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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 29583) of closed subspaces of a Hilbert space. It may be superseded by cmslssbn 24536. (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 1190 | . 2 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → 𝑊 ∈ NrmVec) | |
| 2 | simpl2 1191 | . 2 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → (Scalar‘𝑊) ∈ CMetSp) | |
| 3 | nvcnlm 23860 | . . . . . . . 8 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod) | |
| 4 | nlmngp 23841 | . . . . . . . 8 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp) | |
| 5 | 3, 4 | syl 17 | . . . . . . 7 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmGrp) |
| 6 | nvclmod 23862 | . . . . . . . 8 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ LMod) | |
| 7 | cssbn.s | . . . . . . . . 9 ⊢ 𝑆 = (LSubSp‘𝑊) | |
| 8 | 7 | lsssubg 20219 | . . . . . . . 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 23791 | . . . . . . 7 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝑈 ∈ (SubGrp‘𝑊)) → 𝑋 ∈ NrmGrp) |
| 12 | 5, 9, 11 | syl2an2r 682 | . . . . . 6 ⊢ ((𝑊 ∈ NrmVec ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ NrmGrp) |
| 13 | 12 | 3adant2 1130 | . . . . 5 ⊢ ((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ NrmGrp) |
| 14 | 13 | adantr 481 | . . . 4 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → 𝑋 ∈ NrmGrp) |
| 15 | ngpms 23756 | . . . 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 2738 | . . . . . . . . . 10 ⊢ (dist‘𝑊) = (dist‘𝑊) | |
| 19 | 10, 18 | ressds 17120 | . . . . . . . . 9 ⊢ (𝑈 ∈ 𝑆 → (dist‘𝑊) = (dist‘𝑋)) |
| 20 | 19 | 3ad2ant3 1134 | . . . . . . . 8 ⊢ ((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) → (dist‘𝑊) = (dist‘𝑋)) |
| 21 | 9 | 3adant2 1130 | . . . . . . . . . 10 ⊢ ((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) → 𝑈 ∈ (SubGrp‘𝑊)) |
| 22 | 10 | subgbas 18759 | . . . . . . . . . 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 5621 | . . . . . . . 8 ⊢ ((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) → (𝑈 × 𝑈) = ((Base‘𝑋) × (Base‘𝑋))) |
| 25 | 20, 24 | reseq12d 5892 | . . . . . . 7 ⊢ ((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) → ((dist‘𝑊) ↾ (𝑈 × 𝑈)) = ((dist‘𝑋) ↾ ((Base‘𝑋) × (Base‘𝑋)))) |
| 26 | 17, 25 | eqtrid 2790 | . . . . . 6 ⊢ ((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) → 𝐷 = ((dist‘𝑋) ↾ ((Base‘𝑋) × (Base‘𝑋)))) |
| 27 | 26 | eqcomd 2744 | . . . . 5 ⊢ ((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) → ((dist‘𝑋) ↾ ((Base‘𝑋) × (Base‘𝑋))) = 𝐷) |
| 28 | 27 | adantr 481 | . . . 4 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → ((dist‘𝑋) ↾ ((Base‘𝑋) × (Base‘𝑋))) = 𝐷) |
| 29 | eqid 2738 | . . . . . . . . 9 ⊢ (Base‘𝑋) = (Base‘𝑋) | |
| 30 | eqid 2738 | . . . . . . . . 9 ⊢ ((dist‘𝑋) ↾ ((Base‘𝑋) × (Base‘𝑋))) = ((dist‘𝑋) ↾ ((Base‘𝑋) × (Base‘𝑋))) | |
| 31 | 29, 30 | ngpmet 23759 | . . . . . . . 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 2839 | . . . . . 6 ⊢ ((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) → 𝐷 ∈ (Met‘(Base‘𝑋))) |
| 34 | 33 | adantr 481 | . . . . 5 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → 𝐷 ∈ (Met‘(Base‘𝑋))) |
| 35 | simpr 485 | . . . . 5 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) | |
| 36 | eqid 2738 | . . . . . 6 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷) | |
| 37 | 36 | iscmet2 24458 | . . . . 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 2839 | . . 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 24509 | . . 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 1192 | . 2 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → 𝑈 ∈ 𝑆) | |
| 43 | 10, 7 | cmslssbn 24536 | . 2 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp) ∧ (𝑋 ∈ CMetSp ∧ 𝑈 ∈ 𝑆)) → 𝑋 ∈ Ban) |
| 44 | 1, 2, 41, 42, 43 | syl22anc 836 | 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 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ⊆ wss 3887 × cxp 5587 dom cdm 5589 ↾ cres 5591 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 ↾s cress 16941 Scalarcsca 16965 distcds 16971 SubGrpcsubg 18749 LModclmod 20123 LSubSpclss 20193 Metcmet 20583 MetOpencmopn 20587 ⇝𝑡clm 22377 MetSpcms 23471 NrmGrpcngp 23733 NrmModcnlm 23736 NrmVeccnvc 23737 Cauccau 24417 CMetccmet 24418 CMetSpccms 24496 Bancbn 24497 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cc 10191 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-oadd 8301 df-omul 8302 df-er 8498 df-map 8617 df-pm 8618 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-acn 9700 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ico 13085 df-fz 13240 df-fl 13512 df-seq 13722 df-exp 13783 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-rlim 15198 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-sca 16978 df-vsca 16979 df-tset 16981 df-ds 16984 df-rest 17133 df-topn 17134 df-0g 17152 df-topgen 17154 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-grp 18580 df-minusg 18581 df-sbg 18582 df-subg 18752 df-mgp 19721 df-ur 19738 df-ring 19785 df-lmod 20125 df-lss 20194 df-lvec 20365 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-ntr 22171 df-nei 22249 df-lm 22380 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-nm 23738 df-ngp 23739 df-nlm 23742 df-nvc 23743 df-cfil 24419 df-cau 24420 df-cmet 24421 df-cms 24499 df-bn 24500 |
| This theorem is referenced by: csschl 24540 |
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