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Theorem pclclN 35904
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 35903 . 2 ((𝐾𝑉𝑋𝐴) → (𝑈𝑋) = {𝑦𝑆𝑋𝑦})
51, 2atpsubN 35766 . . . 4 (𝐾𝑉𝐴𝑆)
6 sseq2 3821 . . . . 5 (𝑦 = 𝐴 → (𝑋𝑦𝑋𝐴))
76intminss 4691 . . . 4 ((𝐴𝑆𝑋𝐴) → {𝑦𝑆𝑋𝑦} ⊆ 𝐴)
85, 7sylan 576 . . 3 ((𝐾𝑉𝑋𝐴) → {𝑦𝑆𝑋𝑦} ⊆ 𝐴)
9 r19.26 3243 . . . . . . . 8 (∀𝑦𝑆 ((𝑋𝑦𝑝𝑦) ∧ (𝑋𝑦𝑞𝑦)) ↔ (∀𝑦𝑆 (𝑋𝑦𝑝𝑦) ∧ ∀𝑦𝑆 (𝑋𝑦𝑞𝑦)))
10 jcab 514 . . . . . . . . 9 ((𝑋𝑦 → (𝑝𝑦𝑞𝑦)) ↔ ((𝑋𝑦𝑝𝑦) ∧ (𝑋𝑦𝑞𝑦)))
1110ralbii 3159 . . . . . . . 8 (∀𝑦𝑆 (𝑋𝑦 → (𝑝𝑦𝑞𝑦)) ↔ ∀𝑦𝑆 ((𝑋𝑦𝑝𝑦) ∧ (𝑋𝑦𝑞𝑦)))
12 vex 3386 . . . . . . . . . 10 𝑝 ∈ V
1312elintrab 4677 . . . . . . . . 9 (𝑝 {𝑦𝑆𝑋𝑦} ↔ ∀𝑦𝑆 (𝑋𝑦𝑝𝑦))
14 vex 3386 . . . . . . . . . 10 𝑞 ∈ V
1514elintrab 4677 . . . . . . . . 9 (𝑞 {𝑦𝑆𝑋𝑦} ↔ ∀𝑦𝑆 (𝑋𝑦𝑞𝑦))
1613, 15anbi12i 621 . . . . . . . 8 ((𝑝 {𝑦𝑆𝑋𝑦} ∧ 𝑞 {𝑦𝑆𝑋𝑦}) ↔ (∀𝑦𝑆 (𝑋𝑦𝑝𝑦) ∧ ∀𝑦𝑆 (𝑋𝑦𝑞𝑦)))
179, 11, 163bitr4ri 296 . . . . . . 7 ((𝑝 {𝑦𝑆𝑋𝑦} ∧ 𝑞 {𝑦𝑆𝑋𝑦}) ↔ ∀𝑦𝑆 (𝑋𝑦 → (𝑝𝑦𝑞𝑦)))
18 simpll1 1270 . . . . . . . . . . . . . 14 ((((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) ∧ (𝑝𝑦𝑞𝑦)) → 𝐾𝑉)
19 simplr 786 . . . . . . . . . . . . . 14 ((((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) ∧ (𝑝𝑦𝑞𝑦)) → 𝑦𝑆)
20 simpll3 1274 . . . . . . . . . . . . . 14 ((((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) ∧ (𝑝𝑦𝑞𝑦)) → 𝑟𝐴)
21 simprl 788 . . . . . . . . . . . . . 14 ((((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) ∧ (𝑝𝑦𝑞𝑦)) → 𝑝𝑦)
22 simprr 790 . . . . . . . . . . . . . 14 ((((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) ∧ (𝑝𝑦𝑞𝑦)) → 𝑞𝑦)
23 simpll2 1272 . . . . . . . . . . . . . 14 ((((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) ∧ (𝑝𝑦𝑞𝑦)) → 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞))
24 eqid 2797 . . . . . . . . . . . . . . 15 (le‘𝐾) = (le‘𝐾)
25 eqid 2797 . . . . . . . . . . . . . . 15 (join‘𝐾) = (join‘𝐾)
2624, 25, 1, 2psubspi2N 35761 . . . . . . . . . . . . . 14 (((𝐾𝑉𝑦𝑆𝑟𝐴) ∧ (𝑝𝑦𝑞𝑦𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞))) → 𝑟𝑦)
