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Theorem pclclN 33993
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 33992 . 2 ((𝐾𝑉𝑋𝐴) → (𝑈𝑋) = {𝑦𝑆𝑋𝑦})
51, 2atpsubN 33855 . . . 4 (𝐾𝑉𝐴𝑆)
6 sseq2 3584 . . . . 5 (𝑦 = 𝐴 → (𝑋𝑦𝑋𝐴))
76intminss 4427 . . . 4 ((𝐴𝑆𝑋𝐴) → {𝑦𝑆𝑋𝑦} ⊆ 𝐴)
85, 7sylan 486 . . 3 ((𝐾𝑉𝑋𝐴) → {𝑦𝑆𝑋𝑦} ⊆ 𝐴)
9 r19.26 3040 . . . . . . . 8 (∀𝑦𝑆 ((𝑋𝑦𝑝𝑦) ∧ (𝑋𝑦𝑞𝑦)) ↔ (∀𝑦𝑆 (𝑋𝑦𝑝𝑦) ∧ ∀𝑦𝑆 (𝑋𝑦𝑞𝑦)))
10 jcab 902 . . . . . . . . 9 ((𝑋𝑦 → (𝑝𝑦𝑞𝑦)) ↔ ((𝑋𝑦𝑝𝑦) ∧ (𝑋𝑦𝑞𝑦)))
1110ralbii 2957 . . . . . . . 8 (∀𝑦𝑆 (𝑋𝑦 → (𝑝𝑦𝑞𝑦)) ↔ ∀𝑦𝑆 ((𝑋𝑦𝑝𝑦) ∧ (𝑋𝑦𝑞𝑦)))
12 vex 3170 . . . . . . . . . 10 𝑝 ∈ V
1312elintrab 4412 . . . . . . . . 9 (𝑝 {𝑦𝑆𝑋𝑦} ↔ ∀𝑦𝑆 (𝑋𝑦𝑝𝑦))
14 vex 3170 . . . . . . . . . 10 𝑞 ∈ V
1514elintrab 4412 . . . . . . . . 9 (𝑞 {𝑦𝑆𝑋𝑦} ↔ ∀𝑦𝑆 (𝑋𝑦𝑞𝑦))
1613, 15anbi12i 728 . . . . . . . 8 ((𝑝 {𝑦𝑆𝑋𝑦} ∧ 𝑞 {𝑦𝑆𝑋𝑦}) ↔ (∀𝑦𝑆 (𝑋𝑦𝑝𝑦) ∧ ∀𝑦𝑆 (𝑋𝑦𝑞𝑦)))
179, 11, 163bitr4ri 291 . . . . . . 7 ((𝑝 {𝑦𝑆𝑋𝑦} ∧ 𝑞 {𝑦𝑆𝑋𝑦}) ↔ ∀𝑦𝑆 (𝑋𝑦 → (𝑝𝑦𝑞𝑦)))
18 simpll1 1092 . . . . . . . . . . . . . 14 ((((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) ∧ (𝑝𝑦𝑞𝑦)) → 𝐾𝑉)
19 simplr 787 . . . . . . . . . . . . . 14 ((((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) ∧ (𝑝𝑦𝑞𝑦)) → 𝑦𝑆)
20 simpll3 1094 . . . . . . . . . . . . . 14 ((((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) ∧ (𝑝𝑦𝑞𝑦)) → 𝑟𝐴)
21 simprl 789 . . . . . . . . . . . . . 14 ((((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) ∧ (𝑝𝑦𝑞𝑦)) → 𝑝𝑦)
22 simprr 791 . . . . . . . . . . . . . 14 ((((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) ∧ (𝑝𝑦𝑞𝑦)) → 𝑞𝑦)
23 simpll2 1093 . . . . . . . . . . . . . 14 ((((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) ∧ (𝑝𝑦𝑞𝑦)) → 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞))
24 eqid 2604 . . . . . . . . . . . . . . 15 (le‘𝐾) = (le‘𝐾)
25 eqid 2604 . . . . . . . . . . . . . . 15 (join‘𝐾) = (join‘𝐾)
2624, 25, 1, 2psubspi2N 33850 . . . . . . . . . . . . . 14 (((𝐾𝑉𝑦𝑆𝑟𝐴) ∧ (𝑝𝑦𝑞𝑦𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞))) → 𝑟𝑦)
2718, 19, 20, 21, 22, 23, 26syl33anc 1332 . . . . . . . . . . . . 13 ((((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) ∧ (𝑝𝑦𝑞𝑦)) → 𝑟𝑦)
2827ex 448 . . . . . . . . . . . 12 (((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) → ((𝑝𝑦𝑞𝑦) → 𝑟𝑦))
