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| Mirrors > Home > MPE Home > Th. List > cssbn | Structured version Visualization version GIF version | ||
| Description: A complete subspace of a normed vector space with a complete scalar field is a Banach space. Remark: In contrast to ClSubSp, a complete subspace is defined by "a linear subspace in which all Cauchy sequences converge to a point in the subspace". This is closer to the original, but deprecated definition Cℋ (df-ch 31292) of closed subspaces of a Hilbert space. It may be superseded by cmslssbn 25339. (Contributed by NM, 10-Apr-2008.) (Revised by AV, 6-Oct-2022.) |
| Ref | Expression |
|---|---|
| cssbn.x | ⊢ 𝑋 = (𝑊 ↾s 𝑈) |
| cssbn.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
| cssbn.d | ⊢ 𝐷 = ((dist‘𝑊) ↾ (𝑈 × 𝑈)) |
| Ref | Expression |
|---|---|
| cssbn | ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → 𝑋 ∈ Ban) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simpl1 1193 | . 2 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → 𝑊 ∈ NrmVec) | |
| 2 | simpl2 1194 | . 2 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → (Scalar‘𝑊) ∈ CMetSp) | |
| 3 | nvcnlm 24661 | . . . . . . . 8 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod) | |
| 4 | nlmngp 24642 | . . . . . . . 8 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp) | |
| 5 | 3, 4 | syl 17 | . . . . . . 7 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmGrp) |
| 6 | nvclmod 24663 | . . . . . . . 8 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ LMod) | |
| 7 | cssbn.s | . . . . . . . . 9 ⊢ 𝑆 = (LSubSp‘𝑊) | |
| 8 | 7 | lsssubg 20952 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 𝑈 ∈ (SubGrp‘𝑊)) |
| 9 | 6, 8 | sylan 581 | . . . . . . 7 ⊢ ((𝑊 ∈ NrmVec ∧ 𝑈 ∈ 𝑆) → 𝑈 ∈ (SubGrp‘𝑊)) |
| 10 | cssbn.x | . . . . . . . 8 ⊢ 𝑋 = (𝑊 ↾s 𝑈) | |
| 11 | 10 | subgngp 24600 | . . . . . . 7 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝑈 ∈ (SubGrp‘𝑊)) → 𝑋 ∈ NrmGrp) |
| 12 | 5, 9, 11 | syl2an2r 686 | . . . . . 6 ⊢ ((𝑊 ∈ NrmVec ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ NrmGrp) |
| 13 | 12 | 3adant2 1132 | . . . . 5 ⊢ ((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ NrmGrp) |
| 14 | 13 | adantr 480 | . . . 4 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → 𝑋 ∈ NrmGrp) |
| 15 | ngpms 24565 | . . . 4 ⊢ (𝑋 ∈ NrmGrp → 𝑋 ∈ MetSp) | |
| 16 | 14, 15 | syl 17 | . . 3 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → 𝑋 ∈ MetSp) |
| 17 | cssbn.d | . . . . . . 7 ⊢ 𝐷 = ((dist‘𝑊) ↾ (𝑈 × 𝑈)) | |
| 18 | eqid 2736 | . . . . . . . . . 10 ⊢ (dist‘𝑊) = (dist‘𝑊) | |
| 19 | 10, 18 | ressds 17373 | . . . . . . . . 9 ⊢ (𝑈 ∈ 𝑆 → (dist‘𝑊) = (dist‘𝑋)) |
| 20 | 19 | 3ad2ant3 1136 | . . . . . . . 8 ⊢ ((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) → (dist‘𝑊) = (dist‘𝑋)) |
| 21 | 9 | 3adant2 1132 | . . . . . . . . . 10 ⊢ ((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) → 𝑈 ∈ (SubGrp‘𝑊)) |
| 22 | 10 | subgbas 19106 | . . . . . . . . . 10 ⊢ (𝑈 ∈ (SubGrp‘𝑊) → 𝑈 = (Base‘𝑋)) |
| 23 | 21, 22 | syl 17 | . . . . . . . . 9 ⊢ ((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) → 𝑈 = (Base‘𝑋)) |
| 24 | 23 | sqxpeqd 5663 | . . . . . . . 8 ⊢ ((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) → (𝑈 × 𝑈) = ((Base‘𝑋) × (Base‘𝑋))) |
| 25 | 20, 24 | reseq12d 5945 | . . . . . . 7 ⊢ ((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) → ((dist‘𝑊) ↾ (𝑈 × 𝑈)) = ((dist‘𝑋) ↾ ((Base‘𝑋) × (Base‘𝑋)))) |
| 26 | 17, 25 | eqtrid 2783 | . . . . . 6 ⊢ ((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) → 𝐷 = ((dist‘𝑋) ↾ ((Base‘𝑋) × (Base‘𝑋)))) |
| 27 | 26 | eqcomd 2742 | . . . . 5 ⊢ ((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) → ((dist‘𝑋) ↾ ((Base‘𝑋) × (Base‘𝑋))) = 𝐷) |
