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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 31298) of closed subspaces of a Hilbert space. It may be superseded by cmslssbn 25330. (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 1192 | . 2 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → 𝑊 ∈ NrmVec) | |
| 2 | simpl2 1193 | . 2 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → (Scalar‘𝑊) ∈ CMetSp) | |
| 3 | nvcnlm 24642 | . . . . . . . 8 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod) | |
| 4 | nlmngp 24623 | . . . . . . . 8 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp) | |
| 5 | 3, 4 | syl 17 | . . . . . . 7 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmGrp) |
| 6 | nvclmod 24644 | . . . . . . . 8 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ LMod) | |
| 7 | cssbn.s | . . . . . . . . 9 ⊢ 𝑆 = (LSubSp‘𝑊) | |
| 8 | 7 | lsssubg 20910 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 𝑈 ∈ (SubGrp‘𝑊)) |
| 9 | 6, 8 | sylan 580 | . . . . . . 7 ⊢ ((𝑊 ∈ NrmVec ∧ 𝑈 ∈ 𝑆) → 𝑈 ∈ (SubGrp‘𝑊)) |
| 10 | cssbn.x | . . . . . . . 8 ⊢ 𝑋 = (𝑊 ↾s 𝑈) | |
| 11 | 10 | subgngp 24581 | . . . . . . 7 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝑈 ∈ (SubGrp‘𝑊)) → 𝑋 ∈ NrmGrp) |
| 12 | 5, 9, 11 | syl2an2r 685 | . . . . . 6 ⊢ ((𝑊 ∈ NrmVec ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ NrmGrp) |
| 13 | 12 | 3adant2 1131 | . . . . 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 24546 | . . . 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 17332 | . . . . . . . . 9 ⊢ (𝑈 ∈ 𝑆 → (dist‘𝑊) = (dist‘𝑋)) |
| 20 | 19 | 3ad2ant3 1135 | . . . . . . . 8 ⊢ ((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) → (dist‘𝑊) = (dist‘𝑋)) |
| 21 | 9 | 3adant2 1131 | . . . . . . . . . 10 ⊢ ((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) → 𝑈 ∈ (SubGrp‘𝑊)) |
| 22 | 10 | subgbas 19062 | . . . . . . . . . 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 5656 | . . . . . . . 8 ⊢ ((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) → (𝑈 × 𝑈) = ((Base‘𝑋) × (Base‘𝑋))) |
| 25 | 20, 24 | reseq12d 5939 | . . . . . . 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 24549 | . . . . . . . 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 25252 | . . . . 5 ⊢ (𝐷 ∈ (CMet‘(Base‘𝑋)) ↔ (𝐷 ∈ (Met‘(Base‘𝑋)) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷)))) |
| 38 | 34, 35, 37 | sylanbrc 583 | . . . 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 25303 | . . 3 ⊢ (𝑋 ∈ CMetSp ↔ (𝑋 ∈ MetSp ∧ ((dist‘𝑋) ↾ ((Base‘𝑋) × (Base‘𝑋))) ∈ (CMet‘(Base‘𝑋)))) |
| 41 | 16, 39, 40 | sylanbrc 583 | . 2 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → 𝑋 ∈ CMetSp) |
| 42 | simpl3 1194 | . 2 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → 𝑈 ∈ 𝑆) | |
| 43 | 10, 7 | cmslssbn 25330 | . 2 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp) ∧ (𝑋 ∈ CMetSp ∧ 𝑈 ∈ 𝑆)) → 𝑋 ∈ Ban) |
| 44 | 1, 2, 41, 42, 43 | syl22anc 838 | 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 1086 = wceq 1541 ∈ wcel 2113 ⊆ wss 3901 × cxp 5622 dom cdm 5624 ↾ cres 5626 ‘cfv 6492 (class class class)co 7358 Basecbs 17138 ↾s cress 17159 Scalarcsca 17182 distcds 17188 SubGrpcsubg 19052 LModclmod 20813 LSubSpclss 20884 Metcmet 21297 MetOpencmopn 21301 ⇝𝑡clm 23172 MetSpcms 24264 NrmGrpcngp 24523 NrmModcnlm 24526 NrmVeccnvc 24527 Cauccau 25211 CMetccmet 25212 CMetSpccms 25290 Bancbn 25291 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9552 ax-cc 10347 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 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 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-oadd 8401 df-omul 8402 df-er 8635 df-map 8767 df-pm 8768 df-en 8886 df-dom 8887 df-sdom 8888 df-fin 8889 df-fi 9316 df-sup 9347 df-inf 9348 df-oi 9417 df-card 9853 df-acn 9856 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-7 12215 df-8 12216 df-9 12217 df-n0 12404 df-z 12491 df-dec 12610 df-uz 12754 df-q 12864 df-rp 12908 df-xneg 13028 df-xadd 13029 df-xmul 13030 df-ico 13269 df-fz 13426 df-fl 13714 df-seq 13927 df-exp 13987 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-clim 15413 df-rlim 15414 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-sca 17195 df-vsca 17196 df-tset 17198 df-ds 17201 df-rest 17344 df-topn 17345 df-0g 17363 df-topgen 17365 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-grp 18868 df-minusg 18869 df-sbg 18870 df-subg 19055 df-mgp 20078 df-ur 20119 df-ring 20172 df-lmod 20815 df-lss 20885 df-lvec 21057 df-psmet 21303 df-xmet 21304 df-met 21305 df-bl 21306 df-mopn 21307 df-fbas 21308 df-fg 21309 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22892 df-ntr 22966 df-nei 23044 df-lm 23175 df-fil 23792 df-fm 23884 df-flim 23885 df-flf 23886 df-xms 24266 df-ms 24267 df-nm 24528 df-ngp 24529 df-nlm 24532 df-nvc 24533 df-cfil 25213 df-cau 25214 df-cmet 25215 df-cms 25293 df-bn 25294 |
| This theorem is referenced by: csschl 25334 |
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