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Mirrors > Home > MPE Home > Th. List > lssbn | Structured version Visualization version GIF version |
Description: A subspace of a Banach space is a Banach space iff it is closed. (Contributed by Mario Carneiro, 15-Oct-2015.) |
Ref | Expression |
---|---|
lssbn.x | ⊢ 𝑋 = (𝑊 ↾s 𝑈) |
lssbn.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lssbn.j | ⊢ 𝐽 = (TopOpen‘𝑊) |
Ref | Expression |
---|---|
lssbn | ⊢ ((𝑊 ∈ Ban ∧ 𝑈 ∈ 𝑆) → (𝑋 ∈ Ban ↔ 𝑈 ∈ (Clsd‘𝐽))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnnvc 24702 | . . . 4 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmVec) | |
2 | lssbn.x | . . . . 5 ⊢ 𝑋 = (𝑊 ↾s 𝑈) | |
3 | lssbn.s | . . . . 5 ⊢ 𝑆 = (LSubSp‘𝑊) | |
4 | 2, 3 | lssnvc 24064 | . . . 4 ⊢ ((𝑊 ∈ NrmVec ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ NrmVec) |
5 | 1, 4 | sylan 580 | . . 3 ⊢ ((𝑊 ∈ Ban ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ NrmVec) |
6 | eqid 2736 | . . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
7 | 2, 6 | resssca 17223 | . . . . 5 ⊢ (𝑈 ∈ 𝑆 → (Scalar‘𝑊) = (Scalar‘𝑋)) |
8 | 7 | adantl 482 | . . . 4 ⊢ ((𝑊 ∈ Ban ∧ 𝑈 ∈ 𝑆) → (Scalar‘𝑊) = (Scalar‘𝑋)) |
9 | 6 | bnsca 24701 | . . . . 5 ⊢ (𝑊 ∈ Ban → (Scalar‘𝑊) ∈ CMetSp) |
10 | 9 | adantr 481 | . . . 4 ⊢ ((𝑊 ∈ Ban ∧ 𝑈 ∈ 𝑆) → (Scalar‘𝑊) ∈ CMetSp) |
11 | 8, 10 | eqeltrrd 2839 | . . 3 ⊢ ((𝑊 ∈ Ban ∧ 𝑈 ∈ 𝑆) → (Scalar‘𝑋) ∈ CMetSp) |
12 | eqid 2736 | . . . . . 6 ⊢ (Scalar‘𝑋) = (Scalar‘𝑋) | |
13 | 12 | isbn 24700 | . . . . 5 ⊢ (𝑋 ∈ Ban ↔ (𝑋 ∈ NrmVec ∧ 𝑋 ∈ CMetSp ∧ (Scalar‘𝑋) ∈ CMetSp)) |
14 | 3anan32 1097 | . . . . 5 ⊢ ((𝑋 ∈ NrmVec ∧ 𝑋 ∈ CMetSp ∧ (Scalar‘𝑋) ∈ CMetSp) ↔ ((𝑋 ∈ NrmVec ∧ (Scalar‘𝑋) ∈ CMetSp) ∧ 𝑋 ∈ CMetSp)) | |
15 | 13, 14 | bitri 274 | . . . 4 ⊢ (𝑋 ∈ Ban ↔ ((𝑋 ∈ NrmVec ∧ (Scalar‘𝑋) ∈ CMetSp) ∧ 𝑋 ∈ CMetSp)) |
16 | 15 | baib 536 | . . 3 ⊢ ((𝑋 ∈ NrmVec ∧ (Scalar‘𝑋) ∈ CMetSp) → (𝑋 ∈ Ban ↔ 𝑋 ∈ CMetSp)) |
17 | 5, 11, 16 | syl2anc 584 | . 2 ⊢ ((𝑊 ∈ Ban ∧ 𝑈 ∈ 𝑆) → (𝑋 ∈ Ban ↔ 𝑋 ∈ CMetSp)) |
18 | bncms 24706 | . . 3 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ CMetSp) | |
19 | eqid 2736 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
20 | 19, 3 | lssss 20395 | . . 3 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ (Base‘𝑊)) |
21 | lssbn.j | . . . 4 ⊢ 𝐽 = (TopOpen‘𝑊) | |
22 | 2, 19, 21 | cmsss 24713 | . . 3 ⊢ ((𝑊 ∈ CMetSp ∧ 𝑈 ⊆ (Base‘𝑊)) → (𝑋 ∈ CMetSp ↔ 𝑈 ∈ (Clsd‘𝐽))) |
23 | 18, 20, 22 | syl2an 596 | . 2 ⊢ ((𝑊 ∈ Ban ∧ 𝑈 ∈ 𝑆) → (𝑋 ∈ CMetSp ↔ 𝑈 ∈ (Clsd‘𝐽))) |
24 | 17, 23 | bitrd 278 | 1 ⊢ ((𝑊 ∈ Ban ∧ 𝑈 ∈ 𝑆) → (𝑋 ∈ Ban ↔ 𝑈 ∈ (Clsd‘𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ⊆ wss 3910 ‘cfv 6496 (class class class)co 7356 Basecbs 17082 ↾s cress 17111 Scalarcsca 17135 TopOpenctopn 17302 LSubSpclss 20390 Clsdccld 22365 NrmVeccnvc 23935 CMetSpccms 24694 Bancbn 24695 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 ax-pre-sup 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7802 df-1st 7920 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-1o 8411 df-er 8647 df-map 8766 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-fi 9346 df-sup 9377 df-inf 9378 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-div 11812 df-nn 12153 df-2 12215 df-3 12216 df-4 12217 df-5 12218 df-6 12219 df-7 12220 df-8 12221 df-9 12222 df-n0 12413 df-z 12499 df-dec 12618 df-uz 12763 df-q 12873 df-rp 12915 df-xneg 13032 df-xadd 13033 df-xmul 13034 df-ico 13269 df-icc 13270 df-sets 17035 df-slot 17053 df-ndx 17065 df-base 17083 df-ress 17112 df-plusg 17145 df-sca 17148 df-vsca 17149 df-tset 17151 df-ds 17154 df-rest 17303 df-topn 17304 df-0g 17322 df-topgen 17324 df-mgm 18496 df-sgrp 18545 df-mnd 18556 df-grp 18750 df-minusg 18751 df-sbg 18752 df-subg 18923 df-mgp 19895 df-ur 19912 df-ring 19964 df-lmod 20322 df-lss 20391 df-lvec 20562 df-psmet 20786 df-xmet 20787 df-met 20788 df-bl 20789 df-mopn 20790 df-fbas 20791 df-fg 20792 df-top 22241 df-topon 22258 df-topsp 22280 df-bases 22294 df-cld 22368 df-ntr 22369 df-cls 22370 df-nei 22447 df-haus 22664 df-fil 23195 df-flim 23288 df-xms 23671 df-ms 23672 df-nm 23936 df-ngp 23937 df-nlm 23940 df-nvc 23941 df-cfil 24617 df-cmet 24619 df-cms 24697 df-bn 24698 |
This theorem is referenced by: (None) |
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