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 25212 | . . . 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 24563 | . . . 4 β’ ((π β NrmVec β§ π β π) β π β NrmVec) |
| 5 | 1, 4 | sylan 579 | . . 3 β’ ((π β Ban β§ π β π) β π β NrmVec) |
| 6 | eqid 2724 | . . . . . 6 β’ (Scalarβπ) = (Scalarβπ) | |
| 7 | 2, 6 | resssca 17293 | . . . . 5 β’ (π β π β (Scalarβπ) = (Scalarβπ)) |
| 8 | 7 | adantl 481 | . . . 4 β’ ((π β Ban β§ π β π) β (Scalarβπ) = (Scalarβπ)) |
| 9 | 6 | bnsca 25211 | . . . . 5 β’ (π β Ban β (Scalarβπ) β CMetSp) |
| 10 | 9 | adantr 480 | . . . 4 β’ ((π β Ban β§ π β π) β (Scalarβπ) β CMetSp) |
| 11 | 8, 10 | eqeltrrd 2826 | . . 3 β’ ((π β Ban β§ π β π) β (Scalarβπ) β CMetSp) |
| 12 | eqid 2724 | . . . . . 6 β’ (Scalarβπ) = (Scalarβπ) | |
| 13 | 12 | isbn 25210 | . . . . 5 β’ (π β Ban β (π β NrmVec β§ π β CMetSp β§ (Scalarβπ) β CMetSp)) |
| 14 | 3anan32 1094 | . . . . 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 275 | . . . 4 β’ (π β Ban β ((π β NrmVec β§ (Scalarβπ) β CMetSp) β§ π β CMetSp)) |
| 16 | 15 | baib 535 | . . 3 β’ ((π β NrmVec β§ (Scalarβπ) β CMetSp) β (π β Ban β π β CMetSp)) |
| 17 | 5, 11, 16 | syl2anc 583 | . 2 β’ ((π β Ban β§ π β π) β (π β Ban β π β CMetSp)) |
| 18 | bncms 25216 | . . 3 β’ (π β Ban β π β CMetSp) | |
| 19 | eqid 2724 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
| 20 | 19, 3 | lssss 20779 | . . 3 β’ (π β π β π β (Baseβπ)) |
| 21 | lssbn.j | . . . 4 β’ π½ = (TopOpenβπ) | |
| 22 | 2, 19, 21 | cmsss 25223 | . . 3 β’ ((π β CMetSp β§ π β (Baseβπ)) β (π β CMetSp β π β (Clsdβπ½))) |
| 23 | 18, 20, 22 | syl2an 595 | . 2 β’ ((π β Ban β§ π β π) β (π β CMetSp β π β (Clsdβπ½))) |
| 24 | 17, 23 | bitrd 279 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β wss 3941 βcfv 6534 (class class class)co 7402 Basecbs 17149 βΎs cress 17178 Scalarcsca 17205 TopOpenctopn 17372 LSubSpclss 20774 Clsdccld 22864 NrmVeccnvc 24434 CMetSpccms 25204 Bancbn 25205 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fi 9403 df-sup 9434 df-inf 9435 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-q 12932 df-rp 12976 df-xneg 13093 df-xadd 13094 df-xmul 13095 df-ico 13331 df-icc 13332 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-sca 17218 df-vsca 17219 df-tset 17221 df-ds 17224 df-rest 17373 df-topn 17374 df-0g 17392 df-topgen 17394 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-grp 18862 df-minusg 18863 df-sbg 18864 df-subg 19046 df-mgp 20036 df-ur 20083 df-ring 20136 df-lmod 20704 df-lss 20775 df-lvec 20947 df-psmet 21226 df-xmet 21227 df-met 21228 df-bl 21229 df-mopn 21230 df-fbas 21231 df-fg 21232 df-top 22740 df-topon 22757 df-topsp 22779 df-bases 22793 df-cld 22867 df-ntr 22868 df-cls 22869 df-nei 22946 df-haus 23163 df-fil 23694 df-flim 23787 df-xms 24170 df-ms 24171 df-nm 24435 df-ngp 24436 df-nlm 24439 df-nvc 24440 df-cfil 25127 df-cmet 25129 df-cms 25207 df-bn 25208 |
| This theorem is referenced by: (None) |
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