Theorem pclclN 35177

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 35176	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦})
5	1, 2	atpsubN 35039	. . . 4 ⊢ (𝐾 ∈ 𝑉 → 𝐴 ∈ 𝑆)
6		sseq2 3627	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑋 ⊆ 𝑦 ↔ 𝑋 ⊆ 𝐴))
7	6	intminss 4503	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑋 ⊆ 𝐴) → ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ⊆ 𝐴)
8	5, 7	sylan 488	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ⊆ 𝐴)
9		r19.26 3064	. . . . . . . 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 907	. . . . . . . . 9 ⊢ ((𝑋 ⊆ 𝑦 → (𝑝 ∈ 𝑦 ∧ 𝑞 ∈ 𝑦)) ↔ ((𝑋 ⊆ 𝑦 → 𝑝 ∈ 𝑦) ∧ (𝑋 ⊆ 𝑦 → 𝑞 ∈ 𝑦)))
11	10	ralbii 2980	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑆 (𝑋 ⊆ 𝑦 → (𝑝 ∈ 𝑦 ∧ 𝑞 ∈ 𝑦)) ↔ ∀𝑦 ∈ 𝑆 ((𝑋 ⊆ 𝑦 → 𝑝 ∈ 𝑦) ∧ (𝑋 ⊆ 𝑦 → 𝑞 ∈ 𝑦)))
12		vex 3203	. . . . . . . . . 10 ⊢ 𝑝 ∈ V
13	12	elintrab 4488	. . . . . . . . 9 ⊢ (𝑝 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ↔ ∀𝑦 ∈ 𝑆 (𝑋 ⊆ 𝑦 → 𝑝 ∈ 𝑦))
14		vex 3203	. . . . . . . . . 10 ⊢ 𝑞 ∈ V
15	14	elintrab 4488	. . . . . . . . 9 ⊢ (𝑞 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ↔ ∀𝑦 ∈ 𝑆 (𝑋 ⊆ 𝑦 → 𝑞 ∈ 𝑦))
16	13, 15	anbi12i 733	. . . . . . . 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 293	. . . . . . 7 ⊢ ((𝑝 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ∧ 𝑞 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦}) ↔ ∀𝑦 ∈ 𝑆 (𝑋 ⊆ 𝑦 → (𝑝 ∈ 𝑦 ∧ 𝑞 ∈ 𝑦)))
18		simpll1 1100	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (𝑝 ∈ 𝑦 ∧ 𝑞 ∈ 𝑦)) → 𝐾 ∈ 𝑉)
19		simplr 792	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (𝑝 ∈ 𝑦 ∧ 𝑞 ∈ 𝑦)) → 𝑦 ∈ 𝑆)
20		simpll3 1102	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (𝑝 ∈ 𝑦 ∧ 𝑞 ∈ 𝑦)) → 𝑟 ∈ 𝐴)
21		simprl 794	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (𝑝 ∈ 𝑦 ∧ 𝑞 ∈ 𝑦)) → 𝑝 ∈ 𝑦)
22		simprr 796	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (𝑝 ∈ 𝑦 ∧ 𝑞 ∈ 𝑦)) → 𝑞 ∈ 𝑦)
23		simpll2 1101	. . . . . . . . . . . . . 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 2622	. . . . . . . . . . . . . . 15 ⊢ (le‘𝐾) = (le‘𝐾)
25		eqid 2622	. . . . . . . . . . . . . . 15 ⊢ (join‘𝐾) = (join‘𝐾)
26	24, 25, 1, 2	psubspi2N 35034	. . . . . . . . . . . . . 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 1341	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟 ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (𝑝 ∈ 𝑦 ∧ 𝑞 ∈ 𝑦)) → 𝑟 ∈ 𝑦)
28	27	ex 450	. . . . . . . . . . . 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 2962	. . . . . . . . . 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 3203	. . . . . . . . . . 11 ⊢ 𝑟 ∈ V
32	31	elintrab 4488	. . . . . . . . . 10 ⊢ (𝑟 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ↔ ∀𝑦 ∈ 𝑆 (𝑋 ⊆ 𝑦 → 𝑟 ∈ 𝑦))
33	30, 32	syl6ibr 242	. . . . . . . . 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 1264	. . . . . . . 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 232	. . . . . 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 2968	. . . . 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 2970	. . . 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 35031	. . . 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 957	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ∈ 𝑆)
43	4, 42	eqeltrd 2701	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916   ⊆ wss 3574  ∩ cint 4475   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650  lecple 15948  joincjn 16944  Atomscatm 34550  PSubSpcpsubsp 34782  PClcpclN 35173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-psubsp 34789  df-pclN 35174

This theorem is referenced by:  pclunN  35184  pclfinN  35186