MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mreclatdemoBAD Structured version   Visualization version   GIF version

Theorem mreclatdemoBAD 21623
Description: The closed subspaces of a topology-bearing module form a complete lattice. Demonstration for mreclatBAD 17790. (Contributed by Stefan O'Rear, 31-Jan-2015.) TODO (df-riota 7109 update): This proof uses the old df-clat 17711 and references the required instance of mreclatBAD 17790 as a hypothesis. When mreclatBAD 17790 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)

Proof of Theorem mreclatdemoBAD
StepHypRef Expression
1 fvex 6679 . . . . 5 (TopOpen‘𝑊) ∈ V
21uniex 7458 . . . 4 (TopOpen‘𝑊) ∈ V
3 mremre 16868 . . . 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 4176 . . . . 5 (𝑊 ∈ (TopSp ∩ LMod) → 𝑊 ∈ LMod)
6 eqid 2825 . . . . . 6 (Base‘𝑊) = (Base‘𝑊)
7 eqid 2825 . . . . . 6 (LSubSp‘𝑊) = (LSubSp‘𝑊)
86, 7lssmre 19661 . . . . 5 (𝑊 ∈ LMod → (LSubSp‘𝑊) ∈ (Moore‘(Base‘𝑊)))
95, 8syl 17 . . . 4 (𝑊 ∈ (TopSp ∩ LMod) → (LSubSp‘𝑊) ∈ (Moore‘(Base‘𝑊)))
10 elinel1 4175 . . . . 5 (𝑊 ∈ (TopSp ∩ LMod) → 𝑊 ∈ TopSp)
11 eqid 2825 . . . . . . 7 (TopOpen‘𝑊) = (TopOpen‘𝑊)
126, 11tpsuni 21463 . . . . . 6 (𝑊 ∈ TopSp → (Base‘𝑊) = (TopOpen‘𝑊))
1312fveq2d 6670 . . . . 5 (𝑊 ∈ TopSp → (Moore‘(Base‘𝑊)) = (Moore‘ (TopOpen‘𝑊)))
1410, 13syl 17 . . . 4 (𝑊 ∈ (TopSp ∩ LMod) → (Moore‘(Base‘𝑊)) = (Moore‘ (TopOpen‘𝑊)))
159, 14eleqtrd 2919 . . 3 (𝑊 ∈ (TopSp ∩ LMod) → (LSubSp‘𝑊) ∈ (Moore‘ (TopOpen‘𝑊)))
1611tpstop 21464 . . . 4 (𝑊 ∈ TopSp → (TopOpen‘𝑊) ∈ Top)
17 eqid 2825 . . . . 5 (TopOpen‘𝑊) = (TopOpen‘𝑊)
1817cldmre 21605 . . . 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 16863 . . 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 1365 . 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 1530  wcel 2107  Vcvv 3499  cin 3938  𝒫 cpw 4541   cuni 4836  cfv 6351  Basecbs 16476  TopOpenctopn 16688  Moorecmre 16846  CLatccla 17710  toInccipo 17754  LModclmod 19557  LSubSpclss 19626  Topctop 21420  TopSpctps 21459  Clsdccld 21543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-iin 4919  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8282  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-ndx 16479  df-slot 16480  df-base 16482  df-sets 16483  df-plusg 16571  df-0g 16708  df-mre 16850  df-mgm 17845  df-sgrp 17893  df-mnd 17904  df-grp 18039  df-minusg 18040  df-sbg 18041  df-mgp 19163  df-ur 19175  df-ring 19222  df-lmod 19559  df-lss 19627  df-top 21421  df-topon 21438  df-topsp 21460  df-cld 21546
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator