Theorem pclclN 39852

Description: Closure of the projective subspace closure function. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
pclfval.a	⊢ 𝐴 = (Atoms‘𝐾)
pclfval.s	⊢ 𝑆 = (PSubSp‘𝐾)
pclfval.c	⊢ 𝑈 = (PCl‘𝐾)

Assertion

Ref	Expression
pclclN	⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) ∈ 𝑆)

Proof of Theorem pclclN

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

Step	Hyp	Ref	Expression
1		pclfval.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
2		pclfval.s	. . 3 ⊢ 𝑆 = (PSubSp‘𝐾)
3		pclfval.c	. . 3 ⊢ 𝑈 = (PCl‘𝐾)
4	1, 2, 3	pclvalN 39851	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦})
5	1, 2	atpsubN 39714	. . . 4 ⊢ (𝐾 ∈ 𝑉 → 𝐴 ∈ 𝑆)
6		sseq2 3990	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑋 ⊆ 𝑦 ↔ 𝑋 ⊆ 𝐴))
7	6	intminss 4954	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑋 ⊆ 𝐴) → ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ⊆ 𝐴)
8	5, 7	sylan 580	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ⊆ 𝐴)
9		r19.26 3098	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑆 ((𝑋 ⊆ 𝑦 → 𝑝 ∈ 𝑦) ∧ (𝑋 ⊆ 𝑦 → 𝑞 ∈ 𝑦)) ↔ (∀𝑦 ∈ 𝑆 (𝑋 ⊆ 𝑦 → 𝑝 ∈ 𝑦) ∧ ∀𝑦 ∈ 𝑆 (𝑋 ⊆ 𝑦 → 𝑞 ∈ 𝑦)))
10		jcab 517	. . . . . . . . 9 ⊢ ((𝑋 ⊆ 𝑦 → (𝑝 ∈ 𝑦 ∧ 𝑞 ∈ 𝑦)) ↔ ((𝑋 ⊆ 𝑦 → 𝑝 ∈ 𝑦) ∧ (𝑋 ⊆ 𝑦 → 𝑞 ∈ 𝑦)))
11	10	ralbii 3081	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑆 (𝑋 ⊆ 𝑦 → (𝑝 ∈ 𝑦 ∧ 𝑞 ∈ 𝑦)) ↔ ∀𝑦 ∈ 𝑆 ((𝑋 ⊆ 𝑦 → 𝑝 ∈ 𝑦) ∧ (𝑋 ⊆ 𝑦 → 𝑞 ∈ 𝑦)))
12		vex 3467	. . . . . . . . . 10 ⊢ 𝑝 ∈ V
13	12	elintrab 4940	. . . . . . . . 9 ⊢ (𝑝 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ↔ ∀𝑦 ∈ 𝑆 (𝑋 ⊆ 𝑦 → 𝑝 ∈ 𝑦))
14		vex 3467	. . . . . . . . . 10 ⊢ 𝑞 ∈ V
15	14	elintrab 4940	. . . . . . . . 9 ⊢ (𝑞 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ↔ ∀𝑦 ∈ 𝑆 (𝑋 ⊆ 𝑦 → 𝑞 ∈ 𝑦))
16	13, 15	anbi12i 628	. . . . . . . 8 ⊢ ((𝑝 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ∧ 𝑞 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦}) ↔ (∀𝑦 ∈ 𝑆 (𝑋 ⊆ 𝑦 → 𝑝 ∈ 𝑦) ∧ ∀𝑦 ∈ 𝑆 (𝑋 ⊆ 𝑦 → 𝑞 ∈ 𝑦)))
17	9, 11, 16	3bitr4ri 304	. . . . . . 7 ⊢ ((𝑝 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ∧ 𝑞 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦}) ↔ ∀𝑦 ∈ 𝑆 (𝑋 ⊆ 𝑦 → (𝑝 ∈ 𝑦 ∧ 𝑞 ∈ 𝑦)))
18		simpll1 1212	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (𝑝 ∈ 𝑦 ∧ 𝑞 ∈ 𝑦)) → 𝐾 ∈ 𝑉)
19		simplr 768	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (𝑝 ∈ 𝑦 ∧ 𝑞 ∈ 𝑦)) → 𝑦 ∈ 𝑆)
20		simpll3 1214	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (𝑝 ∈ 𝑦 ∧ 𝑞 ∈ 𝑦)) → 𝑟 ∈ 𝐴)
21		simprl 770	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (𝑝 ∈ 𝑦 ∧ 𝑞 ∈ 𝑦)) → 𝑝 ∈ 𝑦)
22		simprr 772	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (𝑝 ∈ 𝑦 ∧ 𝑞 ∈ 𝑦)) → 𝑞 ∈ 𝑦)
23		simpll2 1213	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (𝑝 ∈ 𝑦 ∧ 𝑞 ∈ 𝑦)) → 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞))
24		eqid 2734	. . . . . . . . . . . . . . 15 ⊢ (le‘𝐾) = (le‘𝐾)
25		eqid 2734	. . . . . . . . . . . . . . 15 ⊢ (join‘𝐾) = (join‘𝐾)
26	24, 25, 1, 2	psubspi2N 39709	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑦 ∈ 𝑆 ∧ 𝑟 ∈ 𝐴) ∧ (𝑝 ∈ 𝑦 ∧ 𝑞 ∈ 𝑦 ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞))) → 𝑟 ∈ 𝑦)
27	18, 19, 20, 21, 22, 23, 26	syl33anc 1386	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (𝑝 ∈ 𝑦 ∧ 𝑞 ∈ 𝑦)) → 𝑟 ∈ 𝑦)
28	27	ex 412	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) → ((𝑝 ∈ 𝑦 ∧ 𝑞 ∈ 𝑦) → 𝑟 ∈ 𝑦))
