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Mirrors > Home > MPE Home > Th. List > bnsscmcl | Structured version Visualization version GIF version |
Description: A subspace of a Banach space is a Banach space iff it is closed in the norm-induced metric of the parent space. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnsscmcl.x | ⊢ 𝑋 = (BaseSet‘𝑈) |
bnsscmcl.d | ⊢ 𝐷 = (IndMet‘𝑈) |
bnsscmcl.j | ⊢ 𝐽 = (MetOpen‘𝐷) |
bnsscmcl.h | ⊢ 𝐻 = (SubSp‘𝑈) |
bnsscmcl.y | ⊢ 𝑌 = (BaseSet‘𝑊) |
Ref | Expression |
---|---|
bnsscmcl | ⊢ ((𝑈 ∈ CBan ∧ 𝑊 ∈ 𝐻) → (𝑊 ∈ CBan ↔ 𝑌 ∈ (Clsd‘𝐽))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnnv 28947 | . . . 4 ⊢ (𝑈 ∈ CBan → 𝑈 ∈ NrmCVec) | |
2 | bnsscmcl.h | . . . . 5 ⊢ 𝐻 = (SubSp‘𝑈) | |
3 | 2 | sspnv 28807 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ NrmCVec) |
4 | 1, 3 | sylan 583 | . . 3 ⊢ ((𝑈 ∈ CBan ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ NrmCVec) |
5 | bnsscmcl.y | . . . . 5 ⊢ 𝑌 = (BaseSet‘𝑊) | |
6 | eqid 2737 | . . . . 5 ⊢ (IndMet‘𝑊) = (IndMet‘𝑊) | |
7 | 5, 6 | iscbn 28945 | . . . 4 ⊢ (𝑊 ∈ CBan ↔ (𝑊 ∈ NrmCVec ∧ (IndMet‘𝑊) ∈ (CMet‘𝑌))) |
8 | 7 | baib 539 | . . 3 ⊢ (𝑊 ∈ NrmCVec → (𝑊 ∈ CBan ↔ (IndMet‘𝑊) ∈ (CMet‘𝑌))) |
9 | 4, 8 | syl 17 | . 2 ⊢ ((𝑈 ∈ CBan ∧ 𝑊 ∈ 𝐻) → (𝑊 ∈ CBan ↔ (IndMet‘𝑊) ∈ (CMet‘𝑌))) |
10 | bnsscmcl.d | . . . . 5 ⊢ 𝐷 = (IndMet‘𝑈) | |
11 | 5, 10, 6, 2 | sspims 28820 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (IndMet‘𝑊) = (𝐷 ↾ (𝑌 × 𝑌))) |
12 | 1, 11 | sylan 583 | . . 3 ⊢ ((𝑈 ∈ CBan ∧ 𝑊 ∈ 𝐻) → (IndMet‘𝑊) = (𝐷 ↾ (𝑌 × 𝑌))) |
13 | 12 | eleq1d 2822 | . 2 ⊢ ((𝑈 ∈ CBan ∧ 𝑊 ∈ 𝐻) → ((IndMet‘𝑊) ∈ (CMet‘𝑌) ↔ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌))) |
14 | bnsscmcl.x | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
15 | 14, 10 | cbncms 28946 | . . . 4 ⊢ (𝑈 ∈ CBan → 𝐷 ∈ (CMet‘𝑋)) |
16 | 15 | adantr 484 | . . 3 ⊢ ((𝑈 ∈ CBan ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ (CMet‘𝑋)) |
17 | bnsscmcl.j | . . . 4 ⊢ 𝐽 = (MetOpen‘𝐷) | |
18 | 17 | cmetss 24213 | . . 3 ⊢ (𝐷 ∈ (CMet‘𝑋) → ((𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌) ↔ 𝑌 ∈ (Clsd‘𝐽))) |
19 | 16, 18 | syl 17 | . 2 ⊢ ((𝑈 ∈ CBan ∧ 𝑊 ∈ 𝐻) → ((𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌) ↔ 𝑌 ∈ (Clsd‘𝐽))) |
20 | 9, 13, 19 | 3bitrd 308 | 1 ⊢ ((𝑈 ∈ CBan ∧ 𝑊 ∈ 𝐻) → (𝑊 ∈ CBan ↔ 𝑌 ∈ (Clsd‘𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 × cxp 5549 ↾ cres 5553 ‘cfv 6380 MetOpencmopn 20353 Clsdccld 21913 CMetccmet 24151 NrmCVeccnv 28665 BaseSetcba 28667 IndMetcims 28672 SubSpcss 28802 CBanccbn 28943 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fi 9027 df-sup 9058 df-inf 9059 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-n0 12091 df-z 12177 df-uz 12439 df-q 12545 df-rp 12587 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-ico 12941 df-icc 12942 df-rest 16927 df-topgen 16948 df-psmet 20355 df-xmet 20356 df-met 20357 df-bl 20358 df-mopn 20359 df-fbas 20360 df-fg 20361 df-top 21791 df-topon 21808 df-bases 21843 df-cld 21916 df-ntr 21917 df-cls 21918 df-nei 21995 df-haus 22212 df-fil 22743 df-flim 22836 df-cfil 24152 df-cmet 24154 df-grpo 28574 df-gid 28575 df-ginv 28576 df-gdiv 28577 df-ablo 28626 df-vc 28640 df-nv 28673 df-va 28676 df-ba 28677 df-sm 28678 df-0v 28679 df-vs 28680 df-nmcv 28681 df-ims 28682 df-ssp 28803 df-cbn 28944 |
This theorem is referenced by: (None) |
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