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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 31311) of closed subspaces of a Hilbert space. It may be superseded by cmslssbn 25353. (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 24675 | . . . . . . . 8 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod) | |
| 4 | nlmngp 24656 | . . . . . . . 8 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp) | |
| 5 | 3, 4 | syl 17 | . . . . . . 7 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmGrp) |
| 6 | nvclmod 24677 | . . . . . . . 8 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ LMod) | |
| 7 | cssbn.s | . . . . . . . . 9 ⊢ 𝑆 = (LSubSp‘𝑊) | |
| 8 | 7 | lsssubg 20947 | . . . . . . . 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 24614 | . . . . . . 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 24579 | . . . 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 2737 | . . . . . . . . . 10 ⊢ (dist‘𝑊) = (dist‘𝑊) | |
| 19 | 10, 18 | ressds 17368 | . . . . . . . . 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 19101 | . . . . . . . . . 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 5658 | . . . . . . . 8 ⊢ ((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) → (𝑈 × 𝑈) = ((Base‘𝑋) × (Base‘𝑋))) |
| 25 | 20, 24 | reseq12d 5941 | . . . . . . 7 ⊢ ((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) → ((dist‘𝑊) ↾ (𝑈 × 𝑈)) = ((dist‘𝑋) ↾ ((Base‘𝑋) × (Base‘𝑋)))) |
| 26 | 17, 25 | eqtrid 2784 | . . . . . 6 ⊢ ((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) → 𝐷 = ((dist‘𝑋) ↾ ((Base‘𝑋) × (Base‘𝑋)))) |
| 27 | 26 | eqcomd 2743 | . . . . 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 2737 | . . . . . . . . 9 ⊢ (Base‘𝑋) = (Base‘𝑋) | |
| 30 | eqid 2737 | . . . . . . . . 9 ⊢ ((dist‘𝑋) ↾ ((Base‘𝑋) × (Base‘𝑋))) = ((dist‘𝑋) ↾ ((Base‘𝑋) × (Base‘𝑋))) | |
| 31 | 29, 30 | ngpmet 24582 | . . . . . . . 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 2837 | . . . . . 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 2737 | . . . . . 6 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷) | |
| 37 | 36 | iscmet2 25275 | . . . . 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 2837 | . . 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 25326 | . . 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 25353 | . 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 3890 × cxp 5624 dom cdm 5626 ↾ cres 5628 ‘cfv 6494 (class class class)co 7362 Basecbs 17174 ↾s cress 17195 Scalarcsca 17218 distcds 17224 SubGrpcsubg 19091 LModclmod 20850 LSubSpclss 20921 Metcmet 21334 MetOpencmopn 21338 ⇝𝑡clm 23205 MetSpcms 24297 NrmGrpcngp 24556 NrmModcnlm 24559 NrmVeccnvc 24560 Cauccau 25234 CMetccmet 25235 CMetSpccms 25313 Bancbn 25314 |
| 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 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5304 ax-pr 5372 ax-un 7684 ax-inf2 9557 ax-cc 10352 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111 |
| 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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5521 df-eprel 5526 df-po 5534 df-so 5535 df-fr 5579 df-se 5580 df-we 5581 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-pred 6261 df-ord 6322 df-on 6323 df-lim 6324 df-suc 6325 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-isom 6503 df-riota 7319 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7813 df-1st 7937 df-2nd 7938 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-1o 8400 df-2o 8401 df-oadd 8404 df-omul 8405 df-er 8638 df-map 8770 df-pm 8771 df-en 8889 df-dom 8890 df-sdom 8891 df-fin 8892 df-fi 9319 df-sup 9350 df-inf 9351 df-oi 9420 df-card 9858 df-acn 9861 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-q 12894 df-rp 12938 df-xneg 13058 df-xadd 13059 df-xmul 13060 df-ico 13299 df-fz 13457 df-fl 13746 df-seq 13959 df-exp 14019 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-clim 15445 df-rlim 15446 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-sca 17231 df-vsca 17232 df-tset 17234 df-ds 17237 df-rest 17380 df-topn 17381 df-0g 17399 df-topgen 17401 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-grp 18907 df-minusg 18908 df-sbg 18909 df-subg 19094 df-mgp 20117 df-ur 20158 df-ring 20211 df-lmod 20852 df-lss 20922 df-lvec 21094 df-psmet 21340 df-xmet 21341 df-met 21342 df-bl 21343 df-mopn 21344 df-fbas 21345 df-fg 21346 df-top 22873 df-topon 22890 df-topsp 22912 df-bases 22925 df-ntr 22999 df-nei 23077 df-lm 23208 df-fil 23825 df-fm 23917 df-flim 23918 df-flf 23919 df-xms 24299 df-ms 24300 df-nm 24561 df-ngp 24562 df-nlm 24565 df-nvc 24566 df-cfil 25236 df-cau 25237 df-cmet 25238 df-cms 25316 df-bn 25317 |
| This theorem is referenced by: csschl 25357 |
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