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Theorem pclclN 38354
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 38353 . 2 ((𝐾𝑉𝑋𝐴) → (𝑈𝑋) = {𝑦𝑆𝑋𝑦})
51, 2atpsubN 38216 . . . 4 (𝐾𝑉𝐴𝑆)
6 sseq2 3970 . . . . 5 (𝑦 = 𝐴 → (𝑋𝑦𝑋𝐴))
76intminss 4935 . . . 4 ((𝐴𝑆𝑋𝐴) → {𝑦𝑆𝑋𝑦} ⊆ 𝐴)
85, 7sylan 580 . . 3 ((𝐾𝑉𝑋𝐴) → {𝑦𝑆𝑋𝑦} ⊆ 𝐴)
9 r19.26 3114 . . . . . . . 8 (∀𝑦𝑆 ((𝑋𝑦𝑝𝑦) ∧ (𝑋𝑦𝑞𝑦)) ↔ (∀𝑦𝑆 (𝑋𝑦𝑝𝑦) ∧ ∀𝑦𝑆 (𝑋𝑦𝑞𝑦)))
10 jcab 518 . . . . . . . . 9 ((𝑋𝑦 → (𝑝𝑦𝑞𝑦)) ↔ ((𝑋𝑦𝑝𝑦) ∧ (𝑋𝑦𝑞𝑦)))
1110ralbii 3096 . . . . . . . 8 (∀𝑦𝑆 (𝑋𝑦 → (𝑝𝑦𝑞𝑦)) ↔ ∀𝑦𝑆 ((𝑋𝑦𝑝𝑦) ∧ (𝑋𝑦𝑞𝑦)))
12 vex 3449 . . . . . . . . . 10 𝑝 ∈ V
1312elintrab 4921 . . . . . . . . 9 (𝑝 {𝑦𝑆𝑋𝑦} ↔ ∀𝑦𝑆 (𝑋𝑦𝑝𝑦))
14 vex 3449 . . . . . . . . . 10 𝑞 ∈ V
1514elintrab 4921 . . . . . . . . 9 (𝑞 {𝑦𝑆𝑋𝑦} ↔ ∀𝑦𝑆 (𝑋𝑦𝑞𝑦))
1613, 15anbi12i 627 . . . . . . . 8 ((𝑝 {𝑦𝑆𝑋𝑦} ∧ 𝑞 {𝑦𝑆𝑋𝑦}) ↔ (∀𝑦𝑆 (𝑋𝑦𝑝𝑦) ∧ ∀𝑦𝑆 (𝑋𝑦𝑞𝑦)))
179, 11, 163bitr4ri 303 . . . . . . 7 ((𝑝 {𝑦𝑆𝑋𝑦} ∧ 𝑞 {𝑦𝑆𝑋𝑦}) ↔ ∀𝑦𝑆 (𝑋𝑦 → (𝑝𝑦𝑞𝑦)))
18 simpll1 1212 . . . . . . . . . . . . . 14 ((((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) ∧ (𝑝𝑦𝑞𝑦)) → 𝐾𝑉)
19 simplr 767 . . . . . . . . . . . . . 14 ((((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) ∧ (𝑝𝑦𝑞𝑦)) → 𝑦𝑆)
20 simpll3 1214 . . . . . . . . . . . . . 14 ((((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) ∧ (𝑝𝑦𝑞𝑦)) → 𝑟𝐴)
21 simprl 769 . . . . . . . . . . . . . 14 ((((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) ∧ (𝑝𝑦𝑞𝑦)) → 𝑝𝑦)
22 simprr 771 . . . . . . . . . . . . . 14 ((((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) ∧ (𝑝𝑦𝑞𝑦)) → 𝑞𝑦)
23 simpll2 1213 . . . . . . . . . . . . . 14 ((((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) ∧ (𝑝𝑦𝑞𝑦)) → 𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞))
24 eqid 2736 . . . . . . . . . . . . . . 15 (le‘𝐾) = (le‘𝐾)
25 eqid 2736 . . . . . . . . . . . . . . 15 (join‘𝐾) = (join‘𝐾)
2624, 25, 1, 2psubspi2N 38211 . . . . . . . . . . . . . 14 (((𝐾𝑉𝑦𝑆𝑟𝐴) ∧ (𝑝𝑦𝑞𝑦𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞))) → 𝑟𝑦)