2928imim2d 54 . . . . . . . . . . 11 (((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) → ((𝑋𝑦 → (𝑝𝑦𝑞𝑦)) → (𝑋𝑦𝑟𝑦)))
3029ralimdva 2939 . . . . . . . . . 10 ((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) → (∀𝑦𝑆 (𝑋𝑦 → (𝑝𝑦𝑞𝑦)) → ∀𝑦𝑆 (𝑋𝑦𝑟𝑦)))
31 vex 3170 . . . . . . . . . . 11 𝑟 ∈ V
3231elintrab 4412 . . . . . . . . . 10 (𝑟 {𝑦𝑆𝑋𝑦} ↔ ∀𝑦𝑆 (𝑋𝑦𝑟𝑦))
3330, 32syl6ibr 240 . . . . . . . . 9 ((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) → (∀𝑦𝑆 (𝑋𝑦 → (𝑝𝑦𝑞𝑦)) → 𝑟 {𝑦𝑆𝑋𝑦}))
34333exp 1255 . . . . . . . 8 (𝐾𝑉 → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → (𝑟𝐴 → (∀𝑦𝑆 (𝑋𝑦 → (𝑝𝑦𝑞𝑦)) → 𝑟 {𝑦𝑆𝑋𝑦}))))
3534com24 92 . . . . . . 7 (𝐾𝑉 → (∀𝑦𝑆 (𝑋𝑦 → (𝑝𝑦𝑞𝑦)) → (𝑟𝐴 → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 {𝑦𝑆𝑋𝑦}))))
3617, 35syl5bi 230 . . . . . 6 (𝐾𝑉 → ((𝑝 {𝑦𝑆𝑋𝑦} ∧ 𝑞 {𝑦𝑆𝑋𝑦}) → (𝑟𝐴 → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 {𝑦𝑆𝑋𝑦}))))
3736ralrimdv 2945 . . . . 5 (𝐾𝑉 → ((𝑝 {𝑦𝑆𝑋𝑦} ∧ 𝑞 {𝑦𝑆𝑋𝑦}) → ∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 {𝑦𝑆𝑋𝑦})))
3837ralrimivv 2947 . . . 4 (𝐾𝑉 → ∀𝑝 {𝑦𝑆𝑋𝑦}∀𝑞 {𝑦𝑆𝑋𝑦}∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 {𝑦𝑆𝑋𝑦}))
3938adantr 479 . . 3 ((𝐾𝑉𝑋𝐴) → ∀𝑝 {𝑦𝑆𝑋𝑦}∀𝑞 {𝑦𝑆𝑋𝑦}∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 {𝑦𝑆𝑋𝑦}))
4024, 25, 1, 2ispsubsp 33847 . . . 4 (𝐾𝑉 → ( {𝑦𝑆𝑋𝑦} ∈ 𝑆 ↔ ( {𝑦𝑆𝑋𝑦} ⊆ 𝐴 ∧ ∀𝑝 {𝑦𝑆𝑋𝑦}∀𝑞 {𝑦𝑆𝑋𝑦}∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 {𝑦𝑆𝑋𝑦}))))
4140adantr 479 . . 3 ((𝐾𝑉𝑋𝐴) → ( {𝑦𝑆𝑋𝑦} ∈ 𝑆 ↔ ( {𝑦𝑆𝑋𝑦} ⊆ 𝐴 ∧ ∀𝑝 {𝑦𝑆𝑋𝑦}∀𝑞 {𝑦𝑆𝑋𝑦}∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 {𝑦𝑆𝑋𝑦}))))
428, 39, 41mpbir2and 958 . 2 ((𝐾𝑉𝑋𝐴) → {𝑦𝑆𝑋𝑦} ∈ 𝑆)
434, 42eqeltrd 2682 1 ((𝐾𝑉𝑋𝐴) → (𝑈𝑋) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1975  wral 2890  {crab 2894  wss 3534   cint 4399   class class class wbr 4572  cfv 5785  (class class class)co 6522  lecple 15716  joincjn 16708  Atomscatm 33366  PSubSpcpsubsp 33598  PClcpclN 33989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-reu 2897  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-int 4400  df-iun 4446  df-br 4573  df-opab 4633  df-mpt 4634  df-id 4938  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-ov 6525  df-psubsp 33605  df-pclN 33990
This theorem is referenced by:  pclunN  34000  pclfinN  34002
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