Nearby theorems
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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 | β’ π = (π βΎs π) |
| lssbn.s | β’ π = (LSubSpβπ) |
| lssbn.j | β’ π½ = (TopOpenβπ) |
| Ref | Expression |
|---|---|
| lssbn | β’ ((π β Ban β§ π β π) β (π β Ban β π β (Clsdβπ½))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bnnvc 24848 | . . . 4 β’ (π β Ban β π β NrmVec) | |
| 2 | lssbn.x | . . . . 5 β’ π = (π βΎs π) | |
| 3 | lssbn.s | . . . . 5 β’ π = (LSubSpβπ) | |
| 4 | 2, 3 | lssnvc 24210 | . . . 4 β’ ((π β NrmVec β§ π β π) β π β NrmVec) |
| 5 | 1, 4 | sylan 580 | . . 3 β’ ((π β Ban β§ π β π) β π β NrmVec) |
| 6 | eqid 2732 | . . . . . 6 β’ (Scalarβπ) = (Scalarβπ) | |
| 7 | 2, 6 | resssca 17284 | . . . . 5 β’ (π β π β (Scalarβπ) = (Scalarβπ)) |
| 8 | 7 | adantl 482 | . . . 4 β’ ((π β Ban β§ π β π) β (Scalarβπ) = (Scalarβπ)) |
| 9 | 6 | bnsca 24847 | . . . . 5 β’ (π β Ban β (Scalarβπ) β CMetSp) |
| 10 | 9 | adantr 481 | . . . 4 β’ ((π β Ban β§ π β π) β (Scalarβπ) β CMetSp) |
| 11 | 8, 10 | eqeltrrd 2834 | . . 3 β’ ((π β Ban β§ π β π) β (Scalarβπ) β CMetSp) |
| 12 | eqid 2732 | . . . . . 6 β’ (Scalarβπ) = (Scalarβπ) | |
| 13 | 12 | isbn 24846 | . . . . 5 β’ (π β Ban β (π β NrmVec β§ π β CMetSp β§ (Scalarβπ) β CMetSp)) |
| 14 | 3anan32 1097 | . . . . 5 β’ ((π β NrmVec β§ π β CMetSp β§ (Scalarβπ) β CMetSp) β ((π β NrmVec β§ (Scalarβπ) β CMetSp) β§ π β CMetSp)) | |
| 15 | 13, 14 | bitri 274 | . . . 4 β’ (π β Ban β ((π β NrmVec β§ (Scalarβπ) β CMetSp) β§ π β CMetSp)) |
| 16 | 15 | baib 536 | . . 3 β’ ((π β NrmVec β§ (Scalarβπ) β CMetSp) β (π β Ban β π β CMetSp)) |
| 17 | 5, 11, 16 | syl2anc 584 | . 2 β’ ((π β Ban β§ π β π) β (π β Ban β π β CMetSp)) |
| 18 | bncms 24852 | . . 3 β’ (π β Ban β π β CMetSp) | |
| 19 | eqid 2732 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
| 20 | 19, 3 | lssss 20539 | . . 3 β’ (π β π β π β (Baseβπ)) |
| 21 | lssbn.j | . . . 4 β’ π½ = (TopOpenβπ) | |
| 22 | 2, 19, 21 | cmsss 24859 | . . 3 β’ ((π β CMetSp β§ π β (Baseβπ)) β (π β CMetSp β π β (Clsdβπ½))) |
| 23 | 18, 20, 22 | syl2an 596 | . 2 β’ ((π β Ban β§ π β π) β (π β CMetSp β π β (Clsdβπ½))) |
| 24 | 17, 23 | bitrd 278 | 1 β’ ((π β Ban β§ π β π) β (π β Ban β π β (Clsdβπ½))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 β wss 3947 βcfv 6540 (class class class)co 7405 Basecbs 17140 βΎs cress 17169 Scalarcsca 17196 TopOpenctopn 17363 LSubSpclss 20534 Clsdccld 22511 NrmVeccnvc 24081 CMetSpccms 24840 Bancbn 24841 |
| 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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
| 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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fi 9402 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ico 13326 df-icc 13327 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-sca 17209 df-vsca 17210 df-tset 17212 df-ds 17215 df-rest 17364 df-topn 17365 df-0g 17383 df-topgen 17385 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-minusg 18819 df-sbg 18820 df-subg 18997 df-mgp 19982 df-ur 19999 df-ring 20051 df-lmod 20465 df-lss 20535 df-lvec 20706 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-haus 22810 df-fil 23341 df-flim 23434 df-xms 23817 df-ms 23818 df-nm 24082 df-ngp 24083 df-nlm 24086 df-nvc 24087 df-cfil 24763 df-cmet 24765 df-cms 24843 df-bn 24844 |
| This theorem is referenced by: (None) |
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