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Theorem pclclN 35672
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 ((𝐾𝑉𝑋𝐴) → (𝑈𝑋) ∈ 𝑆)

Proof of Theorem pclclN
Dummy variables 𝑦 𝑞 𝑝 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pclfval.a . . 3 𝐴 = (Atoms‘𝐾)
2 pclfval.s . . 3 𝑆 = (PSubSp‘𝐾)
3 pclfval.c . . 3 𝑈 = (PCl‘𝐾)
41, 2, 3pclvalN 35671 . 2 ((𝐾𝑉𝑋𝐴) → (𝑈𝑋) = {𝑦𝑆𝑋𝑦})
51, 2atpsubN 35534 . . . 4 (𝐾𝑉𝐴𝑆)
6 sseq2 3760 . . . . 5 (𝑦 = 𝐴 → (𝑋𝑦𝑋𝐴))
76intminss 4647 . . . 4 ((𝐴𝑆𝑋𝐴) → {𝑦𝑆𝑋𝑦} ⊆ 𝐴)
85, 7sylan 489 . . 3 ((𝐾𝑉𝑋𝐴) → {𝑦𝑆𝑋𝑦} ⊆ 𝐴)
9 r19.26 3194 . . . . . . . 8 (∀𝑦𝑆 ((𝑋𝑦𝑝𝑦) ∧ (𝑋𝑦𝑞𝑦)) ↔ (∀𝑦𝑆 (𝑋𝑦𝑝𝑦) ∧ ∀𝑦𝑆 (𝑋𝑦𝑞𝑦)))
10 jcab 943 . . . . . . . . 9 ((𝑋𝑦 → (𝑝𝑦𝑞𝑦)) ↔ ((𝑋𝑦𝑝𝑦) ∧ (𝑋𝑦𝑞𝑦)))
1110ralbii 3110 . . . . . . . 8 (∀𝑦𝑆 (𝑋𝑦 → (𝑝𝑦𝑞𝑦)) ↔ ∀𝑦𝑆 ((𝑋𝑦𝑝𝑦) ∧ (𝑋𝑦𝑞𝑦)))
12 vex 3335 . . . . . . . . . 10 𝑝 ∈ V
1312elintrab 4632 . . . . . . . . 9 (𝑝 {𝑦𝑆𝑋𝑦} ↔ ∀𝑦𝑆 (𝑋𝑦𝑝𝑦))
14 vex 3335 . . . . . . . . . 10 𝑞 ∈ V
1514elintrab 4632 . . . . . . . . 9 (𝑞 {𝑦𝑆𝑋𝑦} ↔ ∀𝑦𝑆 (𝑋𝑦𝑞𝑦))
1613, 15anbi12i 735 . . . . . . . 8 ((𝑝 {𝑦𝑆𝑋𝑦} ∧ 𝑞 {𝑦𝑆𝑋𝑦}) ↔ (∀𝑦𝑆 (𝑋𝑦𝑝𝑦) ∧ ∀𝑦𝑆 (𝑋𝑦𝑞𝑦)))
179, 11, 163bitr4ri 293 . . . . . . 7 ((𝑝 {𝑦𝑆𝑋𝑦} ∧ 𝑞 {𝑦𝑆𝑋𝑦}) ↔ ∀𝑦𝑆 (𝑋𝑦 → (𝑝𝑦𝑞𝑦)))
18 simpll1 1252 . . . . . . . . . . . . . 14 ((((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) ∧ (𝑝𝑦𝑞𝑦)) → 𝐾𝑉)
19 simplr 809 . . . . . . . . . . . . . 14 ((((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) ∧ (𝑝𝑦𝑞𝑦)) → 𝑦𝑆)
20 simpll3 1256 . . . . . . . . . . . . . 14 ((((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) ∧ (𝑝𝑦𝑞𝑦)) → 𝑟𝐴)
21 simprl 811 . . . . . . . . . . . . . 14 ((((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) ∧ (𝑝𝑦𝑞𝑦)) → 𝑝𝑦)
22 simprr 813 . . . . . . . . . . . . . 14 ((((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) ∧ (𝑝𝑦𝑞𝑦)) → 𝑞𝑦)
23 simpll2 1254 . . . . . . . . . . . . . 14 ((((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) ∧ (𝑝𝑦𝑞𝑦)) → 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞))
24 eqid 2752 . . . . . . . . . . . . . . 15 (le‘𝐾) = (le‘𝐾)
25 eqid 2752 . . . . . . . . . . . . . . 15 (join‘𝐾) = (join‘𝐾)
2624, 25, 1, 2psubspi2N 35529 . . . . . . . . . . . . . 14 (((𝐾𝑉𝑦𝑆𝑟𝐴) ∧ (𝑝𝑦𝑞𝑦𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞))) → 𝑟𝑦)
2718, 19, 20, 21, 22, 23, 26syl33anc 1488 . . . . . . . . . . . . 13 ((((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) ∧ (𝑝𝑦𝑞𝑦)) → 𝑟𝑦)
