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Theorem pclclN 38357
Description: Closure of the projective subspace closure function. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
pclfval.a 𝐴 = (Atomsβ€˜πΎ)
pclfval.s 𝑆 = (PSubSpβ€˜πΎ)
pclfval.c π‘ˆ = (PClβ€˜πΎ)
Assertion
Ref Expression
pclclN ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† 𝐴) β†’ (π‘ˆβ€˜π‘‹) ∈ 𝑆)

Proof of Theorem pclclN
Dummy variables 𝑦 π‘ž 𝑝 π‘Ÿ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pclfval.a . . 3 𝐴 = (Atomsβ€˜πΎ)
2 pclfval.s . . 3 𝑆 = (PSubSpβ€˜πΎ)
3 pclfval.c . . 3 π‘ˆ = (PClβ€˜πΎ)
41, 2, 3pclvalN 38356 . 2 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† 𝐴) β†’ (π‘ˆβ€˜π‘‹) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦})
51, 2atpsubN 38219 . . . 4 (𝐾 ∈ 𝑉 β†’ 𝐴 ∈ 𝑆)
6 sseq2 3971 . . . . 5 (𝑦 = 𝐴 β†’ (𝑋 βŠ† 𝑦 ↔ 𝑋 βŠ† 𝐴))
76intminss 4936 . . . 4 ((𝐴 ∈ 𝑆 ∧ 𝑋 βŠ† 𝐴) β†’ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦} βŠ† 𝐴)
85, 7sylan 581 . . 3 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† 𝐴) β†’ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦} βŠ† 𝐴)
9 r19.26 3115 . . . . . . . 8 (βˆ€π‘¦ ∈ 𝑆 ((𝑋 βŠ† 𝑦 β†’ 𝑝 ∈ 𝑦) ∧ (𝑋 βŠ† 𝑦 β†’ π‘ž ∈ 𝑦)) ↔ (βˆ€π‘¦ ∈ 𝑆 (𝑋 βŠ† 𝑦 β†’ 𝑝 ∈ 𝑦) ∧ βˆ€π‘¦ ∈ 𝑆 (𝑋 βŠ† 𝑦 β†’ π‘ž ∈ 𝑦)))
10 jcab 519 . . . . . . . . 9 ((𝑋 βŠ† 𝑦 β†’ (𝑝 ∈ 𝑦 ∧ π‘ž ∈ 𝑦)) ↔ ((𝑋 βŠ† 𝑦 β†’ 𝑝 ∈ 𝑦) ∧ (𝑋 βŠ† 𝑦 β†’ π‘ž ∈ 𝑦)))
1110ralbii 3097 . . . . . . . 8 (βˆ€π‘¦ ∈ 𝑆 (𝑋 βŠ† 𝑦 β†’ (𝑝 ∈ 𝑦 ∧ π‘ž ∈ 𝑦)) ↔ βˆ€π‘¦ ∈ 𝑆 ((𝑋 βŠ† 𝑦 β†’ 𝑝 ∈ 𝑦) ∧ (𝑋 βŠ† 𝑦 β†’ π‘ž ∈ 𝑦)))
12 vex 3450 . . . . . . . . . 10 𝑝 ∈ V
1312elintrab 4922 . . . . . . . . 9 (𝑝 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦} ↔ βˆ€π‘¦ ∈ 𝑆 (𝑋 βŠ† 𝑦 β†’ 𝑝 ∈ 𝑦))
14 vex 3450 . . . . . . . . . 10 π‘ž ∈ V
1514elintrab 4922 . . . . . . . . 9 (π‘ž ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦} ↔ βˆ€π‘¦ ∈ 𝑆 (𝑋 βŠ† 𝑦 β†’ π‘ž ∈ 𝑦))
1613, 15anbi12i 628 . . . . . . . 8 ((𝑝 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦} ∧ π‘ž ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦}) ↔ (βˆ€π‘¦ ∈ 𝑆 (𝑋 βŠ† 𝑦 β†’ 𝑝 ∈ 𝑦) ∧ βˆ€π‘¦ ∈ 𝑆 (𝑋 βŠ† 𝑦 β†’ π‘ž ∈ 𝑦)))
179, 11, 163bitr4ri 304 . . . . . . 7 ((𝑝 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦} ∧ π‘ž ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦}) ↔ βˆ€π‘¦ ∈ 𝑆 (𝑋 βŠ† 𝑦 β†’ (𝑝 ∈ 𝑦 ∧ π‘ž ∈ 𝑦)))
18 simpll1 1213 . . . . . . . . . . . . . 14 ((((𝐾 ∈ 𝑉 ∧ π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž) ∧ π‘Ÿ ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (𝑝 ∈ 𝑦 ∧ π‘ž ∈ 𝑦)) β†’ 𝐾 ∈ 𝑉)
19 simplr 768 . . . . . . . . . . . . . 14 ((((𝐾 ∈ 𝑉 ∧ π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž) ∧ π‘Ÿ ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (𝑝 ∈ 𝑦 ∧ π‘ž ∈ 𝑦)) β†’ 𝑦 ∈ 𝑆)
