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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 31425) of closed subspaces of a Hilbert space. It may be superseded by cmslssbn 25435. (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 1206 | . 2 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → 𝑊 ∈ NrmVec) | |
| 2 | simpl2 1207 | . 2 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → (Scalar‘𝑊) ∈ CMetSp) | |
| 3 | nvcnlm 24757 | . . . . . . . 8 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod) | |
| 4 | nlmngp 24738 | . . . . . . . 8 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp) | |
| 5 | 3, 4 | syl 17 | . . . . . . 7 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmGrp) |
| 6 | nvclmod 24759 | . . . . . . . 8 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ LMod) | |
| 7 | cssbn.s | . . . . . . . . 9 ⊢ 𝑆 = (LSubSp‘𝑊) | |
| 8 | 7 | lsssubg 21025 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 𝑈 ∈ (SubGrp‘𝑊)) |
| 9 | 6, 8 | sylan 589 | . . . . . . 7 ⊢ ((𝑊 ∈ NrmVec ∧ 𝑈 ∈ 𝑆) → 𝑈 ∈ (SubGrp‘𝑊)) |
| 10 | cssbn.x | . . . . . . . 8 ⊢ 𝑋 = (𝑊 ↾s 𝑈) | |
| 11 | 10 | subgngp 24696 | . . . . . . 7 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝑈 ∈ (SubGrp‘𝑊)) → 𝑋 ∈ NrmGrp) |
| 12 | 5, 9, 11 | syl2an2r 695 | . . . . . 6 ⊢ ((𝑊 ∈ NrmVec ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ NrmGrp) |
| 13 | 12 | 3adant2 1145 | . . . . 5 ⊢ ((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ NrmGrp) |
| 14 | 13 | adantr 484 | . . . 4 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → 𝑋 ∈ NrmGrp) |
| 15 | ngpms 24661 | . . . 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 2763 | . . . . . . . . . 10 ⊢ (dist‘𝑊) = (dist‘𝑊) | |
| 19 | 10, 18 | ressds 17440 | . . . . . . . . 9 ⊢ (𝑈 ∈ 𝑆 → (dist‘𝑊) = (dist‘𝑋)) |
| 20 | 19 | 3ad2ant3 1149 | . . . . . . . 8 ⊢ ((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) → (dist‘𝑊) = (dist‘𝑋)) |
| 21 | 9 | 3adant2 1145 | . . . . . . . . . 10 ⊢ ((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) → 𝑈 ∈ (SubGrp‘𝑊)) |
| 22 | 10 | subgbas 19173 | . . . . . . . . . 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 5680 | . . . . . . . 8 ⊢ ((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) → (𝑈 × 𝑈) = ((Base‘𝑋) × (Base‘𝑋))) |
| 25 | 20, 24 | reseq12d 5967 | . . . . . . 7 ⊢ ((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) → ((dist‘𝑊) ↾ (𝑈 × 𝑈)) = ((dist‘𝑋) ↾ ((Base‘𝑋) × (Base‘𝑋)))) |
| 26 | 17, 25 | eqtrid 2810 | . . . . . 6 ⊢ ((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) → 𝐷 = ((dist‘𝑋) ↾ ((Base‘𝑋) × (Base‘𝑋)))) |
| 27 | 26 | eqcomd 2769 | . . . . 5 ⊢ ((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) → ((dist‘𝑋) ↾ ((Base‘𝑋) × (Base‘𝑋))) = 𝐷) |
| 28 | 27 | adantr 484 | . . . 4 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → ((dist‘𝑋) ↾ ((Base‘𝑋) × (Base‘𝑋))) = 𝐷) |
| 29 | eqid 2763 | . . . . . . . . 9 ⊢ (Base‘𝑋) = (Base‘𝑋) | |
| 30 | eqid 2763 | . . . . . . . . 9 ⊢ ((dist‘𝑋) ↾ ((Base‘𝑋) × (Base‘𝑋))) = ((dist‘𝑋) ↾ ((Base‘𝑋) × (Base‘𝑋))) | |
| 31 | 29, 30 | ngpmet 24664 | . . . . . . . 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 2863 | . . . . . 6 ⊢ ((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) → 𝐷 ∈ (Met‘(Base‘𝑋))) |
| 34 | 33 | adantr 484 | . . . . 5 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → 𝐷 ∈ (Met‘(Base‘𝑋))) |
| 35 | simpr 488 | . . . . 5 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) | |
| 36 | eqid 2763 | . . . . . 6 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷) | |
| 37 | 36 | iscmet2 25357 | . . . . 5 ⊢ (𝐷 ∈ (CMet‘(Base‘𝑋)) ↔ (𝐷 ∈ (Met‘(Base‘𝑋)) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷)))) |
