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Theorem pclclN 37069
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 37068 . 2 ((𝐾𝑉𝑋𝐴) → (𝑈𝑋) = {𝑦𝑆𝑋𝑦})
51, 2atpsubN 36931 . . . 4 (𝐾𝑉𝐴𝑆)
6 sseq2 3969 . . . . 5 (𝑦 = 𝐴 → (𝑋𝑦𝑋𝐴))
76intminss 4875 . . . 4 ((𝐴𝑆𝑋𝐴) → {𝑦𝑆𝑋𝑦} ⊆ 𝐴)
85, 7sylan 583 . . 3 ((𝐾𝑉𝑋𝐴) → {𝑦𝑆𝑋𝑦} ⊆ 𝐴)
9 r19.26 3158 . . . . . . . 8 (∀𝑦𝑆 ((𝑋𝑦𝑝𝑦) ∧ (𝑋𝑦𝑞𝑦)) ↔ (∀𝑦𝑆 (𝑋𝑦𝑝𝑦) ∧ ∀𝑦𝑆 (𝑋𝑦𝑞𝑦)))
10 jcab 521 . . . . . . . . 9 ((𝑋𝑦 → (𝑝𝑦𝑞𝑦)) ↔ ((𝑋𝑦𝑝𝑦) ∧ (𝑋𝑦𝑞𝑦)))
1110ralbii 3153 . . . . . . . 8 (∀𝑦𝑆 (𝑋𝑦 → (𝑝𝑦𝑞𝑦)) ↔ ∀𝑦𝑆 ((𝑋𝑦𝑝𝑦) ∧ (𝑋𝑦𝑞𝑦)))
12 vex 3474 . . . . . . . . . 10 𝑝 ∈ V
1312elintrab 4861 . . . . . . . . 9 (𝑝 {𝑦𝑆𝑋𝑦} ↔ ∀𝑦𝑆 (𝑋𝑦𝑝𝑦))
14 vex 3474 . . . . . . . . . 10 𝑞 ∈ V
1514elintrab 4861 . . . . . . . . 9 (𝑞 {𝑦𝑆𝑋𝑦} ↔ ∀𝑦𝑆 (𝑋𝑦𝑞𝑦))
1613, 15anbi12i 629 . . . . . . . 8 ((𝑝 {𝑦𝑆𝑋𝑦} ∧ 𝑞 {𝑦𝑆𝑋𝑦}) ↔ (∀𝑦𝑆 (𝑋𝑦𝑝𝑦) ∧ ∀𝑦𝑆 (𝑋𝑦𝑞𝑦)))
179, 11, 163bitr4ri 307 . . . . . . 7 ((𝑝 {𝑦𝑆𝑋𝑦} ∧ 𝑞 {𝑦𝑆𝑋𝑦}) ↔ ∀𝑦𝑆 (𝑋𝑦 → (𝑝𝑦𝑞𝑦)))
18 simpll1 1209 . . . . . . . . . . . . . 14 ((((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) ∧ (𝑝𝑦𝑞𝑦)) → 𝐾𝑉)
19 simplr 768 . . . . . . . . . . . . . 14 ((((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) ∧ (𝑝𝑦𝑞𝑦)) → 𝑦𝑆)
20 simpll3 1211 . . . . . . . . . . . . . 14 ((((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) ∧ (𝑝𝑦𝑞𝑦)) → 𝑟𝐴)
21 simprl 770 . . . . . . . . . . . . . 14 ((((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) ∧ (𝑝𝑦𝑞𝑦)) → 𝑝𝑦)
22 simprr 772 . . . . . . . . . . . . . 14 ((((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) ∧ (𝑝𝑦𝑞𝑦)) → 𝑞𝑦)
23 simpll2 1210 . . . . . . . . . . . . . 14 ((((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) ∧ (𝑝𝑦𝑞𝑦)) → 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞))
24 eqid 2821 . . . . . . . . . . . . . . 15 (le‘𝐾) = (le‘𝐾)
25 eqid 2821 . . . . . . . . . . . . . . 15 (join‘𝐾) = (join‘𝐾)
2624, 25, 1, 2psubspi2N 36926 . . . . . . . . . . . . . 14 (((𝐾𝑉𝑦𝑆𝑟𝐴) ∧ (𝑝𝑦𝑞𝑦𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞))) → 𝑟𝑦)
