Theorem mreclatdemoBAD 23222

Description: The closed subspaces of a topology-bearing module form a complete lattice. Demonstration for mreclatBAD 18619. (Contributed by Stefan O'Rear, 31-Jan-2015.) TODO (df-riota 7368 update): This proof uses the old df-clat 18555 and references the required instance of mreclatBAD 18619 as a hypothesis. When mreclatBAD 18619 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

Step	Hyp	Ref	Expression
1		fvex 6895	. . . . 5 ⊢ (TopOpen‘𝑊) ∈ V
2	1	uniex 7740	. . . 4 ⊢ ∪ (TopOpen‘𝑊) ∈ V
3		mremre 17656	. . . 4 ⊢ (∪ (TopOpen‘𝑊) ∈ V → (Moore‘∪ (TopOpen‘𝑊)) ∈ (Moore‘𝒫 ∪ (TopOpen‘𝑊)))
4	2, 3	mp1i 14	. . 3 ⊢ (𝑊 ∈ (TopSp ∩ LMod) → (Moore‘∪ (TopOpen‘𝑊)) ∈ (Moore‘𝒫 ∪ (TopOpen‘𝑊)))
5		elinel2 4163	. . . . 5 ⊢ (𝑊 ∈ (TopSp ∩ LMod) → 𝑊 ∈ LMod)
6		eqid 2769	. . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊)
7		eqid 2769	. . . . . 6 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
8	6, 7	lssmre 21065	. . . . 5 ⊢ (𝑊 ∈ LMod → (LSubSp‘𝑊) ∈ (Moore‘(Base‘𝑊)))
9	5, 8	syl 18	. . . 4 ⊢ (𝑊 ∈ (TopSp ∩ LMod) → (LSubSp‘𝑊) ∈ (Moore‘(Base‘𝑊)))
10		elinel1 4162	. . . . 5 ⊢ (𝑊 ∈ (TopSp ∩ LMod) → 𝑊 ∈ TopSp)
11		eqid 2769	. . . . . . 7 ⊢ (TopOpen‘𝑊) = (TopOpen‘𝑊)
12	6, 11	tpsuni 23062	. . . . . 6 ⊢ (𝑊 ∈ TopSp → (Base‘𝑊) = ∪ (TopOpen‘𝑊))
13	12	fveq2d 6886	. . . . 5 ⊢ (𝑊 ∈ TopSp → (Moore‘(Base‘𝑊)) = (Moore‘∪ (TopOpen‘𝑊)))
14	10, 13	syl 18	. . . 4 ⊢ (𝑊 ∈ (TopSp ∩ LMod) → (Moore‘(Base‘𝑊)) = (Moore‘∪ (TopOpen‘𝑊)))
15	9, 14	eleqtrd 2871	. . 3 ⊢ (𝑊 ∈ (TopSp ∩ LMod) → (LSubSp‘𝑊) ∈ (Moore‘∪ (TopOpen‘𝑊)))
16	11	tpstop 23063	. . . 4 ⊢ (𝑊 ∈ TopSp → (TopOpen‘𝑊) ∈ Top)
17		eqid 2769	. . . . 5 ⊢ ∪ (TopOpen‘𝑊) = ∪ (TopOpen‘𝑊)
18	17	cldmre 23204	. . . 4 ⊢ ((TopOpen‘𝑊) ∈ Top → (Clsd‘(TopOpen‘𝑊)) ∈ (Moore‘∪ (TopOpen‘𝑊)))
19	10, 16, 18	3syl 19	. . 3 ⊢ (𝑊 ∈ (TopSp ∩ LMod) → (Clsd‘(TopOpen‘𝑊)) ∈ (Moore‘∪ (TopOpen‘𝑊)))
20		mreincl 17651	. . 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)))
21	4, 15, 19, 20	syl3anc 1396	. 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)
23	21, 22	syl 18	1 ⊢ (𝑊 ∈ (TopSp ∩ LMod) → (toInc‘((LSubSp‘𝑊) ∩ (Clsd‘(TopOpen‘𝑊)))) ∈ CLat)

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  Vcvv 3463   ∩ cin 3912  𝒫 cpw 4567  ∪ cuni 4876  ‘cfv 6537  Basecbs 17269  TopOpenctopn 17474  Moorecmre 17634  CLatccla 18554  toInccipo 18583  LModclmod 20959  LSubSpclss 21030  Topctop 23019  TopSpctps 23058  Clsdccld 23142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-iin 4963  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-er 8694  df-en 8944  df-dom 8945  df-sdom 8946  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-2 12303  df-sets 17224  df-slot 17242  df-ndx 17254  df-base 17270  df-plusg 17323  df-0g 17494  df-mre 17638  df-mgm 18698  df-sgrp 18777  df-mnd 18793  df-grp 19003  df-minusg 19004  df-sbg 19005  df-mgp 20217  df-ur 20264  df-ring 20317  df-lmod 20961  df-lss 21031  df-top 23020  df-topon 23037  df-topsp 23059  df-cld 23145

This theorem is referenced by: (None)