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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 17785. (Contributed by Stefan O'Rear, 31-Jan-2015.) TODO (df-riota 7103 update): This proof uses the old df-clat 17706 and references the required instance of mreclatBAD 17785 as a hypothesis. When mreclatBAD 17785 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 6676 | . . . . 5 ⊢ (TopOpen‘𝑊) ∈ V | |
2 | 1 | uniex 7454 | . . . 4 ⊢ ∪ (TopOpen‘𝑊) ∈ V |
3 | mremre 16863 | . . . 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 4170 | . . . . 5 ⊢ (𝑊 ∈ (TopSp ∩ LMod) → 𝑊 ∈ LMod) | |
6 | eqid 2818 | . . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
7 | eqid 2818 | . . . . . 6 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
8 | 6, 7 | lssmre 19667 | . . . . 5 ⊢ (𝑊 ∈ LMod → (LSubSp‘𝑊) ∈ (Moore‘(Base‘𝑊))) |
9 | 5, 8 | syl 17 | . . . 4 ⊢ (𝑊 ∈ (TopSp ∩ LMod) → (LSubSp‘𝑊) ∈ (Moore‘(Base‘𝑊))) |
10 | elinel1 4169 | . . . . 5 ⊢ (𝑊 ∈ (TopSp ∩ LMod) → 𝑊 ∈ TopSp) | |
11 | eqid 2818 | . . . . . . 7 ⊢ (TopOpen‘𝑊) = (TopOpen‘𝑊) | |
12 | 6, 11 | tpsuni 21472 | . . . . . 6 ⊢ (𝑊 ∈ TopSp → (Base‘𝑊) = ∪ (TopOpen‘𝑊)) |
13 | 12 | fveq2d 6667 | . . . . 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 2912 | . . 3 ⊢ (𝑊 ∈ (TopSp ∩ LMod) → (LSubSp‘𝑊) ∈ (Moore‘∪ (TopOpen‘𝑊))) |
16 | 11 | tpstop 21473 | . . . 4 ⊢ (𝑊 ∈ TopSp → (TopOpen‘𝑊) ∈ Top) |
17 | eqid 2818 | . . . . 5 ⊢ ∪ (TopOpen‘𝑊) = ∪ (TopOpen‘𝑊) | |
18 | 17 | cldmre 21614 | . . . 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 16858 | . . 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 1363 | . 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 1528 ∈ wcel 2105 Vcvv 3492 ∩ cin 3932 𝒫 cpw 4535 ∪ cuni 4830 ‘cfv 6348 Basecbs 16471 TopOpenctopn 16683 Moorecmre 16841 CLatccla 17705 toInccipo 17749 LModclmod 19563 LSubSpclss 19632 Topctop 21429 TopSpctps 21468 Clsdccld 21552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-plusg 16566 df-0g 16703 df-mre 16845 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-sbg 18046 df-mgp 19169 df-ur 19181 df-ring 19228 df-lmod 19565 df-lss 19633 df-top 21430 df-topon 21447 df-topsp 21469 df-cld 21555 |
This theorem is referenced by: (None) |
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