Theorem pclclN 39301

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 39300	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦})
5	1, 2	atpsubN 39163	. . . 4 ⊢ (𝐾 ∈ 𝑉 → 𝐴 ∈ 𝑆)
6		sseq2 4004	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑋 ⊆ 𝑦 ↔ 𝑋 ⊆ 𝐴))
7	6	intminss 4972	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑋 ⊆ 𝐴) → ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ⊆ 𝐴)
8	5, 7	sylan 579	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ⊆ 𝐴)
9		r19.26 3106	. . . . . . . 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 3088	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑆 (𝑋 ⊆ 𝑦 → (𝑝 ∈ 𝑦 ∧ 𝑞 ∈ 𝑦)) ↔ ∀𝑦 ∈ 𝑆 ((𝑋 ⊆ 𝑦 → 𝑝 ∈ 𝑦) ∧ (𝑋 ⊆ 𝑦 → 𝑞 ∈ 𝑦)))
12		vex 3473	. . . . . . . . . 10 ⊢ 𝑝 ∈ V
13	12	elintrab 4958	. . . . . . . . 9 ⊢ (𝑝 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ↔ ∀𝑦 ∈ 𝑆 (𝑋 ⊆ 𝑦 → 𝑝 ∈ 𝑦))
14		vex 3473	. . . . . . . . . 10 ⊢ 𝑞 ∈ V
15	14	elintrab 4958	. . . . . . . . 9 ⊢ (𝑞 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ↔ ∀𝑦 ∈ 𝑆 (𝑋 ⊆ 𝑦 → 𝑞 ∈ 𝑦))
16	13, 15	anbi12i 626	. . . . . . . 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 1210	. . . . . . . . . . . . . 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 1212	. . . . . . . . . . . . . 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 1211	. . . . . . . . . . . . . 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 2727	. . . . . . . . . . . . . . 15 ⊢ (le‘𝐾) = (le‘𝐾)
25		eqid 2727	. . . . . . . . . . . . . . 15 ⊢ (join‘𝐾) = (join‘𝐾)
26	24, 25, 1, 2	psubspi2N 39158	. . . . . . . . . . . . . 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 1383	. . . . . . . . . . . . 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 3162	. . . . . . . . . 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 3473	. . . . . . . . . . 11 ⊢ 𝑟 ∈ V
32	31	elintrab 4958	. . . . . . . . . 10 ⊢ (𝑟 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ↔ ∀𝑦 ∈ 𝑆 (𝑋 ⊆ 𝑦 → 𝑟 ∈ 𝑦))
33	30, 32	imbitrrdi 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 1117	. . . . . . . 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 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 3147	. . . . 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 3193	. . . 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 39155	. . . 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 712	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦} ∈ 𝑆)
43	4, 42	eqeltrd 2828	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  ∀wral 3056  {crab 3427   ⊆ wss 3944  ∩ cint 4944   class class class wbr 5142  ‘cfv 6542  (class class class)co 7414  lecple 17231  joincjn 18294  Atomscatm 38672  PSubSpcpsubsp 38906  PClcpclN 39297

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-psubsp 38913  df-pclN 39298

This theorem is referenced by:  pclunN  39308  pclfinN  39310