19, 20	anbi12i 629	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ {𝑃} ∧ 𝑞 ∈ {𝑃}) ↔ (𝑝 = 𝑃 ∧ 𝑞 = 𝑃))
22	21	anbi1i 626	. . . . . . . . . . 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 7144	. . . . . . . . . . . . 13 ⊢ ((𝑝 = 𝑃 ∧ 𝑞 = 𝑃) → (𝑝(join‘𝐾)𝑞) = (𝑃(join‘𝐾)𝑃))
24	23	breq2d 5042	. . . . . . . . . . . 12 ⊢ ((𝑝 = 𝑃 ∧ 𝑞 = 𝑃) → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ↔ 𝑟(le‘𝐾)(𝑃(join‘𝐾)𝑃)))
25	24	pm5.32i 578	. . . . . . . . . . 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 278	. . . . . . . . . 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 4541	. . . . . . . . . 10 ⊢ (𝑟 ∈ {𝑃} ↔ 𝑟 = 𝑃)
28	18, 26, 27	3imtr4g 299	. . . . . . . . 9 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) ∧ 𝑟 ∈ 𝐴) → (((𝑝 ∈ {𝑃} ∧ 𝑞 ∈ {𝑃}) ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞)) → 𝑟 ∈ {𝑃}))
29	28	exp4b 434	. . . . . . . 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 3153	. . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → ((𝑝 ∈ {𝑃} ∧ 𝑞 ∈ {𝑃}) → ∀𝑟 ∈ 𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑃})))
32	31	ralrimivv 3155	. . . . 5 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → ∀𝑝 ∈ {𝑃}∀𝑞 ∈ {𝑃}∀𝑟 ∈ 𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑃}))
33	2, 32	jca 515	. . . 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 416	. . 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 37041	. . 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 262	. 2 ⊢ (𝐾 ∈ AtLat → (𝑃 ∈ 𝐴 → {𝑃} ∈ 𝑆))
38	37	imp 410	1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → {𝑃} ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ⊆ wss 3881  {csn 4525   class class class wbr 5030  ‘cfv 6324  (class class class)co 7135  Basecbs 16475  lecple 16564  joincjn 17546  Latclat 17647  Atomscatm 36559  AtLatcal 36560  PSubSpcpsubsp 36792

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-lat 17648  df-covers 36562  df-ats 36563  df-atl 36594  df-psubsp 36799

This theorem is referenced by:  pointpsubN  37047  pclfinN  37196  pclfinclN  37246