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Mirrors > Home > MPE Home > Th. List > mreclatdemoBAD | Structured version Visualization version GIF version |
Description: The closed subspaces of a topology-bearing module form a complete lattice. Demonstration for mreclatBAD 18498. (Contributed by Stefan O'Rear, 31-Jan-2015.) TODO (df-riota 7349 update): This proof uses the old df-clat 18434 and references the required instance of mreclatBAD 18498 as a hypothesis. When mreclatBAD 18498 is corrected to become mreclat, delete this theorem and uncomment the mreclatdemo below. |
Ref | Expression |
---|---|
mreclatBAD. | ⊢ (((LSubSp‘𝑊) ∩ (Clsd‘(TopOpen‘𝑊))) ∈ (Moore‘∪ (TopOpen‘𝑊)) → (toInc‘((LSubSp‘𝑊) ∩ (Clsd‘(TopOpen‘𝑊)))) ∈ CLat) |
Ref | Expression |
---|---|
mreclatdemoBAD | ⊢ (𝑊 ∈ (TopSp ∩ LMod) → (toInc‘((LSubSp‘𝑊) ∩ (Clsd‘(TopOpen‘𝑊)))) ∈ CLat) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6891 | . . . . 5 ⊢ (TopOpen‘𝑊) ∈ V | |
2 | 1 | uniex 7714 | . . . 4 ⊢ ∪ (TopOpen‘𝑊) ∈ V |
3 | mremre 17530 | . . . 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 4192 | . . . . 5 ⊢ (𝑊 ∈ (TopSp ∩ LMod) → 𝑊 ∈ LMod) | |
6 | eqid 2731 | . . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
7 | eqid 2731 | . . . . . 6 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
8 | 6, 7 | lssmre 20526 | . . . . 5 ⊢ (𝑊 ∈ LMod → (LSubSp‘𝑊) ∈ (Moore‘(Base‘𝑊))) |
9 | 5, 8 | syl 17 | . . . 4 ⊢ (𝑊 ∈ (TopSp ∩ LMod) → (LSubSp‘𝑊) ∈ (Moore‘(Base‘𝑊))) |
10 | elinel1 4191 | . . . . 5 ⊢ (𝑊 ∈ (TopSp ∩ LMod) → 𝑊 ∈ TopSp) | |
11 | eqid 2731 | . . . . . . 7 ⊢ (TopOpen‘𝑊) = (TopOpen‘𝑊) | |
12 | 6, 11 | tpsuni 22367 | . . . . . 6 ⊢ (𝑊 ∈ TopSp → (Base‘𝑊) = ∪ (TopOpen‘𝑊)) |
13 | 12 | fveq2d 6882 | . . . . 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 2834 | . . 3 ⊢ (𝑊 ∈ (TopSp ∩ LMod) → (LSubSp‘𝑊) ∈ (Moore‘∪ (TopOpen‘𝑊))) |
16 | 11 | tpstop 22368 | . . . 4 ⊢ (𝑊 ∈ TopSp → (TopOpen‘𝑊) ∈ Top) |
17 | eqid 2731 | . . . . 5 ⊢ ∪ (TopOpen‘𝑊) = ∪ (TopOpen‘𝑊) | |
18 | 17 | cldmre 22511 | . . . 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 17525 | . . 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 1371 | . 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) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 Vcvv 3473 ∩ cin 3943 𝒫 cpw 4596 ∪ cuni 4901 ‘cfv 6532 Basecbs 17126 TopOpenctopn 17349 Moorecmre 17508 CLatccla 18433 toInccipo 18462 LModclmod 20420 LSubSpclss 20491 Topctop 22324 TopSpctps 22363 Clsdccld 22449 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-2 12257 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-plusg 17192 df-0g 17369 df-mre 17512 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-grp 18797 df-minusg 18798 df-sbg 18799 df-mgp 19947 df-ur 19964 df-ring 20016 df-lmod 20422 df-lss 20492 df-top 22325 df-topon 22342 df-topsp 22364 df-cld 22452 |
This theorem is referenced by: (None) |
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