Theorem mreclatdemoBAD 23074

Description: The closed subspaces of a topology-bearing module form a complete lattice. Demonstration for mreclatBAD 18523. (Contributed by Stefan O'Rear, 31-Jan-2015.) TODO (df-riota 7318 update): This proof uses the old df-clat 18459 and references the required instance of mreclatBAD 18523 as a hypothesis. When mreclatBAD 18523 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 6848	. . . . 5 ⊢ (TopOpen‘𝑊) ∈ V
2	1	uniex 7689	. . . 4 ⊢ ∪ (TopOpen‘𝑊) ∈ V
3		mremre 17560	. . . 4 ⊢ (∪ (TopOpen‘𝑊) ∈ V → (Moore‘∪ (TopOpen‘𝑊)) ∈ (Moore‘𝒫 ∪ (TopOpen‘𝑊)))
4	2, 3	mp1i 13	. . 3 ⊢ (𝑊 ∈ (TopSp ∩ LMod) → (Moore‘∪ (TopOpen‘𝑊)) ∈ (Moore‘𝒫 ∪ (TopOpen‘𝑊)))
5		elinel2 4143	. . . . 5 ⊢ (𝑊 ∈ (TopSp ∩ LMod) → 𝑊 ∈ LMod)
6		eqid 2737	. . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊)
7		eqid 2737	. . . . . 6 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
8	6, 7	lssmre 20955	. . . . 5 ⊢ (𝑊 ∈ LMod → (LSubSp‘𝑊) ∈ (Moore‘(Base‘𝑊)))
9	5, 8	syl 17	. . . 4 ⊢ (𝑊 ∈ (TopSp ∩ LMod) → (LSubSp‘𝑊) ∈ (Moore‘(Base‘𝑊)))
10		elinel1 4142	. . . . 5 ⊢ (𝑊 ∈ (TopSp ∩ LMod) → 𝑊 ∈ TopSp)
11		eqid 2737	. . . . . . 7 ⊢ (TopOpen‘𝑊) = (TopOpen‘𝑊)
12	6, 11	tpsuni 22914	. . . . . 6 ⊢ (𝑊 ∈ TopSp → (Base‘𝑊) = ∪ (TopOpen‘𝑊))
13	12	fveq2d 6839	. . . . 5 ⊢ (𝑊 ∈ TopSp → (Moore‘(Base‘𝑊)) = (Moore‘∪ (TopOpen‘𝑊)))
14	10, 13	syl 17	. . . 4 ⊢ (𝑊 ∈ (TopSp ∩ LMod) → (Moore‘(Base‘𝑊)) = (Moore‘∪ (TopOpen‘𝑊)))
15	9, 14	eleqtrd 2839	. . 3 ⊢ (𝑊 ∈ (TopSp ∩ LMod) → (LSubSp‘𝑊) ∈ (Moore‘∪ (TopOpen‘𝑊)))
16	11	tpstop 22915	. . . 4 ⊢ (𝑊 ∈ TopSp → (TopOpen‘𝑊) ∈ Top)
17		eqid 2737	. . . . 5 ⊢ ∪ (TopOpen‘𝑊) = ∪ (TopOpen‘𝑊)
18	17	cldmre 23056	. . . 4 ⊢ ((TopOpen‘𝑊) ∈ Top → (Clsd‘(TopOpen‘𝑊)) ∈ (Moore‘∪ (TopOpen‘𝑊)))
19	10, 16, 18	3syl 18	. . 3 ⊢ (𝑊 ∈ (TopSp ∩ LMod) → (Clsd‘(TopOpen‘𝑊)) ∈ (Moore‘∪ (TopOpen‘𝑊)))
20		mreincl 17555	. . 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 1374	. 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 17	1 ⊢ (𝑊 ∈ (TopSp ∩ LMod) → (toInc‘((LSubSp‘𝑊) ∩ (Clsd‘(TopOpen‘𝑊)))) ∈ CLat)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3430   ∩ cin 3889  𝒫 cpw 4542  ∪ cuni 4851  ‘cfv 6493  Basecbs 17173  TopOpenctopn 17378  Moorecmre 17538  CLatccla 18458  toInccipo 18487  LModclmod 20849  LSubSpclss 20920  Topctop 22871  TopSpctps 22910  Clsdccld 22994

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-nn 12169  df-2 12238  df-sets 17128  df-slot 17146  df-ndx 17158  df-base 17174  df-plusg 17227  df-0g 17398  df-mre 17542  df-mgm 18602  df-sgrp 18681  df-mnd 18697  df-grp 18906  df-minusg 18907  df-sbg 18908  df-mgp 20116  df-ur 20157  df-ring 20210  df-lmod 20851  df-lss 20921  df-top 22872  df-topon 22889  df-topsp 22911  df-cld 22997

This theorem is referenced by: (None)