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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 25218 | . . . 4 β’ (π β Ban β π β NrmVec) | |
| 2 | eqid 2726 | . . . . 5 β’ (Scalarβπ) = (Scalarβπ) | |
| 3 | 2 | bnsca 25217 | . . . 4 β’ (π β Ban β (Scalarβπ) β CMetSp) |
| 4 | 1, 3 | jca 511 | . . 3 β’ (π β Ban β (π β NrmVec β§ (Scalarβπ) β CMetSp)) |
| 5 | 4 | ad2antrr 723 | . 2 β’ (((π β Ban β§ π β βPreHil) β§ π β π) β (π β NrmVec β§ (Scalarβπ) β CMetSp)) |
| 6 | bncms 25222 | . . 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 25251 | . . 3 β’ (((π β CMetSp β§ π β βPreHil) β§ π β π) β π β CMetSp) |
| 10 | 6, 9 | sylanl1 677 | . 2 β’ (((π β Ban β§ π β βPreHil) β§ π β π) β π β CMetSp) |
| 11 | cphphl 25049 | . . . 4 β’ (π β βPreHil β π β PreHil) | |
| 12 | 11 | adantl 481 | . . 3 β’ ((π β Ban β§ π β βPreHil) β π β PreHil) |
| 13 | eqid 2726 | . . . 4 β’ (LSubSpβπ) = (LSubSpβπ) | |
| 14 | 8, 13 | csslss 21579 | . . 3 β’ ((π β PreHil β§ π β π) β π β (LSubSpβπ)) |
| 15 | 12, 14 | sylan 579 | . 2 β’ (((π β Ban β§ π β βPreHil) β§ π β π) β π β (LSubSpβπ)) |
| 16 | 7, 13 | cmslssbn 25250 | . 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 834 | 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 395 = wceq 1533 β wcel 2098 βcfv 6536 (class class class)co 7404 βΎs cress 17179 Scalarcsca 17206 LSubSpclss 20775 PreHilcphl 21512 ClSubSpccss 21549 NrmVeccnvc 24440 βPreHilccph 25044 CMetSpccms 25210 Bancbn 25211 |
| 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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 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 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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8144 df-tpos 8209 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-2o 8465 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 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-q 12934 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-ico 13333 df-icc 13334 df-fz 13488 df-fzo 13631 df-seq 13970 df-exp 14030 df-hash 14293 df-cj 15049 df-re 15050 df-im 15051 df-sqrt 15185 df-abs 15186 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-mulr 17217 df-starv 17218 df-sca 17219 df-vsca 17220 df-ip 17221 df-tset 17222 df-ple 17223 df-ds 17225 df-unif 17226 df-hom 17227 df-cco 17228 df-rest 17374 df-topn 17375 df-0g 17393 df-gsum 17394 df-topgen 17395 df-pt 17396 df-prds 17399 df-xrs 17454 df-qtop 17459 df-imas 17460 df-xps 17462 df-mre 17536 df-mrc 17537 df-acs 17539 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-mhm 18710 df-submnd 18711 df-grp 18863 df-minusg 18864 df-sbg 18865 df-mulg 18993 df-subg 19047 df-ghm 19136 df-cntz 19230 df-cmn 19699 df-abl 19700 df-mgp 20037 df-rng 20055 df-ur 20084 df-ring 20137 df-cring 20138 df-oppr 20233 df-dvdsr 20256 df-unit 20257 df-invr 20287 df-dvr 20300 df-rhm 20371 df-subrng 20443 df-subrg 20468 df-drng 20586 df-staf 20685 df-srng 20686 df-lmod 20705 df-lss 20776 df-lmhm 20867 df-lvec 20948 df-sra 21018 df-rgmod 21019 df-psmet 21227 df-xmet 21228 df-met 21229 df-bl 21230 df-mopn 21231 df-fbas 21232 df-fg 21233 df-cnfld 21236 df-phl 21514 df-ipf 21515 df-ocv 21551 df-css 21552 df-top 22746 df-topon 22763 df-topsp 22785 df-bases 22799 df-cld 22873 df-ntr 22874 df-cls 22875 df-nei 22952 df-cn 23081 df-cnp 23082 df-t1 23168 df-haus 23169 df-tx 23416 df-hmeo 23609 df-fil 23700 df-flim 23793 df-xms 24176 df-ms 24177 df-tms 24178 df-nm 24441 df-ngp 24442 df-tng 24443 df-nlm 24445 df-nvc 24446 df-clm 24940 df-cph 25046 df-tcph 25047 df-cfil 25133 df-cmet 25135 df-cms 25213 df-bn 25214 |
| This theorem is referenced by: chlcsschl 25256 |
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