Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pclvalN Structured version   Visualization version   GIF version

Theorem pclvalN 35494
 Description: Value of the projective subspace closure function. (Contributed by NM, 7-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
pclvalN ((𝐾𝑉𝑋𝐴) → (𝑈𝑋) = {𝑦𝑆𝑋𝑦})
Distinct variable groups:   𝑦,𝐴   𝑦,𝐾   𝑦,𝑆   𝑦,𝑋
Allowed substitution hints:   𝑈(𝑦)   𝑉(𝑦)

Proof of Theorem pclvalN
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 pclfval.a . . . 4 𝐴 = (Atoms‘𝐾)
2 fvex 6239 . . . 4 (Atoms‘𝐾) ∈ V
31, 2eqeltri 2726 . . 3 𝐴 ∈ V
43elpw2 4858 . 2 (𝑋 ∈ 𝒫 𝐴𝑋𝐴)
5 pclfval.s . . . . . 6 𝑆 = (PSubSp‘𝐾)
6 pclfval.c . . . . . 6 𝑈 = (PCl‘𝐾)
71, 5, 6pclfvalN 35493 . . . . 5 (𝐾𝑉𝑈 = (𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦}))
87fveq1d 6231 . . . 4 (𝐾𝑉 → (𝑈𝑋) = ((𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦})‘𝑋))
98adantr 480 . . 3 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → (𝑈𝑋) = ((𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦})‘𝑋))
10 simpr 476 . . . 4 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → 𝑋 ∈ 𝒫 𝐴)
11 elpwi 4201 . . . . . . . 8 (𝑋 ∈ 𝒫 𝐴𝑋𝐴)
1211adantl 481 . . . . . . 7 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → 𝑋𝐴)
131, 5atpsubN 35357 . . . . . . . . 9 (𝐾𝑉𝐴𝑆)
1413adantr 480 . . . . . . . 8 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → 𝐴𝑆)
15 sseq2 3660 . . . . . . . . 9 (𝑦 = 𝐴 → (𝑋𝑦𝑋𝐴))
1615elrab3 3397 . . . . . . . 8 (𝐴𝑆 → (𝐴 ∈ {𝑦𝑆𝑋𝑦} ↔ 𝑋𝐴))
1714, 16syl 17 . . . . . . 7 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → (𝐴 ∈ {𝑦𝑆𝑋𝑦} ↔ 𝑋𝐴))
1812, 17mpbird 247 . . . . . 6 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → 𝐴 ∈ {𝑦𝑆𝑋𝑦})
19 ne0i 3954 . . . . . 6 (𝐴 ∈ {𝑦𝑆𝑋𝑦} → {𝑦𝑆𝑋𝑦} ≠ ∅)
2018, 19syl 17 . . . . 5 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → {𝑦𝑆𝑋𝑦} ≠ ∅)
21 intex 4850 . . . . 5 ({𝑦𝑆𝑋𝑦} ≠ ∅ ↔ {𝑦𝑆𝑋𝑦} ∈ V)
2220, 21sylib 208 . . . 4 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → {𝑦𝑆𝑋𝑦} ∈ V)
23 sseq1 3659 . . . . . . 7 (𝑥 = 𝑋 → (𝑥𝑦𝑋𝑦))
2423rabbidv 3220 . . . . . 6 (𝑥 = 𝑋 → {𝑦𝑆𝑥𝑦} = {𝑦𝑆𝑋𝑦})
2524inteqd 4512 . . . . 5 (𝑥 = 𝑋 {𝑦𝑆𝑥𝑦} = {𝑦𝑆𝑋𝑦})
26 eqid 2651 . . . . 5 (𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦}) = (𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦})
2725, 26fvmptg 6319 . . . 4 ((𝑋 ∈ 𝒫 𝐴 {𝑦𝑆𝑋𝑦} ∈ V) → ((𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦})‘𝑋) = {𝑦𝑆𝑋𝑦})
2810, 22, 27syl2anc 694 . . 3 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → ((𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦})‘𝑋) = {𝑦𝑆𝑋𝑦})
299, 28eqtrd 2685 . 2 ((𝐾𝑉𝑋 ∈ 𝒫 𝐴) → (𝑈𝑋) = {𝑦𝑆𝑋𝑦})
304, 29sylan2br 492 1 ((𝐾𝑉𝑋𝐴) → (𝑈𝑋) = {𝑦𝑆𝑋𝑦})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  {crab 2945  Vcvv 3231   ⊆ wss 3607  ∅c0 3948  𝒫 cpw 4191  ∩ cint 4507   ↦ cmpt 4762  ‘cfv 5926  Atomscatm 34868  PSubSpcpsubsp 35100  PClcpclN 35491 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-psubsp 35107  df-pclN 35492 This theorem is referenced by:  pclclN  35495  elpclN  35496  elpcliN  35497  pclssN  35498  pclssidN  35499  pclidN  35500
 Copyright terms: Public domain W3C validator