2718, 19, 20, 21, 22, 23, 26syl33anc 1505 . . . . . . . . . . . . 13 ((((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) ∧ (𝑝𝑦𝑞𝑦)) → 𝑟𝑦)
2827ex 402 . . . . . . . . . . . 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 3141 . . . . . . . . . 10 ((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) → (∀𝑦𝑆 (𝑋𝑦 → (𝑝𝑦𝑞𝑦)) → ∀𝑦𝑆 (𝑋𝑦𝑟𝑦)))
31 vex 3386 . . . . . . . . . . 11 𝑟 ∈ V
3231elintrab 4677 . . . . . . . . . 10 (𝑟 {𝑦𝑆𝑋𝑦} ↔ ∀𝑦𝑆 (𝑋𝑦𝑟𝑦))
3330, 32syl6ibr 244 . . . . . . . . 9 ((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) → (∀𝑦𝑆 (𝑋𝑦 → (𝑝𝑦𝑞𝑦)) → 𝑟 {𝑦𝑆𝑋𝑦}))
34333exp 1149 . . . . . . . 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 234 . . . . . 6 (𝐾𝑉 → ((𝑝 {𝑦𝑆𝑋𝑦} ∧ 𝑞 {𝑦𝑆𝑋𝑦}) → (𝑟𝐴 → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 {𝑦𝑆𝑋𝑦}))))
3736ralrimdv 3147 . . . . 5 (𝐾𝑉 → ((𝑝 {𝑦𝑆𝑋𝑦} ∧ 𝑞 {𝑦𝑆𝑋𝑦}) → ∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 {𝑦𝑆𝑋𝑦})))
3837ralrimivv 3149 . . . 4 (𝐾𝑉 → ∀𝑝 {𝑦𝑆𝑋𝑦}∀𝑞 {𝑦𝑆𝑋𝑦}∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 {𝑦𝑆𝑋𝑦}))
3938adantr 473 . . 3 ((𝐾𝑉𝑋𝐴) → ∀𝑝 {𝑦𝑆𝑋𝑦}∀𝑞 {𝑦𝑆𝑋𝑦}∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 {𝑦𝑆𝑋𝑦}))
4024, 25, 1, 2ispsubsp 35758 . . . 4 (𝐾𝑉 → ( {𝑦𝑆𝑋𝑦} ∈ 𝑆 ↔ ( {𝑦𝑆𝑋𝑦} ⊆ 𝐴 ∧ ∀𝑝 {𝑦𝑆𝑋𝑦}∀𝑞 {𝑦𝑆𝑋𝑦}∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 {𝑦𝑆𝑋𝑦}))))
4140adantr 473 . . 3 ((𝐾𝑉𝑋𝐴) → ( {𝑦𝑆𝑋𝑦} ∈ 𝑆 ↔ ( {𝑦𝑆𝑋𝑦} ⊆ 𝐴 ∧ ∀𝑝 {𝑦𝑆𝑋𝑦}∀𝑞 {𝑦𝑆𝑋𝑦}∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 {𝑦𝑆𝑋𝑦}))))
428, 39, 41mpbir2and 705 . 2 ((𝐾𝑉𝑋𝐴) → {𝑦𝑆𝑋𝑦} ∈ 𝑆)
434, 42eqeltrd 2876 1 ((𝐾𝑉𝑋𝐴) → (𝑈𝑋) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  wral 3087  {crab 3091  wss 3767   cint 4665   class class class wbr 4841  cfv 6099  (class class class)co 6876  lecple 16271  joincjn 17256  Atomscatm 35276  PSubSpcpsubsp 35509  PClcpclN 35900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-int 4666  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-ov 6879  df-psubsp 35516  df-pclN 35901
This theorem is referenced by:  pclunN  35911  pclfinN  35913
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