Theorem pclclN 37905

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 37904	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦})
5	1, 2	atpsubN 37767	. . . 4 ⊢ (𝐾 ∈ 𝑉 → 𝐴 ∈ 𝑆)
6		sseq2 3947	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑋 ⊆ 𝑦 ↔ 𝑋 ⊆ 𝐴))
7	6	intminss 4905	. . . 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 3095	. . . . . . . 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 518	. . . . . . . . 9 ⊢ ((𝑋 ⊆ 𝑦 → (𝑝 ∈ 𝑦 ∧ 𝑞 ∈ 𝑦)) ↔ ((𝑋 ⊆ 𝑦 → 𝑝 ∈ 𝑦) ∧ (𝑋 ⊆ 𝑦 → 𝑞 ∈ 𝑦)))
11	10	ralbii 3092	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑆 (𝑋 ⊆ 𝑦 → (𝑝 ∈ 𝑦 ∧ 𝑞 ∈ 𝑦)) ↔ ∀𝑦 ∈ 𝑆 ((𝑋 ⊆ 𝑦 → 𝑝 ∈ 𝑦) ∧ (𝑋 ⊆ 𝑦 → 𝑞 ∈ 𝑦)))
12		vex 3436	. . . . . . . . . 10 ⊢ 𝑝 ∈ V
13	12	elintrab 4891	. . . . . . . . 9 ⊢ (𝑝 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ↔ ∀𝑦 ∈ 𝑆 (𝑋 ⊆ 𝑦 → 𝑝 ∈ 𝑦))
14		vex 3436	. . . . . . . . . 10 ⊢ 𝑞 ∈ V
15	14	elintrab 4891	. . . . . . . . 9 ⊢ (𝑞 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ↔ ∀𝑦 ∈ 𝑆 (𝑋 ⊆ 𝑦 → 𝑞 ∈ 𝑦))
16	13, 15	anbi12i 627	. . . . . . . 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 1211	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (𝑝 ∈ 𝑦 ∧ 𝑞 ∈ 𝑦)) → 𝐾 ∈ 𝑉)
19		simplr 766	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (𝑝 ∈ 𝑦 ∧ 𝑞 ∈ 𝑦)) → 𝑦 ∈ 𝑆)
20		simpll3 1213	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (𝑝 ∈ 𝑦 ∧ 𝑞 ∈ 𝑦)) → 𝑟 ∈ 𝐴)
21		simprl 768	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (𝑝 ∈ 𝑦 ∧ 𝑞 ∈ 𝑦)) → 𝑝 ∈ 𝑦)
22		simprr 770	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (𝑝 ∈ 𝑦 ∧ 𝑞 ∈ 𝑦)) → 𝑞 ∈ 𝑦)
23		simpll2 1212	. . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . 15 ⊢ (le‘𝐾) = (le‘𝐾)
25		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ (join‘𝐾) = (join‘𝐾)
26	24, 25, 1, 2	psubspi2N 37762	. . . . . . . . . . . . . 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 1384	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (𝑝 ∈ 𝑦 ∧ 𝑞 ∈ 𝑦)) → 𝑟 ∈ 𝑦)
28	27	ex 413	. . . . . . . . . . . 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 3108	. . . . . . . . . 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 3436	. . . . . . . . . . 11 ⊢ 𝑟 ∈ V
32	31	elintrab 4891	. . . . . . . . . 10 ⊢ (𝑟 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ↔ ∀𝑦 ∈ 𝑆 (𝑋 ⊆ 𝑦 → 𝑟 ∈ 𝑦))
33	30, 32	syl6ibr 251	. . . . . . . . 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 1118	. . . . . . . 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	syl5bi 241	. . . . . 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 3105	. . . . 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 3122	. . . 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 481	. . 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 37759	. . . 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 481	. . 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 710	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ∈ 𝑆)
43	4, 42	eqeltrd 2839	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  {crab 3068   ⊆ wss 3887  ∩ cint 4879   class class class wbr 5074  ‘cfv 6433  (class class class)co 7275  lecple 16969  joincjn 18029  Atomscatm 37277  PSubSpcpsubsp 37510  PClcpclN 37901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-psubsp 37517  df-pclN 37902

This theorem is referenced by:  pclunN  37912  pclfinN  37914