20 simpll3 1215 . . . . . . . . . . . . . 14 ((((𝐾 ∈ 𝑉 ∧ π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž) ∧ π‘Ÿ ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (𝑝 ∈ 𝑦 ∧ π‘ž ∈ 𝑦)) β†’ π‘Ÿ ∈ 𝐴)
21 simprl 770 . . . . . . . . . . . . . 14 ((((𝐾 ∈ 𝑉 ∧ π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž) ∧ π‘Ÿ ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (𝑝 ∈ 𝑦 ∧ π‘ž ∈ 𝑦)) β†’ 𝑝 ∈ 𝑦)
22 simprr 772 . . . . . . . . . . . . . 14 ((((𝐾 ∈ 𝑉 ∧ π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž) ∧ π‘Ÿ ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (𝑝 ∈ 𝑦 ∧ π‘ž ∈ 𝑦)) β†’ π‘ž ∈ 𝑦)
23 simpll2 1214 . . . . . . . . . . . . . 14 ((((𝐾 ∈ 𝑉 ∧ π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž) ∧ π‘Ÿ ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (𝑝 ∈ 𝑦 ∧ π‘ž ∈ 𝑦)) β†’ π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž))
24 eqid 2737 . . . . . . . . . . . . . . 15 (leβ€˜πΎ) = (leβ€˜πΎ)
25 eqid 2737 . . . . . . . . . . . . . . 15 (joinβ€˜πΎ) = (joinβ€˜πΎ)
2624, 25, 1, 2psubspi2N 38214 . . . . . . . . . . . . . 14 (((𝐾 ∈ 𝑉 ∧ 𝑦 ∈ 𝑆 ∧ π‘Ÿ ∈ 𝐴) ∧ (𝑝 ∈ 𝑦 ∧ π‘ž ∈ 𝑦 ∧ π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž))) β†’ π‘Ÿ ∈ 𝑦)
2718, 19, 20, 21, 22, 23, 26syl33anc 1386 . . . . . . . . . . . . 13 ((((𝐾 ∈ 𝑉 ∧ π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž) ∧ π‘Ÿ ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) ∧ (𝑝 ∈ 𝑦 ∧ π‘ž ∈ 𝑦)) β†’ π‘Ÿ ∈ 𝑦)
2827ex 414 . . . . . . . . . . . 12 (((𝐾 ∈ 𝑉 ∧ π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž) ∧ π‘Ÿ ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) β†’ ((𝑝 ∈ 𝑦 ∧ π‘ž ∈ 𝑦) β†’ π‘Ÿ ∈ 𝑦))
2928imim2d 57 . . . . . . . . . . 11 (((𝐾 ∈ 𝑉 ∧ π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž) ∧ π‘Ÿ ∈ 𝐴) ∧ 𝑦 ∈ 𝑆) β†’ ((𝑋 βŠ† 𝑦 β†’ (𝑝 ∈ 𝑦 ∧ π‘ž ∈ 𝑦)) β†’ (𝑋 βŠ† 𝑦 β†’ π‘Ÿ ∈ 𝑦)))
3029ralimdva 3165 . . . . . . . . . 10 ((𝐾 ∈ 𝑉 ∧ π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž) ∧ π‘Ÿ ∈ 𝐴) β†’ (βˆ€π‘¦ ∈ 𝑆 (𝑋 βŠ† 𝑦 β†’ (𝑝 ∈ 𝑦 ∧ π‘ž ∈ 𝑦)) β†’ βˆ€π‘¦ ∈ 𝑆 (𝑋 βŠ† 𝑦 β†’ π‘Ÿ ∈ 𝑦)))
31 vex 3450 . . . . . . . . . . 11 π‘Ÿ ∈ V
3231elintrab 4922 . . . . . . . . . 10 (π‘Ÿ ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦} ↔ βˆ€π‘¦ ∈ 𝑆 (𝑋 βŠ† 𝑦 β†’ π‘Ÿ ∈ 𝑦))
3330, 32syl6ibr 252 . . . . . . . . 9 ((𝐾 ∈ 𝑉 ∧ π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž) ∧ π‘Ÿ ∈ 𝐴) β†’ (βˆ€π‘¦ ∈ 𝑆 (𝑋 βŠ† 𝑦 β†’ (𝑝 ∈ 𝑦 ∧ π‘ž ∈ 𝑦)) β†’ π‘Ÿ ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦}))
34333exp 1120 . . . . . . . 8 (𝐾 ∈ 𝑉 β†’ (π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž) β†’ (π‘Ÿ ∈ 𝐴 β†’ (βˆ€π‘¦ ∈ 𝑆 (𝑋 βŠ† 𝑦 β†’ (𝑝 ∈ 𝑦 ∧ π‘ž ∈ 𝑦)) β†’ π‘Ÿ ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦}))))
3534com24 95 . . . . . . 7 (𝐾 ∈ 𝑉 β†’ (βˆ€π‘¦ ∈ 𝑆 (𝑋 βŠ† 𝑦 β†’ (𝑝 ∈ 𝑦 ∧ π‘ž ∈ 𝑦)) β†’ (π‘Ÿ ∈ 𝐴 β†’ (π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž) β†’ π‘Ÿ ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦}))))