| 38 | 34, 35, 37 | sylanbrc 592 | . . . 4 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → 𝐷 ∈ (CMet‘(Base‘𝑋))) |
| 39 | 28, 38 | eqeltrd 2863 | . . 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 25408 | . . 3 ⊢ (𝑋 ∈ CMetSp ↔ (𝑋 ∈ MetSp ∧ ((dist‘𝑋) ↾ ((Base‘𝑋) × (Base‘𝑋))) ∈ (CMet‘(Base‘𝑋)))) |
| 41 | 16, 39, 40 | sylanbrc 592 | . 2 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → 𝑋 ∈ CMetSp) |
| 42 | simpl3 1208 | . 2 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → 𝑈 ∈ 𝑆) | |
| 43 | 10, 7 | cmslssbn 25435 | . 2 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp) ∧ (𝑋 ∈ CMetSp ∧ 𝑈 ∈ 𝑆)) → 𝑋 ∈ Ban) |
| 44 | 1, 2, 41, 42, 43 | syl22anc 849 | 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 399 ∧ w3a 1099 = wceq 1561 ∈ wcel 2143 ⊆ wss 3905 × cxp 5646 dom cdm 5648 ↾ cres 5650 ‘cfv 6522 (class class class)co 7397 Basecbs 17246 ↾s cress 17267 Scalarcsca 17290 distcds 17296 SubGrpcsubg 19163 LModclmod 20928 LSubSpclss 20999 Metcmet 21411 MetOpencmopn 21415 ⇝𝑡clm 23287 MetSpcms 24379 NrmGrpcngp 24638 NrmModcnlm 24641 NrmVeccnvc 24642 Cauccau 25316 CMetccmet 25317 CMetSpccms 25395 Bancbn 25396 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1816 ax-4 1830 ax-5 1931 ax-6 1988 ax-7 2029 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5228 ax-sep 5247 ax-nul 5257 ax-pow 5323 ax-pr 5391 ax-un 7719 ax-inf2 9597 ax-cc 10393 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151 ax-pre-sup 11152 |
| This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1564 df-fal 1574 df-ex 1801 df-nf 1805 df-sb 2092 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3457 df-sbc 3746 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5102 df-opab 5164 df-mpt 5183 df-tr 5209 df-id 5543 df-eprel 5548 df-po 5556 df-so 5557 df-fr 5601 df-se 5602 df-we 5603 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-pred 6289 df-ord 6350 df-on 6351 df-lim 6352 df-suc 6353 df-iota 6478 df-fun 6524 df-fn 6525 df-f 6526 df-f1 6527 df-fo 6528 df-f1o 6529 df-fv 6530 df-isom 6531 df-riota 7354 df-ov 7400 df-oprab 7401 df-mpo 7402 df-om 7848 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8382 df-1o 8438 df-2o 8439 df-oadd 8442 df-omul 8443 df-er 8679 df-map 8811 df-pm 8812 df-en 8929 df-dom 8930 df-sdom 8931 df-fin 8932 df-fi 9358 df-sup 9389 df-inf 9390 df-oi 9459 df-card 9898 df-acn 9901 df-pnf 11219 df-mnf 11220 df-xr 11221 df-ltxr 11222 df-le 11223 df-sub 11417 df-neg 11418 df-div 11846 df-nn 12212 df-2 12281 df-3 12282 df-4 12283 df-5 12284 df-6 12285 df-7 12286 df-8 12287 df-9 12288 df-n0 12483 df-z 12570 df-dec 12690 df-uz 12841 df-q 12951 df-rp 12995 df-xneg 13115 df-xadd 13116 df-xmul 13117 df-ico 13356 df-fz 13514 df-fl 13803 df-seq 14016 df-exp 14076 df-cj 15127 df-re 15128 df-im 15129 df-sqrt 15263 df-abs 15264 df-clim 15516 df-rlim 15517 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17247 df-ress 17268 df-plusg 17300 df-sca 17303 df-vsca 17304 df-tset 17306 df-ds 17309 df-rest 17452 df-topn 17453 df-0g 17471 df-topgen 17473 df-mgm 18675 df-sgrp 18754 df-mnd 18770 df-grp 18979 df-minusg 18980 df-sbg 18981 df-subg 19166 df-mgp 20188 df-ur 20233 df-ring 20286 df-lmod 20930 df-lss 21000 df-lvec 21171 df-psmet 21417 df-xmet 21418 df-met 21419 df-bl 21420 df-mopn 21421 df-fbas 21422 df-fg 21423 df-top 22955 df-topon 22972 df-topsp 22994 df-bases 23007 df-ntr 23081 df-nei 23159 df-lm 23290 df-fil 23907 df-fm 23999 df-flim 24000 df-flf 24001 df-xms 24381 df-ms 24382 df-nm 24643 df-ngp 24644 df-nlm 24647 df-nvc 24648 df-cfil 25318 df-cau 25319 df-cmet 25320 df-cms 25398 df-bn 25399 |
| This theorem is referenced by: csschl 25439 |
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