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 Description: The closed subspaces of a topology-bearing module form a complete lattice. Demonstration for mreclatBAD 17234. (Contributed by Stefan O'Rear, 31-Jan-2015.) TODO (df-riota 6651 update): This proof uses the old df-clat 17155 and references the required instance of mreclatBAD 17234 as a hypothesis. When mreclatBAD 17234 is corrected to become mreclat, delete this theorem and uncomment the mreclatdemo below.
Hypothesis
Ref Expression
mreclatBAD. (((LSubSp‘𝑊) ∩ (Clsd‘(TopOpen‘𝑊))) ∈ (Moore‘ (TopOpen‘𝑊)) → (toInc‘((LSubSp‘𝑊) ∩ (Clsd‘(TopOpen‘𝑊)))) ∈ CLat)
Assertion
Ref Expression
mreclatdemoBAD (𝑊 ∈ (TopSp ∩ LMod) → (toInc‘((LSubSp‘𝑊) ∩ (Clsd‘(TopOpen‘𝑊)))) ∈ CLat)

StepHypRef Expression
1 fvex 6239 . . . . 5 (TopOpen‘𝑊) ∈ V
21uniex 6995 . . . 4 (TopOpen‘𝑊) ∈ V
3 mremre 16311 . . . 4 ( (TopOpen‘𝑊) ∈ V → (Moore‘ (TopOpen‘𝑊)) ∈ (Moore‘𝒫 (TopOpen‘𝑊)))
42, 3mp1i 13 . . 3 (𝑊 ∈ (TopSp ∩ LMod) → (Moore‘ (TopOpen‘𝑊)) ∈ (Moore‘𝒫 (TopOpen‘𝑊)))
5 inss2 3867 . . . . . 6 (TopSp ∩ LMod) ⊆ LMod
65sseli 3632 . . . . 5 (𝑊 ∈ (TopSp ∩ LMod) → 𝑊 ∈ LMod)
7 eqid 2651 . . . . . 6 (Base‘𝑊) = (Base‘𝑊)
8 eqid 2651 . . . . . 6 (LSubSp‘𝑊) = (LSubSp‘𝑊)
97, 8lssmre 19014 . . . . 5 (𝑊 ∈ LMod → (LSubSp‘𝑊) ∈ (Moore‘(Base‘𝑊)))
106, 9syl 17 . . . 4 (𝑊 ∈ (TopSp ∩ LMod) → (LSubSp‘𝑊) ∈ (Moore‘(Base‘𝑊)))
11 inss1 3866 . . . . . 6 (TopSp ∩ LMod) ⊆ TopSp
1211sseli 3632 . . . . 5 (𝑊 ∈ (TopSp ∩ LMod) → 𝑊 ∈ TopSp)
13 eqid 2651 . . . . . . 7 (TopOpen‘𝑊) = (TopOpen‘𝑊)
147, 13tpsuni 20788 . . . . . 6 (𝑊 ∈ TopSp → (Base‘𝑊) = (TopOpen‘𝑊))
1514fveq2d 6233 . . . . 5 (𝑊 ∈ TopSp → (Moore‘(Base‘𝑊)) = (Moore‘ (TopOpen‘𝑊)))
1612, 15syl 17 . . . 4 (𝑊 ∈ (TopSp ∩ LMod) → (Moore‘(Base‘𝑊)) = (Moore‘ (TopOpen‘𝑊)))
1710, 16eleqtrd 2732 . . 3 (𝑊 ∈ (TopSp ∩ LMod) → (LSubSp‘𝑊) ∈ (Moore‘ (TopOpen‘𝑊)))
1813tpstop 20789 . . . 4 (𝑊 ∈ TopSp → (TopOpen‘𝑊) ∈ Top)
19 eqid 2651 . . . . 5 (TopOpen‘𝑊) = (TopOpen‘𝑊)
2019cldmre 20930 . . . 4 ((TopOpen‘𝑊) ∈ Top → (Clsd‘(TopOpen‘𝑊)) ∈ (Moore‘ (TopOpen‘𝑊)))
2112, 18, 203syl 18 . . 3 (𝑊 ∈ (TopSp ∩ LMod) → (Clsd‘(TopOpen‘𝑊)) ∈ (Moore‘ (TopOpen‘𝑊)))
22 mreincl 16306 . . 3 (((Moore‘ (TopOpen‘𝑊)) ∈ (Moore‘𝒫 (TopOpen‘𝑊)) ∧ (LSubSp‘𝑊) ∈ (Moore‘ (TopOpen‘𝑊)) ∧ (Clsd‘(TopOpen‘𝑊)) ∈ (Moore‘ (TopOpen‘𝑊))) → ((LSubSp‘𝑊) ∩ (Clsd‘(TopOpen‘𝑊))) ∈ (Moore‘ (TopOpen‘𝑊)))
234, 17, 21, 22syl3anc 1366 . 2 (𝑊 ∈ (TopSp ∩ LMod) → ((LSubSp‘𝑊) ∩ (Clsd‘(TopOpen‘𝑊))) ∈ (Moore‘ (TopOpen‘𝑊)))
24 mreclatBAD. . 2 (((LSubSp‘𝑊) ∩ (Clsd‘(TopOpen‘𝑊))) ∈ (Moore‘ (TopOpen‘𝑊)) → (toInc‘((LSubSp‘𝑊) ∩ (Clsd‘(TopOpen‘𝑊)))) ∈ CLat)
2523, 24syl 17 1 (𝑊 ∈ (TopSp ∩ LMod) → (toInc‘((LSubSp‘𝑊) ∩ (Clsd‘(TopOpen‘𝑊)))) ∈ CLat)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1523   ∈ wcel 2030  Vcvv 3231   ∩ cin 3606  𝒫 cpw 4191  ∪ cuni 4468  ‘cfv 5926  Basecbs 15904  TopOpenctopn 16129  Moorecmre 16289  CLatccla 17154  toInccipo 17198  LModclmod 18911  LSubSpclss 18980  Topctop 20746  TopSpctps 20784  Clsdccld 20868 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  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  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-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  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-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  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-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  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-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-plusg 16001  df-0g 16149  df-mre 16293  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473  df-sbg 17474  df-mgp 18536  df-ur 18548  df-ring 18595  df-lmod 18913  df-lss 18981  df-top 20747  df-topon 20764  df-topsp 20785  df-cld 20871 This theorem is referenced by: (None)
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