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| Mirrors > Home > MPE Home > Th. List > bncssbn | Structured version Visualization version GIF version | ||
| Description: A closed subspace of a Banach space which is also a subcomplex pre-Hilbert space is a Banach space. Remark: the assumption that the Banach space must be a (subcomplex) pre-Hilbert space is required because the definition of ClSubSp is based on an inner product. If ClSubSp was generalized for arbitrary topological spaces, this assuption could be omitted. (Contributed by AV, 8-Oct-2022.) |
| Ref | Expression |
|---|---|
| cmslssbn.x | β’ π = (π βΎs π) |
| cmscsscms.s | β’ π = (ClSubSpβπ) |
| Ref | Expression |
|---|---|
| bncssbn | β’ (((π β Ban β§ π β βPreHil) β§ π β π) β π β Ban) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bnnvc 24856 | . . . 4 β’ (π β Ban β π β NrmVec) | |
| 2 | eqid 2732 | . . . . 5 β’ (Scalarβπ) = (Scalarβπ) | |
| 3 | 2 | bnsca 24855 | . . . 4 β’ (π β Ban β (Scalarβπ) β CMetSp) |
| 4 | 1, 3 | jca 512 | . . 3 β’ (π β Ban β (π β NrmVec β§ (Scalarβπ) β CMetSp)) |
| 5 | 4 | ad2antrr 724 | . 2 β’ (((π β Ban β§ π β βPreHil) β§ π β π) β (π β NrmVec β§ (Scalarβπ) β CMetSp)) |
| 6 | bncms 24860 | . . 3 β’ (π β Ban β π β CMetSp) | |
| 7 | cmslssbn.x | . . . 4 β’ π = (π βΎs π) | |
| 8 | cmscsscms.s | . . . 4 β’ π = (ClSubSpβπ) | |
| 9 | 7, 8 | cmscsscms 24889 | . . 3 β’ (((π β CMetSp β§ π β βPreHil) β§ π β π) β π β CMetSp) |
| 10 | 6, 9 | sylanl1 678 | . 2 β’ (((π β Ban β§ π β βPreHil) β§ π β π) β π β CMetSp) |
| 11 | cphphl 24687 | . . . 4 β’ (π β βPreHil β π β PreHil) | |
| 12 | 11 | adantl 482 | . . 3 β’ ((π β Ban β§ π β βPreHil) β π β PreHil) |
| 13 | eqid 2732 | . . . 4 β’ (LSubSpβπ) = (LSubSpβπ) | |
| 14 | 8, 13 | csslss 21243 | . . 3 β’ ((π β PreHil β§ π β π) β π β (LSubSpβπ)) |
| 15 | 12, 14 | sylan 580 | . 2 β’ (((π β Ban β§ π β βPreHil) β§ π β π) β π β (LSubSpβπ)) |
| 16 | 7, 13 | cmslssbn 24888 | . 2 β’ (((π β NrmVec β§ (Scalarβπ) β CMetSp) β§ (π β CMetSp β§ π β (LSubSpβπ))) β π β Ban) |
| 17 | 5, 10, 15, 16 | syl12anc 835 | 1 β’ (((π β Ban β§ π β βPreHil) β§ π β π) β π β Ban) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 βcfv 6543 (class class class)co 7408 βΎs cress 17172 Scalarcsca 17199 LSubSpclss 20541 PreHilcphl 21176 ClSubSpccss 21213 NrmVeccnvc 24089 βPreHilccph 24682 CMetSpccms 24848 Bancbn 24849 |
| 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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 ax-mulf 11189 |
| 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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7669 df-om 7855 df-1st 7974 df-2nd 7975 df-supp 8146 df-tpos 8210 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-2o 8466 df-er 8702 df-map 8821 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-fi 9405 df-sup 9436 df-inf 9437 df-oi 9504 df-card 9933 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 12974 df-xneg 13091 df-xadd 13092 df-xmul 13093 df-ico 13329 df-icc 13330 df-fz 13484 df-fzo 13627 df-seq 13966 df-exp 14027 df-hash 14290 df-cj 15045 df-re 15046 df-im 15047 df-sqrt 15181 df-abs 15182 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17367 df-topn 17368 df-0g 17386 df-gsum 17387 df-topgen 17388 df-pt 17389 df-prds 17392 df-xrs 17447 df-qtop 17452 df-imas 17453 df-xps 17455 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-mhm 18670 df-submnd 18671 df-grp 18821 df-minusg 18822 df-sbg 18823 df-mulg 18950 df-subg 19002 df-ghm 19089 df-cntz 19180 df-cmn 19649 df-abl 19650 df-mgp 19987 df-ur 20004 df-ring 20057 df-cring 20058 df-oppr 20149 df-dvdsr 20170 df-unit 20171 df-invr 20201 df-dvr 20214 df-rnghom 20250 df-subrg 20316 df-drng 20358 df-staf 20452 df-srng 20453 df-lmod 20472 df-lss 20542 df-lmhm 20632 df-lvec 20713 df-sra 20784 df-rgmod 20785 df-psmet 20935 df-xmet 20936 df-met 20937 df-bl 20938 df-mopn 20939 df-fbas 20940 df-fg 20941 df-cnfld 20944 df-phl 21178 df-ipf 21179 df-ocv 21215 df-css 21216 df-top 22395 df-topon 22412 df-topsp 22434 df-bases 22448 df-cld 22522 df-ntr 22523 df-cls 22524 df-nei 22601 df-cn 22730 df-cnp 22731 df-t1 22817 df-haus 22818 df-tx 23065 df-hmeo 23258 df-fil 23349 df-flim 23442 df-xms 23825 df-ms 23826 df-tms 23827 df-nm 24090 df-ngp 24091 df-tng 24092 df-nlm 24094 df-nvc 24095 df-clm 24578 df-cph 24684 df-tcph 24685 df-cfil 24771 df-cmet 24773 df-cms 24851 df-bn 24852 |
| This theorem is referenced by: chlcsschl 24894 |
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