3617, 35biimtrid 241 . . . . . 6 (𝐾 ∈ 𝑉 β†’ ((𝑝 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦} ∧ π‘ž ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦}) β†’ (π‘Ÿ ∈ 𝐴 β†’ (π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž) β†’ π‘Ÿ ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦}))))
3736ralrimdv 3150 . . . . 5 (𝐾 ∈ 𝑉 β†’ ((𝑝 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦} ∧ π‘ž ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦}) β†’ βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž) β†’ π‘Ÿ ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦})))
3837ralrimivv 3196 . . . 4 (𝐾 ∈ 𝑉 β†’ βˆ€π‘ ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦}βˆ€π‘ž ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦}βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž) β†’ π‘Ÿ ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦}))
3938adantr 482 . . 3 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† 𝐴) β†’ βˆ€π‘ ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦}βˆ€π‘ž ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦}βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž) β†’ π‘Ÿ ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦}))
4024, 25, 1, 2ispsubsp 38211 . . . 4 (𝐾 ∈ 𝑉 β†’ (∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦} ∈ 𝑆 ↔ (∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦} βŠ† 𝐴 ∧ βˆ€π‘ ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦}βˆ€π‘ž ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦}βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž) β†’ π‘Ÿ ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦}))))
4140adantr 482 . . 3 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† 𝐴) β†’ (∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦} ∈ 𝑆 ↔ (∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦} βŠ† 𝐴 ∧ βˆ€π‘ ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦}βˆ€π‘ž ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦}βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž) β†’ π‘Ÿ ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦}))))
428, 39, 41mpbir2and 712 . 2 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† 𝐴) β†’ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦} ∈ 𝑆)
434, 42eqeltrd 2838 1 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† 𝐴) β†’ (π‘ˆβ€˜π‘‹) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  {crab 3408   βŠ† wss 3911  βˆ© cint 4908   class class class wbr 5106  β€˜cfv 6497  (class class class)co 7358  lecple 17141  joincjn 18201  Atomscatm 37728  PSubSpcpsubsp 37962  PClcpclN 38353
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-psubsp 37969  df-pclN 38354
This theorem is referenced by:  pclunN  38364  pclfinN  38366
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