| 28 | 27 | adantr 480 | . . . 4 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → ((dist‘𝑋) ↾ ((Base‘𝑋) × (Base‘𝑋))) = 𝐷) |
| 29 | eqid 2736 | . . . . . . . . 9 ⊢ (Base‘𝑋) = (Base‘𝑋) | |
| 30 | eqid 2736 | . . . . . . . . 9 ⊢ ((dist‘𝑋) ↾ ((Base‘𝑋) × (Base‘𝑋))) = ((dist‘𝑋) ↾ ((Base‘𝑋) × (Base‘𝑋))) | |
| 31 | 29, 30 | ngpmet 24568 | . . . . . . . 8 ⊢ (𝑋 ∈ NrmGrp → ((dist‘𝑋) ↾ ((Base‘𝑋) × (Base‘𝑋))) ∈ (Met‘(Base‘𝑋))) |
| 32 | 13, 31 | syl 17 | . . . . . . 7 ⊢ ((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) → ((dist‘𝑋) ↾ ((Base‘𝑋) × (Base‘𝑋))) ∈ (Met‘(Base‘𝑋))) |
| 33 | 26, 32 | eqeltrd 2836 | . . . . . 6 ⊢ ((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) → 𝐷 ∈ (Met‘(Base‘𝑋))) |
| 34 | 33 | adantr 480 | . . . . 5 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → 𝐷 ∈ (Met‘(Base‘𝑋))) |
| 35 | simpr 484 | . . . . 5 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) | |
| 36 | eqid 2736 | . . . . . 6 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷) | |
| 37 | 36 | iscmet2 25261 | . . . . 5 ⊢ (𝐷 ∈ (CMet‘(Base‘𝑋)) ↔ (𝐷 ∈ (Met‘(Base‘𝑋)) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷)))) |
| 38 | 34, 35, 37 | sylanbrc 584 | . . . 4 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → 𝐷 ∈ (CMet‘(Base‘𝑋))) |
| 39 | 28, 38 | eqeltrd 2836 | . . 3 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → ((dist‘𝑋) ↾ ((Base‘𝑋) × (Base‘𝑋))) ∈ (CMet‘(Base‘𝑋))) |
| 40 | 29, 30 | iscms 25312 | . . 3 ⊢ (𝑋 ∈ CMetSp ↔ (𝑋 ∈ MetSp ∧ ((dist‘𝑋) ↾ ((Base‘𝑋) × (Base‘𝑋))) ∈ (CMet‘(Base‘𝑋)))) |
| 41 | 16, 39, 40 | sylanbrc 584 | . 2 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → 𝑋 ∈ CMetSp) |
| 42 | simpl3 1195 | . 2 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → 𝑈 ∈ 𝑆) | |
| 43 | 10, 7 | cmslssbn 25339 | . 2 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp) ∧ (𝑋 ∈ CMetSp ∧ 𝑈 ∈ 𝑆)) → 𝑋 ∈ Ban) |
| 44 | 1, 2, 41, 42, 43 | syl22anc 839 | 1 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → 𝑋 ∈ Ban) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ⊆ wss 3889 × cxp 5629 dom cdm 5631 ↾ cres 5633 ‘cfv 6498 (class class class)co 7367 Basecbs 17179 ↾s cress 17200 Scalarcsca 17223 distcds 17229 SubGrpcsubg 19096 LModclmod 20855 LSubSpclss 20926 Metcmet 21338 MetOpencmopn 21342 ⇝𝑡clm 23191 MetSpcms 24283 NrmGrpcngp 24542 NrmModcnlm 24545 NrmVeccnvc 24546 Cauccau 25220 CMetccmet 25221 CMetSpccms 25299 Bancbn 25300 |
| 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 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cc 10357 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 |
| 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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-oadd 8409 df-omul 8410 df-er 8643 df-map 8775 df-pm 8776 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-acn 9866 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ico 13304 df-fz 13462 df-fl 13751 df-seq 13964 df-exp 14024 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-clim 15450 df-rlim 15451 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-sca 17236 df-vsca 17237 df-tset 17239 df-ds 17242 df-rest 17385 df-topn 17386 df-0g 17404 df-topgen 17406 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-grp 18912 df-minusg 18913 df-sbg 18914 df-subg 19099 df-mgp 20122 df-ur 20163 df-ring 20216 df-lmod 20857 df-lss 20927 df-lvec 21098 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-ntr 22985 df-nei 23063 df-lm 23194 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-nm 24547 df-ngp 24548 df-nlm 24551 df-nvc 24552 df-cfil 25222 df-cau 25223 df-cmet 25224 df-cms 25302 df-bn 25303 |
| This theorem is referenced by: csschl 25343 |
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