2718, 19, 20, 21, 22, 23, 26syl33anc 1385 . . . . . . . . . . . . 13 ((((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) ∧ 𝑦𝑆) ∧ (𝑝𝑦𝑞𝑦)) → 𝑟𝑦)
2827ex 413 . . . . . . . . . . . 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 3164 . . . . . . . . . 10 ((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) → (∀𝑦𝑆 (𝑋𝑦 → (𝑝𝑦𝑞𝑦)) → ∀𝑦𝑆 (𝑋𝑦𝑟𝑦)))
31 vex 3449 . . . . . . . . . . 11 𝑟 ∈ V
3231elintrab 4921 . . . . . . . . . 10 (𝑟 {𝑦𝑆𝑋𝑦} ↔ ∀𝑦𝑆 (𝑋𝑦𝑟𝑦))
3330, 32syl6ibr 251 . . . . . . . . 9 ((𝐾𝑉𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ 𝑟𝐴) → (∀𝑦𝑆 (𝑋𝑦 → (𝑝𝑦𝑞𝑦)) → 𝑟 {𝑦𝑆𝑋𝑦}))
34333exp 1119 . . . . . . . 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, 35biimtrid 241 . . . . . 6 (𝐾𝑉 → ((𝑝 {𝑦𝑆𝑋𝑦} ∧ 𝑞 {𝑦𝑆𝑋𝑦}) → (𝑟𝐴 → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 {𝑦𝑆𝑋𝑦}))))
3736ralrimdv 3149 . . . . 5 (𝐾𝑉 → ((𝑝 {𝑦𝑆𝑋𝑦} ∧ 𝑞 {𝑦𝑆𝑋𝑦}) → ∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 {𝑦𝑆𝑋𝑦})))
3837ralrimivv 3195 . . . 4 (𝐾𝑉 → ∀𝑝 {𝑦𝑆𝑋𝑦}∀𝑞 {𝑦𝑆𝑋𝑦}∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 {𝑦𝑆𝑋𝑦}))
3938adantr 481 . . 3 ((𝐾𝑉𝑋𝐴) → ∀𝑝 {𝑦𝑆𝑋𝑦}∀𝑞 {𝑦𝑆𝑋𝑦}∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 {𝑦𝑆𝑋𝑦}))
4024, 25, 1, 2ispsubsp 38208 . . . 4 (𝐾𝑉 → ( {𝑦𝑆𝑋𝑦} ∈ 𝑆 ↔ ( {𝑦𝑆𝑋𝑦} ⊆ 𝐴 ∧ ∀𝑝 {𝑦𝑆𝑋𝑦}∀𝑞 {𝑦𝑆𝑋𝑦}∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 {𝑦𝑆𝑋𝑦}))))
4140adantr 481 . . 3 ((𝐾𝑉𝑋𝐴) → ( {𝑦𝑆𝑋𝑦} ∈ 𝑆 ↔ ( {𝑦𝑆𝑋𝑦} ⊆ 𝐴 ∧ ∀𝑝 {𝑦𝑆𝑋𝑦}∀𝑞 {𝑦𝑆𝑋𝑦}∀𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 {𝑦𝑆𝑋𝑦}))))
428, 39, 41mpbir2and 711 . 2 ((𝐾𝑉𝑋𝐴) → {𝑦𝑆𝑋𝑦} ∈ 𝑆)
434, 42eqeltrd 2838 1 ((𝐾𝑉𝑋𝐴) → (𝑈𝑋) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  {crab 3407  wss 3910   cint 4907   class class class wbr 5105  cfv 6496  (class class class)co 7357  lecple 17140  joincjn 18200  Atomscatm 37725  PSubSpcpsubsp 37959  PClcpclN 38350
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-psubsp 37966  df-pclN 38351
This theorem is referenced by:  pclunN  38361  pclfinN  38363
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