Theorem mreclatdemoBAD 23469

Description: The closed subspaces of a topology-bearing module form a complete lattice. Demonstration for mreclatBAD 18787. (Contributed by Stefan O'Rear, 31-Jan-2015.) TODO (df-riota 7408 update): This proof uses the old df-clat 18723 and references the required instance of mreclatBAD 18787 as a hypothesis. When mreclatBAD 18787 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 6929	. . . . 5 ⊢ (TopOpen‘𝑊) ∈ V
2	1	uniex 7790	. . . 4 ⊢ ∪ (TopOpen‘𝑊) ∈ V
3		mremre 17824	. . . 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 4102	. . . . 5 ⊢ (𝑊 ∈ (TopSp ∩ LMod) → 𝑊 ∈ LMod)
6		eqid 2763	. . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊)
7		eqid 2763	. . . . . 6 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
8	6, 7	lssmre 21292	. . . . 5 ⊢ (𝑊 ∈ LMod → (LSubSp‘𝑊) ∈ (Moore‘(Base‘𝑊)))
9	5, 8	syl 18	. . . 4 ⊢ (𝑊 ∈ (TopSp ∩ LMod) → (LSubSp‘𝑊) ∈ (Moore‘(Base‘𝑊)))
10		elinel1 4101	. . . . 5 ⊢ (𝑊 ∈ (TopSp ∩ LMod) → 𝑊 ∈ TopSp)
11		eqid 2763	. . . . . . 7 ⊢ (TopOpen‘𝑊) = (TopOpen‘𝑊)
12	6, 11	tpsuni 23309	. . . . . 6 ⊢ (𝑊 ∈ TopSp → (Base‘𝑊) = ∪ (TopOpen‘𝑊))
13	12	fveq2d 6920	. . . . 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 2866	. . 3 ⊢ (𝑊 ∈ (TopSp ∩ LMod) → (LSubSp‘𝑊) ∈ (Moore‘∪ (TopOpen‘𝑊)))
16	11	tpstop 23310	. . . 4 ⊢ (𝑊 ∈ TopSp → (TopOpen‘𝑊) ∈ Top)
17		eqid 2763	. . . . 5 ⊢ ∪ (TopOpen‘𝑊) = ∪ (TopOpen‘𝑊)
18	17	cldmre 23451	. . . 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 17819	. . 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 1399	. 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)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ∩ cin 4092  𝒫 cpw 4560  ∪ cuni 4872  ‘cfv 6566  Basecbs 17431  TopOpenctopn 17642  Moorecmre 17802  CLatccla 18722  toInccipo 18751  LModclmod 21186  LSubSpclss 21257  Topctop 23266  TopSpctps 23305  Clsdccld 23389

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-er 8755  df-en 9012  df-dom 9013  df-sdom 9014  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-2 12445  df-sets 17386  df-slot 17404  df-ndx 17416  df-base 17432  df-plusg 17486  df-0g 17662  df-mre 17806  df-mgm 18866  df-sgrp 18958  df-mnd 18974  df-grp 19197  df-minusg 19198  df-sbg 19199  df-mgp 20412  df-ur 20459  df-ring 20512  df-lmod 21188  df-lss 21258  df-top 23267  df-topon 23284  df-topsp 23306  df-cld 23392

This theorem is used by: (None)