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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 25288 | . . . 4 β’ (π β Ban β π β NrmVec) | |
| 2 | eqid 2728 | . . . . 5 β’ (Scalarβπ) = (Scalarβπ) | |
| 3 | 2 | bnsca 25287 | . . . 4 β’ (π β Ban β (Scalarβπ) β CMetSp) |
| 4 | 1, 3 | jca 510 | . . 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 25292 | . . 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 25321 | . . 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 25119 | . . . 4 β’ (π β βPreHil β π β PreHil) | |
| 12 | 11 | adantl 480 | . . 3 β’ ((π β Ban β§ π β βPreHil) β π β PreHil) |
| 13 | eqid 2728 | . . . 4 β’ (LSubSpβπ) = (LSubSpβπ) | |
| 14 | 8, 13 | csslss 21630 | . . 3 β’ ((π β PreHil β§ π β π) β π β (LSubSpβπ)) |
| 15 | 12, 14 | sylan 578 | . 2 β’ (((π β Ban β§ π β βPreHil) β§ π β π) β π β (LSubSpβπ)) |
| 16 | 7, 13 | cmslssbn 25320 | . 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 394 = wceq 1533 β wcel 2098 βcfv 6553 (class class class)co 7426 βΎs cress 17216 Scalarcsca 17243 LSubSpclss 20822 PreHilcphl 21563 ClSubSpccss 21600 NrmVeccnvc 24510 βPreHilccph 25114 CMetSpccms 25280 Bancbn 25281 |
| 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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225 ax-mulf 11226 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 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 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-tpos 8238 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-er 8731 df-map 8853 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-fi 9442 df-sup 9473 df-inf 9474 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-seq 14007 df-exp 14067 df-hash 14330 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-hom 17264 df-cco 17265 df-rest 17411 df-topn 17412 df-0g 17430 df-gsum 17431 df-topgen 17432 df-pt 17433 df-prds 17436 df-xrs 17491 df-qtop 17496 df-imas 17497 df-xps 17499 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-mhm 18747 df-submnd 18748 df-grp 18900 df-minusg 18901 df-sbg 18902 df-mulg 19031 df-subg 19085 df-ghm 19175 df-cntz 19275 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-cring 20183 df-oppr 20280 df-dvdsr 20303 df-unit 20304 df-invr 20334 df-dvr 20347 df-rhm 20418 df-subrng 20490 df-subrg 20515 df-drng 20633 df-staf 20732 df-srng 20733 df-lmod 20752 df-lss 20823 df-lmhm 20914 df-lvec 20995 df-sra 21065 df-rgmod 21066 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-fbas 21283 df-fg 21284 df-cnfld 21287 df-phl 21565 df-ipf 21566 df-ocv 21602 df-css 21603 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 df-cld 22943 df-ntr 22944 df-cls 22945 df-nei 23022 df-cn 23151 df-cnp 23152 df-t1 23238 df-haus 23239 df-tx 23486 df-hmeo 23679 df-fil 23770 df-flim 23863 df-xms 24246 df-ms 24247 df-tms 24248 df-nm 24511 df-ngp 24512 df-tng 24513 df-nlm 24515 df-nvc 24516 df-clm 25010 df-cph 25116 df-tcph 25117 df-cfil 25203 df-cmet 25205 df-cms 25283 df-bn 25284 |
| This theorem is referenced by: chlcsschl 25326 |
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