29	28	imim2d 57	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) → ((𝑋 ⊆ 𝑦 → (𝑝 ∈ 𝑦 ∧ 𝑞 ∈ 𝑦)) → (𝑋 ⊆ 𝑦 → 𝑟 ∈ 𝑦)))
30	29	ralimdva 3154	. . . . . . . . . 10 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟 ∈ 𝐴) → (∀𝑦 ∈ 𝑆 (𝑋 ⊆ 𝑦 → (𝑝 ∈ 𝑦 ∧ 𝑞 ∈ 𝑦)) → ∀𝑦 ∈ 𝑆 (𝑋 ⊆ 𝑦 → 𝑟 ∈ 𝑦)))
31		vex 3467	. . . . . . . . . . 11 ⊢ 𝑟 ∈ V
32	31	elintrab 4940	. . . . . . . . . 10 ⊢ (𝑟 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ↔ ∀𝑦 ∈ 𝑆 (𝑋 ⊆ 𝑦 → 𝑟 ∈ 𝑦))
33	30, 32	imbitrrdi 252	. . . . . . . . 9 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟 ∈ 𝐴) → (∀𝑦 ∈ 𝑆 (𝑋 ⊆ 𝑦 → (𝑝 ∈ 𝑦 ∧ 𝑞 ∈ 𝑦)) → 𝑟 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦}))
34	33	3exp 1119	. . . . . . . 8 ⊢ (𝐾 ∈ 𝑉 → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → (𝑟 ∈ 𝐴 → (∀𝑦 ∈ 𝑆 (𝑋 ⊆ 𝑦 → (𝑝 ∈ 𝑦 ∧ 𝑞 ∈ 𝑦)) → 𝑟 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦}))))
35	34	com24 95	. . . . . . 7 ⊢ (𝐾 ∈ 𝑉 → (∀𝑦 ∈ 𝑆 (𝑋 ⊆ 𝑦 → (𝑝 ∈ 𝑦 ∧ 𝑞 ∈ 𝑦)) → (𝑟 ∈ 𝐴 → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦}))))
36	17, 35	biimtrid 242	. . . . . 6 ⊢ (𝐾 ∈ 𝑉 → ((𝑝 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ∧ 𝑞 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦}) → (𝑟 ∈ 𝐴 → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦}))))
37	36	ralrimdv 3139	. . . . 5 ⊢ (𝐾 ∈ 𝑉 → ((𝑝 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ∧ 𝑞 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦}) → ∀𝑟 ∈ 𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦})))
38	37	ralrimivv 3187	. . . 4 ⊢ (𝐾 ∈ 𝑉 → ∀𝑝 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦}∀𝑞 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦}∀𝑟 ∈ 𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦}))
39	38	adantr 480	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → ∀𝑝 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦}∀𝑞 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦}∀𝑟 ∈ 𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦}))
40	24, 25, 1, 2	ispsubsp 39706	. . . 4 ⊢ (𝐾 ∈ 𝑉 → (∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ∈ 𝑆 ↔ (∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ⊆ 𝐴 ∧ ∀𝑝 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦}∀𝑞 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦}∀𝑟 ∈ 𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦}))))
41	40	adantr 480	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ∈ 𝑆 ↔ (∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ⊆ 𝐴 ∧ ∀𝑝 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦}∀𝑞 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦}∀𝑟 ∈ 𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦}))))
42	8, 39, 41	mpbir2and 713	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ∈ 𝑆)
43	4, 42	eqeltrd 2833	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ∀wral 3050  {crab 3419   ⊆ wss 3931  ∩ cint 4926   class class class wbr 5123  ‘cfv 6541  (class class class)co 7413  lecple 17280  joincjn 18327  Atomscatm 39223  PSubSpcpsubsp 39457  PClcpclN 39848

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-int 4927  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7416  df-psubsp 39464  df-pclN 39849

This theorem is referenced by:  pclunN  39859  pclfinN  39861