2718, 19, 20, 21, 22, 23, 26syl33anc 1382 . . . . . . . . . . . . 13 ((((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) ∧ (𝑝𝑦𝑞𝑦)) → 𝑟𝑦)
2827ex 416 . . . . . . . . . . . 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 3165 . . . . . . . . . 10 ((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) → (∀𝑦𝑆 (𝑋𝑦 → (𝑝𝑦𝑞𝑦)) → ∀𝑦𝑆 (𝑋𝑦𝑟𝑦)))
31 vex 3474 . . . . . . . . . . 11 𝑟 ∈ V
3231elintrab 4861 . . . . . . . . . 10 (𝑟 {𝑦𝑆𝑋𝑦} ↔ ∀𝑦𝑆 (𝑋𝑦𝑟𝑦))
3330, 32syl6ibr 255 . . . . . . . . 9 ((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) → (∀𝑦𝑆 (𝑋𝑦 → (𝑝𝑦𝑞𝑦)) → 𝑟 {𝑦𝑆𝑋𝑦}))
34333exp 1116 . . . . . . . 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 245 . . . . . 6 (𝐾𝑉 → ((𝑝 {𝑦𝑆𝑋𝑦} ∧ 𝑞 {𝑦𝑆𝑋𝑦}) → (𝑟𝐴 → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 {𝑦𝑆𝑋𝑦}))))
3736ralrimdv 3176 . . . . 5 (𝐾𝑉 → ((𝑝 {𝑦𝑆𝑋𝑦} ∧ 𝑞 {𝑦𝑆𝑋𝑦}) → ∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 {𝑦𝑆𝑋𝑦})))
3837ralrimivv 3178 . . . 4 (𝐾𝑉 → ∀𝑝 {𝑦𝑆𝑋𝑦}∀𝑞 {𝑦𝑆𝑋𝑦}∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 {𝑦𝑆𝑋𝑦}))
3938adantr 484 . . 3 ((𝐾𝑉𝑋𝐴) → ∀𝑝 {𝑦𝑆𝑋𝑦}∀𝑞 {𝑦𝑆𝑋𝑦}∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 {𝑦𝑆𝑋𝑦}))
4024, 25, 1, 2ispsubsp 36923 . . . 4 (𝐾𝑉 → ( {𝑦𝑆𝑋𝑦} ∈ 𝑆 ↔ ( {𝑦𝑆𝑋𝑦} ⊆ 𝐴 ∧ ∀𝑝 {𝑦𝑆𝑋𝑦}∀𝑞 {𝑦𝑆𝑋𝑦}∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 {𝑦𝑆𝑋𝑦}))))
4140adantr 484 . . 3 ((𝐾𝑉𝑋𝐴) → ( {𝑦𝑆𝑋𝑦} ∈ 𝑆 ↔ ( {𝑦𝑆𝑋𝑦} ⊆ 𝐴 ∧ ∀𝑝 {𝑦𝑆𝑋𝑦}∀𝑞 {𝑦𝑆𝑋𝑦}∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 {𝑦𝑆𝑋𝑦}))))
428, 39, 41mpbir2and 712 . 2 ((𝐾𝑉𝑋𝐴) → {𝑦𝑆𝑋𝑦} ∈ 𝑆)
434, 42eqeltrd 2912 1 ((𝐾𝑉𝑋𝐴) → (𝑈𝑋) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3126  {crab 3130  wss 3910   cint 4849   class class class wbr 5039  cfv 6328  (class class class)co 7130  lecple 16551  joincjn 17533  Atomscatm 36441  PSubSpcpsubsp 36674  PClcpclN 37065
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-psubsp 36681  df-pclN 37066
This theorem is referenced by:  pclunN  37076  pclfinN  37078
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