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Mirrors > Home > MPE Home > Th. List > lssbn | Structured version Visualization version GIF version |
Description: A subspace of a Banach space is a Banach space iff it is closed. (Contributed by Mario Carneiro, 15-Oct-2015.) |
Ref | Expression |
---|---|
lssbn.x | ⊢ 𝑋 = (𝑊 ↾s 𝑈) |
lssbn.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lssbn.j | ⊢ 𝐽 = (TopOpen‘𝑊) |
Ref | Expression |
---|---|
lssbn | ⊢ ((𝑊 ∈ Ban ∧ 𝑈 ∈ 𝑆) → (𝑋 ∈ Ban ↔ 𝑈 ∈ (Clsd‘𝐽))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnnvc 23546 | . . . 4 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmVec) | |
2 | lssbn.x | . . . . 5 ⊢ 𝑋 = (𝑊 ↾s 𝑈) | |
3 | lssbn.s | . . . . 5 ⊢ 𝑆 = (LSubSp‘𝑊) | |
4 | 2, 3 | lssnvc 22914 | . . . 4 ⊢ ((𝑊 ∈ NrmVec ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ NrmVec) |
5 | 1, 4 | sylan 575 | . . 3 ⊢ ((𝑊 ∈ Ban ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ NrmVec) |
6 | eqid 2778 | . . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
7 | 2, 6 | resssca 16423 | . . . . 5 ⊢ (𝑈 ∈ 𝑆 → (Scalar‘𝑊) = (Scalar‘𝑋)) |
8 | 7 | adantl 475 | . . . 4 ⊢ ((𝑊 ∈ Ban ∧ 𝑈 ∈ 𝑆) → (Scalar‘𝑊) = (Scalar‘𝑋)) |
9 | 6 | bnsca 23545 | . . . . 5 ⊢ (𝑊 ∈ Ban → (Scalar‘𝑊) ∈ CMetSp) |
10 | 9 | adantr 474 | . . . 4 ⊢ ((𝑊 ∈ Ban ∧ 𝑈 ∈ 𝑆) → (Scalar‘𝑊) ∈ CMetSp) |
11 | 8, 10 | eqeltrrd 2860 | . . 3 ⊢ ((𝑊 ∈ Ban ∧ 𝑈 ∈ 𝑆) → (Scalar‘𝑋) ∈ CMetSp) |
12 | eqid 2778 | . . . . . 6 ⊢ (Scalar‘𝑋) = (Scalar‘𝑋) | |
13 | 12 | isbn 23544 | . . . . 5 ⊢ (𝑋 ∈ Ban ↔ (𝑋 ∈ NrmVec ∧ 𝑋 ∈ CMetSp ∧ (Scalar‘𝑋) ∈ CMetSp)) |
14 | 3anan32 1081 | . . . . 5 ⊢ ((𝑋 ∈ NrmVec ∧ 𝑋 ∈ CMetSp ∧ (Scalar‘𝑋) ∈ CMetSp) ↔ ((𝑋 ∈ NrmVec ∧ (Scalar‘𝑋) ∈ CMetSp) ∧ 𝑋 ∈ CMetSp)) | |
15 | 13, 14 | bitri 267 | . . . 4 ⊢ (𝑋 ∈ Ban ↔ ((𝑋 ∈ NrmVec ∧ (Scalar‘𝑋) ∈ CMetSp) ∧ 𝑋 ∈ CMetSp)) |
16 | 15 | baib 531 | . . 3 ⊢ ((𝑋 ∈ NrmVec ∧ (Scalar‘𝑋) ∈ CMetSp) → (𝑋 ∈ Ban ↔ 𝑋 ∈ CMetSp)) |
17 | 5, 11, 16 | syl2anc 579 | . 2 ⊢ ((𝑊 ∈ Ban ∧ 𝑈 ∈ 𝑆) → (𝑋 ∈ Ban ↔ 𝑋 ∈ CMetSp)) |
18 | bncms 23550 | . . 3 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ CMetSp) | |
19 | eqid 2778 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
20 | 19, 3 | lssss 19329 | . . 3 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ (Base‘𝑊)) |
21 | lssbn.j | . . . 4 ⊢ 𝐽 = (TopOpen‘𝑊) | |
22 | 2, 19, 21 | cmsss 23557 | . . 3 ⊢ ((𝑊 ∈ CMetSp ∧ 𝑈 ⊆ (Base‘𝑊)) → (𝑋 ∈ CMetSp ↔ 𝑈 ∈ (Clsd‘𝐽))) |
23 | 18, 20, 22 | syl2an 589 | . 2 ⊢ ((𝑊 ∈ Ban ∧ 𝑈 ∈ 𝑆) → (𝑋 ∈ CMetSp ↔ 𝑈 ∈ (Clsd‘𝐽))) |
24 | 17, 23 | bitrd 271 | 1 ⊢ ((𝑊 ∈ Ban ∧ 𝑈 ∈ 𝑆) → (𝑋 ∈ Ban ↔ 𝑈 ∈ (Clsd‘𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ⊆ wss 3792 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 ↾s cress 16256 Scalarcsca 16341 TopOpenctopn 16468 LSubSpclss 19324 Clsdccld 21228 NrmVeccnvc 22794 CMetSpccms 23538 Bancbn 23539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fi 8605 df-sup 8636 df-inf 8637 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-ico 12493 df-icc 12494 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-sca 16354 df-vsca 16355 df-tset 16357 df-ds 16360 df-rest 16469 df-topn 16470 df-0g 16488 df-topgen 16490 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-grp 17812 df-minusg 17813 df-sbg 17814 df-subg 17975 df-mgp 18877 df-ur 18889 df-ring 18936 df-lmod 19257 df-lss 19325 df-lvec 19498 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-mopn 20138 df-fbas 20139 df-fg 20140 df-top 21106 df-topon 21123 df-topsp 21145 df-bases 21158 df-cld 21231 df-ntr 21232 df-cls 21233 df-nei 21310 df-haus 21527 df-fil 22058 df-flim 22151 df-xms 22533 df-ms 22534 df-nm 22795 df-ngp 22796 df-nlm 22799 df-nvc 22800 df-cfil 23461 df-cmet 23463 df-cms 23541 df-bn 23542 |
This theorem is referenced by: (None) |
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