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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 25267 | . . . 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 24618 | . . . 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 2728 | . . . . . 6 β’ (Scalarβπ) = (Scalarβπ) | |
| 7 | 2, 6 | resssca 17323 | . . . . 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 25266 | . . . . 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 2830 | . . 3 β’ ((π β Ban β§ π β π) β (Scalarβπ) β CMetSp) |
| 12 | eqid 2728 | . . . . . 6 β’ (Scalarβπ) = (Scalarβπ) | |
| 13 | 12 | isbn 25265 | . . . . 5 β’ (π β Ban β (π β NrmVec β§ π β CMetSp β§ (Scalarβπ) β CMetSp)) |
| 14 | 3anan32 1095 | . . . . 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 25271 | . . 3 β’ (π β Ban β π β CMetSp) | |
| 19 | eqid 2728 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
| 20 | 19, 3 | lssss 20819 | . . 3 β’ (π β π β π β (Baseβπ)) |
| 21 | lssbn.j | . . . 4 β’ π½ = (TopOpenβπ) | |
| 22 | 2, 19, 21 | cmsss 25278 | . . 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 1085 = wceq 1534 β wcel 2099 β wss 3947 βcfv 6548 (class class class)co 7420 Basecbs 17179 βΎs cress 17208 Scalarcsca 17235 TopOpenctopn 17402 LSubSpclss 20814 Clsdccld 22919 NrmVeccnvc 24489 CMetSpccms 25259 Bancbn 25260 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-map 8846 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-fi 9434 df-sup 9465 df-inf 9466 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-ico 13362 df-icc 13363 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-sca 17248 df-vsca 17249 df-tset 17251 df-ds 17254 df-rest 17403 df-topn 17404 df-0g 17422 df-topgen 17424 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18892 df-minusg 18893 df-sbg 18894 df-subg 19077 df-mgp 20074 df-ur 20121 df-ring 20174 df-lmod 20744 df-lss 20815 df-lvec 20987 df-psmet 21270 df-xmet 21271 df-met 21272 df-bl 21273 df-mopn 21274 df-fbas 21275 df-fg 21276 df-top 22795 df-topon 22812 df-topsp 22834 df-bases 22848 df-cld 22922 df-ntr 22923 df-cls 22924 df-nei 23001 df-haus 23218 df-fil 23749 df-flim 23842 df-xms 24225 df-ms 24226 df-nm 24490 df-ngp 24491 df-nlm 24494 df-nvc 24495 df-cfil 25182 df-cmet 25184 df-cms 25262 df-bn 25263 |
| This theorem is referenced by: (None) |
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