2827ex 449 . . . . . . . . . . . 12 (((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) → ((𝑝𝑦𝑞𝑦) → 𝑟𝑦))
2928imim2d 57 . . . . . . . . . . 11 (((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) → ((𝑋𝑦 → (𝑝𝑦𝑞𝑦)) → (𝑋𝑦𝑟𝑦)))
3029ralimdva 3092 . . . . . . . . . 10 ((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) → (∀𝑦𝑆 (𝑋𝑦 → (𝑝𝑦𝑞𝑦)) → ∀𝑦𝑆 (𝑋𝑦𝑟𝑦)))
31 vex 3335 . . . . . . . . . . 11 𝑟 ∈ V
3231elintrab 4632 . . . . . . . . . 10 (𝑟 {𝑦𝑆𝑋𝑦} ↔ ∀𝑦𝑆 (𝑋𝑦𝑟𝑦))
3330, 32syl6ibr 242 . . . . . . . . 9 ((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) → (∀𝑦𝑆 (𝑋𝑦 → (𝑝𝑦𝑞𝑦)) → 𝑟 {𝑦𝑆𝑋𝑦}))
34333exp 1112 . . . . . . . 8 (𝐾𝑉 → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → (𝑟𝐴 → (∀𝑦𝑆 (𝑋𝑦 → (𝑝𝑦𝑞𝑦)) → 𝑟 {𝑦𝑆𝑋𝑦}))))
3534com24 95 . . . . . . 7 (𝐾𝑉 → (∀𝑦𝑆 (𝑋𝑦 → (𝑝𝑦𝑞𝑦)) → (𝑟𝐴 → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 {𝑦𝑆𝑋𝑦}))))
3617, 35syl5bi 232 . . . . . 6 (𝐾𝑉 → ((𝑝 {𝑦𝑆𝑋𝑦} ∧ 𝑞 {𝑦𝑆𝑋𝑦}) → (𝑟𝐴 → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 {𝑦𝑆𝑋𝑦}))))
3736ralrimdv 3098 . . . . 5 (𝐾𝑉 → ((𝑝 {𝑦𝑆𝑋𝑦} ∧ 𝑞 {𝑦𝑆𝑋𝑦}) → ∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 {𝑦𝑆𝑋𝑦})))
3837ralrimivv 3100 . . . 4 (𝐾𝑉 → ∀𝑝 {𝑦𝑆𝑋𝑦}∀𝑞 {𝑦𝑆𝑋𝑦}∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 {𝑦𝑆𝑋𝑦}))
3938adantr 472 . . 3 ((𝐾𝑉𝑋𝐴) → ∀𝑝 {𝑦𝑆𝑋𝑦}∀𝑞 {𝑦𝑆𝑋𝑦}∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 {𝑦𝑆𝑋𝑦}))
4024, 25, 1, 2ispsubsp 35526 . . . 4 (𝐾𝑉 → ( {𝑦𝑆𝑋𝑦} ∈ 𝑆 ↔ ( {𝑦𝑆𝑋𝑦} ⊆ 𝐴 ∧ ∀𝑝 {𝑦𝑆𝑋𝑦}∀𝑞 {𝑦𝑆𝑋𝑦}∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 {𝑦𝑆𝑋𝑦}))))
4140adantr 472 . . 3 ((𝐾𝑉𝑋𝐴) → ( {𝑦𝑆𝑋𝑦} ∈ 𝑆 ↔ ( {𝑦𝑆𝑋𝑦} ⊆ 𝐴 ∧ ∀𝑝 {𝑦𝑆𝑋𝑦}∀𝑞 {𝑦𝑆𝑋𝑦}∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 {𝑦𝑆𝑋𝑦}))))
428, 39, 41mpbir2and 995 . 2 ((𝐾𝑉𝑋𝐴) → {𝑦𝑆𝑋𝑦} ∈ 𝑆)
434, 42eqeltrd 2831 1 ((𝐾𝑉𝑋𝐴) → (𝑈𝑋) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1624  wcel 2131  wral 3042  {crab 3046  wss 3707   cint 4619   class class class wbr 4796  cfv 6041  (class class class)co 6805  lecple 16142  joincjn 17137  Atomscatm 35045  PSubSpcpsubsp 35277  PClcpclN 35668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-ov 6808  df-psubsp 35284  df-pclN 35669
This theorem is referenced by:  pclunN  35679  pclfinN  35681
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