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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 24707 | . . . 4 β’ (π β Ban β π β NrmVec) | |
2 | eqid 2737 | . . . . 5 β’ (Scalarβπ) = (Scalarβπ) | |
3 | 2 | bnsca 24706 | . . . 4 β’ (π β Ban β (Scalarβπ) β CMetSp) |
4 | 1, 3 | jca 513 | . . 3 β’ (π β Ban β (π β NrmVec β§ (Scalarβπ) β CMetSp)) |
5 | 4 | ad2antrr 725 | . 2 β’ (((π β Ban β§ π β βPreHil) β§ π β π) β (π β NrmVec β§ (Scalarβπ) β CMetSp)) |
6 | bncms 24711 | . . 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 24740 | . . 3 β’ (((π β CMetSp β§ π β βPreHil) β§ π β π) β π β CMetSp) |
10 | 6, 9 | sylanl1 679 | . 2 β’ (((π β Ban β§ π β βPreHil) β§ π β π) β π β CMetSp) |
11 | cphphl 24538 | . . . 4 β’ (π β βPreHil β π β PreHil) | |
12 | 11 | adantl 483 | . . 3 β’ ((π β Ban β§ π β βPreHil) β π β PreHil) |
13 | eqid 2737 | . . . 4 β’ (LSubSpβπ) = (LSubSpβπ) | |
14 | 8, 13 | csslss 21098 | . . 3 β’ ((π β PreHil β§ π β π) β π β (LSubSpβπ)) |
15 | 12, 14 | sylan 581 | . 2 β’ (((π β Ban β§ π β βPreHil) β§ π β π) β π β (LSubSpβπ)) |
16 | 7, 13 | cmslssbn 24739 | . 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 836 | 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 397 = wceq 1542 β wcel 2107 βcfv 6497 (class class class)co 7358 βΎs cress 17113 Scalarcsca 17137 LSubSpclss 20395 PreHilcphl 21031 ClSubSpccss 21068 NrmVeccnvc 23940 βPreHilccph 24533 CMetSpccms 24699 Bancbn 24700 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 ax-pre-sup 11130 ax-addf 11131 ax-mulf 11132 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7618 df-om 7804 df-1st 7922 df-2nd 7923 df-supp 8094 df-tpos 8158 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-2o 8414 df-er 8649 df-map 8768 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9307 df-fi 9348 df-sup 9379 df-inf 9380 df-oi 9447 df-card 9876 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-div 11814 df-nn 12155 df-2 12217 df-3 12218 df-4 12219 df-5 12220 df-6 12221 df-7 12222 df-8 12223 df-9 12224 df-n0 12415 df-z 12501 df-dec 12620 df-uz 12765 df-q 12875 df-rp 12917 df-xneg 13034 df-xadd 13035 df-xmul 13036 df-ico 13271 df-icc 13272 df-fz 13426 df-fzo 13569 df-seq 13908 df-exp 13969 df-hash 14232 df-cj 14985 df-re 14986 df-im 14987 df-sqrt 15121 df-abs 15122 df-struct 17020 df-sets 17037 df-slot 17055 df-ndx 17067 df-base 17085 df-ress 17114 df-plusg 17147 df-mulr 17148 df-starv 17149 df-sca 17150 df-vsca 17151 df-ip 17152 df-tset 17153 df-ple 17154 df-ds 17156 df-unif 17157 df-hom 17158 df-cco 17159 df-rest 17305 df-topn 17306 df-0g 17324 df-gsum 17325 df-topgen 17326 df-pt 17327 df-prds 17330 df-xrs 17385 df-qtop 17390 df-imas 17391 df-xps 17393 df-mre 17467 df-mrc 17468 df-acs 17470 df-mgm 18498 df-sgrp 18547 df-mnd 18558 df-mhm 18602 df-submnd 18603 df-grp 18752 df-minusg 18753 df-sbg 18754 df-mulg 18874 df-subg 18926 df-ghm 19007 df-cntz 19098 df-cmn 19565 df-abl 19566 df-mgp 19898 df-ur 19915 df-ring 19967 df-cring 19968 df-oppr 20050 df-dvdsr 20071 df-unit 20072 df-invr 20102 df-dvr 20113 df-rnghom 20147 df-drng 20188 df-subrg 20223 df-staf 20307 df-srng 20308 df-lmod 20327 df-lss 20396 df-lmhm 20486 df-lvec 20567 df-sra 20636 df-rgmod 20637 df-psmet 20791 df-xmet 20792 df-met 20793 df-bl 20794 df-mopn 20795 df-fbas 20796 df-fg 20797 df-cnfld 20800 df-phl 21033 df-ipf 21034 df-ocv 21070 df-css 21071 df-top 22246 df-topon 22263 df-topsp 22285 df-bases 22299 df-cld 22373 df-ntr 22374 df-cls 22375 df-nei 22452 df-cn 22581 df-cnp 22582 df-t1 22668 df-haus 22669 df-tx 22916 df-hmeo 23109 df-fil 23200 df-flim 23293 df-xms 23676 df-ms 23677 df-tms 23678 df-nm 23941 df-ngp 23942 df-tng 23943 df-nlm 23945 df-nvc 23946 df-clm 24429 df-cph 24535 df-tcph 24536 df-cfil 24622 df-cmet 24624 df-cms 24702 df-bn 24703 |
This theorem is referenced by: chlcsschl 24745 |
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