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Mirrors > Home > MPE Home > Th. List > csschl | Structured version Visualization version GIF version |
Description: A complete subspace of a complex pre-Hilbert space is a complex Hilbert space. Remarks: (a) In contrast to ClSubSp, a complete subspace is defined by "a linear subspace in which all Cauchy sequences converge to a point in the subspace". This is closer to the original, but deprecated definition Cℋ (df-ch 31244) of closed subspaces of a Hilbert space. (b) This theorem does not hold for arbitrary subcomplex (pre-)Hilbert spaces, because the scalar field as restriction of the field of the complex numbers need not be closed. (Contributed by NM, 10-Apr-2008.) (Revised by AV, 6-Oct-2022.) |
Ref | Expression |
---|---|
cssbn.x | ⊢ 𝑋 = (𝑊 ↾s 𝑈) |
cssbn.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
cssbn.d | ⊢ 𝐷 = ((dist‘𝑊) ↾ (𝑈 × 𝑈)) |
csschl.c | ⊢ (Scalar‘𝑊) = ℂfld |
Ref | Expression |
---|---|
csschl | ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑈 ∈ 𝑆 ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → (𝑋 ∈ ℂHil ∧ (Scalar‘𝑋) = ℂfld)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cphnvc 25222 | . . . . 5 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmVec) | |
2 | 1 | 3ad2ant1 1133 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑈 ∈ 𝑆 ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → 𝑊 ∈ NrmVec) |
3 | csschl.c | . . . . 5 ⊢ (Scalar‘𝑊) = ℂfld | |
4 | cncms 25401 | . . . . . 6 ⊢ ℂfld ∈ CMetSp | |
5 | eleq1 2826 | . . . . . 6 ⊢ ((Scalar‘𝑊) = ℂfld → ((Scalar‘𝑊) ∈ CMetSp ↔ ℂfld ∈ CMetSp)) | |
6 | 4, 5 | mpbiri 258 | . . . . 5 ⊢ ((Scalar‘𝑊) = ℂfld → (Scalar‘𝑊) ∈ CMetSp) |
7 | 3, 6 | mp1i 13 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑈 ∈ 𝑆 ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → (Scalar‘𝑊) ∈ CMetSp) |
8 | simp2 1137 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑈 ∈ 𝑆 ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → 𝑈 ∈ 𝑆) | |
9 | simp3 1138 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑈 ∈ 𝑆 ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) | |
10 | cssbn.x | . . . . 5 ⊢ 𝑋 = (𝑊 ↾s 𝑈) | |
11 | cssbn.s | . . . . 5 ⊢ 𝑆 = (LSubSp‘𝑊) | |
12 | cssbn.d | . . . . 5 ⊢ 𝐷 = ((dist‘𝑊) ↾ (𝑈 × 𝑈)) | |
13 | 10, 11, 12 | cssbn 25421 | . . . 4 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → 𝑋 ∈ Ban) |
14 | 2, 7, 8, 9, 13 | syl31anc 1373 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑈 ∈ 𝑆 ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → 𝑋 ∈ Ban) |
15 | 10, 11 | cphssphl 25417 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑈 ∈ 𝑆 ∧ 𝑋 ∈ Ban) → 𝑋 ∈ ℂHil) |
16 | 14, 15 | syld3an3 1409 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑈 ∈ 𝑆 ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → 𝑋 ∈ ℂHil) |
17 | eqid 2734 | . . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
18 | 10, 17 | resssca 17397 | . . . 4 ⊢ (𝑈 ∈ 𝑆 → (Scalar‘𝑊) = (Scalar‘𝑋)) |
19 | 18, 3 | eqtr3di 2789 | . . 3 ⊢ (𝑈 ∈ 𝑆 → (Scalar‘𝑋) = ℂfld) |
20 | 19 | 3ad2ant2 1134 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑈 ∈ 𝑆 ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → (Scalar‘𝑋) = ℂfld) |
21 | 16, 20 | jca 511 | 1 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑈 ∈ 𝑆 ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → (𝑋 ∈ ℂHil ∧ (Scalar‘𝑋) = ℂfld)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2103 ⊆ wss 3970 × cxp 5697 dom cdm 5699 ↾ cres 5701 ‘cfv 6572 (class class class)co 7445 ↾s cress 17282 Scalarcsca 17309 distcds 17315 LSubSpclss 20947 MetOpencmopn 21372 ℂfldccnfld 21382 ⇝𝑡clm 23248 NrmVeccnvc 24608 ℂPreHilccph 25212 Cauccau 25299 CMetSpccms 25378 Bancbn 25379 ℂHilchl 25380 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-inf2 9706 ax-cc 10500 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 ax-pre-sup 11258 ax-addf 11259 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4973 df-iun 5021 df-iin 5022 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-se 5655 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-isom 6581 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-of 7710 df-om 7900 df-1st 8026 df-2nd 8027 df-supp 8198 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-2o 8519 df-oadd 8522 df-omul 8523 df-er 8759 df-map 8882 df-pm 8883 df-ixp 8952 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-fsupp 9428 df-fi 9476 df-sup 9507 df-inf 9508 df-oi 9575 df-card 10004 df-acn 10007 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-div 11944 df-nn 12290 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12550 df-z 12636 df-dec 12755 df-uz 12900 df-q 13010 df-rp 13054 df-xneg 13171 df-xadd 13172 df-xmul 13173 df-ioo 13407 df-ico 13409 df-icc 13410 df-fz 13564 df-fzo 13708 df-fl 13839 df-seq 14049 df-exp 14109 df-hash 14376 df-cj 15144 df-re 15145 df-im 15146 df-sqrt 15280 df-abs 15281 df-clim 15530 df-rlim 15531 df-struct 17189 df-sets 17206 df-slot 17224 df-ndx 17236 df-base 17254 df-ress 17283 df-plusg 17319 df-mulr 17320 df-starv 17321 df-sca 17322 df-vsca 17323 df-ip 17324 df-tset 17325 df-ple 17326 df-ds 17328 df-unif 17329 df-hom 17330 df-cco 17331 df-rest 17477 df-topn 17478 df-0g 17496 df-gsum 17497 df-topgen 17498 df-pt 17499 df-prds 17502 df-xrs 17557 df-qtop 17562 df-imas 17563 df-xps 17565 df-mre 17639 df-mrc 17640 df-acs 17642 df-mgm 18673 df-sgrp 18752 df-mnd 18768 df-submnd 18814 df-grp 18971 df-minusg 18972 df-sbg 18973 df-mulg 19103 df-subg 19158 df-ghm 19248 df-cntz 19352 df-cmn 19819 df-abl 19820 df-mgp 20157 df-rng 20175 df-ur 20204 df-ring 20257 df-subrg 20592 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lmhm 21039 df-lvec 21120 df-sra 21190 df-rgmod 21191 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-phl 21662 df-top 22914 df-topon 22931 df-topsp 22953 df-bases 22967 df-cld 23041 df-ntr 23042 df-cls 23043 df-nei 23120 df-cn 23249 df-cnp 23250 df-lm 23251 df-haus 23337 df-cmp 23409 df-tx 23584 df-hmeo 23777 df-fil 23868 df-fm 23960 df-flim 23961 df-flf 23962 df-fcls 23963 df-xms 24344 df-ms 24345 df-tms 24346 df-nm 24609 df-ngp 24610 df-nlm 24613 df-nvc 24614 df-cncf 24916 df-cph 25214 df-cfil 25301 df-cau 25302 df-cmet 25303 df-cms 25381 df-bn 25382 df-hl 25383 |
This theorem is referenced by: (None) |
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