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Theorem pclclN 38750
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 38749 . 2 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† 𝐴) β†’ (π‘ˆβ€˜π‘‹) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦})
51, 2atpsubN 38612 . . . 4 (𝐾 ∈ 𝑉 β†’ 𝐴 ∈ 𝑆)
6 sseq2 4007 . . . . 5 (𝑦 = 𝐴 β†’ (𝑋 βŠ† 𝑦 ↔ 𝑋 βŠ† 𝐴))
76intminss 4977 . . . 4 ((𝐴 ∈ 𝑆 ∧ 𝑋 βŠ† 𝐴) β†’ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦} βŠ† 𝐴)
85, 7sylan 580 . . 3 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† 𝐴) β†’ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦} βŠ† 𝐴)
9 r19.26 3111 . . . . . . . 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 518 . . . . . . . . 9 ((𝑋 βŠ† 𝑦 β†’ (𝑝 ∈ 𝑦 ∧ π‘ž ∈ 𝑦)) ↔ ((𝑋 βŠ† 𝑦 β†’ 𝑝 ∈ 𝑦) ∧ (𝑋 βŠ† 𝑦 β†’ π‘ž ∈ 𝑦)))
1110ralbii 3093 . . . . . . . 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 3478 . . . . . . . . . 10 𝑝 ∈ V
1312elintrab 4963 . . . . . . . . 9 (𝑝 ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦} ↔ βˆ€π‘¦ ∈ 𝑆 (𝑋 βŠ† 𝑦 β†’ 𝑝 ∈ 𝑦))
14 vex 3478 . . . . . . . . . 10 π‘ž ∈ V
1514elintrab 4963 . . . . . . . . 9 (π‘ž ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦} ↔ βˆ€π‘¦ ∈ 𝑆 (𝑋 βŠ† 𝑦 β†’ π‘ž ∈ 𝑦))
1613, 15anbi12i 627 . . . . . . . 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 303 . . . . . . 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 1212 . . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . 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 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 ∈ 𝐴)
21 simprl 769 . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . 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 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†’ π‘Ÿ(leβ€˜πΎ)(𝑝(joinβ€˜πΎ)π‘ž))
24 eqid 2732 . . . . . . . . . . . . . . 15 (leβ€˜πΎ) = (leβ€˜πΎ)
25 eqid 2732 . . . . . . . . . . . . . . 15 (joinβ€˜πΎ) = (joinβ€˜πΎ)
2624, 25, 1, 2psubspi2N 38607 . . . . . . . . . . . . . 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 1385 . . . . . . . . . . . . 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 413 . . . . . . . . . . . 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 3167 . . . . . . . . . 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 3478 . . . . . . . . . . 11 π‘Ÿ ∈ V
3231elintrab 4963 . . . . . . . . . 10 (π‘Ÿ ∈ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦} ↔ βˆ€π‘¦ ∈ 𝑆 (𝑋 βŠ† 𝑦 β†’ π‘Ÿ ∈ 𝑦))
3330, 32syl6ibr 251 . . . . . . . . 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 1119 . . . . . . . 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 3152 . . . . 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 3198 . . . 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 481 . . 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 38604 . . . 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 481 . . 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 711 . 2 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† 𝐴) β†’ ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦} ∈ 𝑆)
434, 42eqeltrd 2833 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  {crab 3432   βŠ† wss 3947  βˆ© cint 4949   class class class wbr 5147  β€˜cfv 6540  (class class class)co 7405  lecple 17200  joincjn 18260  Atomscatm 38121  PSubSpcpsubsp 38355  PClcpclN 38746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-psubsp 38362  df-pclN 38747
This theorem is referenced by:  pclunN  38757  pclfinN  38759
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