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 Description: The closed subspaces of a topology-bearing module form a complete lattice. Demonstration for mreclatBAD 17863. (Contributed by Stefan O'Rear, 31-Jan-2015.) TODO (df-riota 7108 update): This proof uses the old df-clat 17784 and references the required instance of mreclatBAD 17863 as a hypothesis. When mreclatBAD 17863 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 6671 . . . . 5 (TopOpen‘𝑊) ∈ V
21uniex 7465 . . . 4 (TopOpen‘𝑊) ∈ V
3 mremre 16933 . . . 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 elinel2 4101 . . . . 5 (𝑊 ∈ (TopSp ∩ LMod) → 𝑊 ∈ LMod)
6 eqid 2758 . . . . . 6 (Base‘𝑊) = (Base‘𝑊)
7 eqid 2758 . . . . . 6 (LSubSp‘𝑊) = (LSubSp‘𝑊)
86, 7lssmre 19806 . . . . 5 (𝑊 ∈ LMod → (LSubSp‘𝑊) ∈ (Moore‘(Base‘𝑊)))
95, 8syl 17 . . . 4 (𝑊 ∈ (TopSp ∩ LMod) → (LSubSp‘𝑊) ∈ (Moore‘(Base‘𝑊)))
10 elinel1 4100 . . . . 5 (𝑊 ∈ (TopSp ∩ LMod) → 𝑊 ∈ TopSp)
11 eqid 2758 . . . . . . 7 (TopOpen‘𝑊) = (TopOpen‘𝑊)
126, 11tpsuni 21636 . . . . . 6 (𝑊 ∈ TopSp → (Base‘𝑊) = (TopOpen‘𝑊))
1312fveq2d 6662 . . . . 5 (𝑊 ∈ TopSp → (Moore‘(Base‘𝑊)) = (Moore‘ (TopOpen‘𝑊)))
1410, 13syl 17 . . . 4 (𝑊 ∈ (TopSp ∩ LMod) → (Moore‘(Base‘𝑊)) = (Moore‘ (TopOpen‘𝑊)))
159, 14eleqtrd 2854 . . 3 (𝑊 ∈ (TopSp ∩ LMod) → (LSubSp‘𝑊) ∈ (Moore‘ (TopOpen‘𝑊)))
1611tpstop 21637 . . . 4 (𝑊 ∈ TopSp → (TopOpen‘𝑊) ∈ Top)
17 eqid 2758 . . . . 5 (TopOpen‘𝑊) = (TopOpen‘𝑊)
1817cldmre 21778 . . . 4 ((TopOpen‘𝑊) ∈ Top → (Clsd‘(TopOpen‘𝑊)) ∈ (Moore‘ (TopOpen‘𝑊)))
1910, 16, 183syl 18 . . 3 (𝑊 ∈ (TopSp ∩ LMod) → (Clsd‘(TopOpen‘𝑊)) ∈ (Moore‘ (TopOpen‘𝑊)))
20 mreincl 16928 . . 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)))
214, 15, 19, 20syl3anc 1368 . 2 (𝑊 ∈ (TopSp ∩ LMod) → ((LSubSp‘𝑊) ∩ (Clsd‘(TopOpen‘𝑊))) ∈ (Moore‘ (TopOpen‘𝑊)))
22 mreclatBAD. . 2 (((LSubSp‘𝑊) ∩ (Clsd‘(TopOpen‘𝑊))) ∈ (Moore‘ (TopOpen‘𝑊)) → (toInc‘((LSubSp‘𝑊) ∩ (Clsd‘(TopOpen‘𝑊)))) ∈ CLat)
2321, 22syl 17 1 (𝑊 ∈ (TopSp ∩ LMod) → (toInc‘((LSubSp‘𝑊) ∩ (Clsd‘(TopOpen‘𝑊)))) ∈ CLat)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Vcvv 3409   ∩ cin 3857  𝒫 cpw 4494  ∪ cuni 4798  ‘cfv 6335  Basecbs 16541  TopOpenctopn 16753  Moorecmre 16911  CLatccla 17783  toInccipo 17827  LModclmod 19702  LSubSpclss 19771  Topctop 21593  TopSpctps 21632  Clsdccld 21716 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-iin 4886  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-1st 7693  df-2nd 7694  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-er 8299  df-en 8528  df-dom 8529  df-sdom 8530  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-nn 11675  df-2 11737  df-ndx 16544  df-slot 16545  df-base 16547  df-sets 16548  df-plusg 16636  df-0g 16773  df-mre 16915  df-mgm 17918  df-sgrp 17967  df-mnd 17978  df-grp 18172  df-minusg 18173  df-sbg 18174  df-mgp 19308  df-ur 19320  df-ring 19367  df-lmod 19704  df-lss 19772  df-top 21594  df-topon 21611  df-topsp 21633  df-cld 21719 This theorem is referenced by: (None)
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