Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 29156) 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 23928 | . . . . 5 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmVec) | |
2 | 1 | 3ad2ant1 1134 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑈 ∈ 𝑆 ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → 𝑊 ∈ NrmVec) |
3 | csschl.c | . . . . 5 ⊢ (Scalar‘𝑊) = ℂfld | |
4 | cncms 24107 | . . . . . 6 ⊢ ℂfld ∈ CMetSp | |
5 | eleq1 2820 | . . . . . 6 ⊢ ((Scalar‘𝑊) = ℂfld → ((Scalar‘𝑊) ∈ CMetSp ↔ ℂfld ∈ CMetSp)) | |
6 | 4, 5 | mpbiri 261 | . . . . 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 1138 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑈 ∈ 𝑆 ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → 𝑈 ∈ 𝑆) | |
9 | simp3 1139 | . . . 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 24127 | . . . 4 ⊢ (((𝑊 ∈ NrmVec ∧ (Scalar‘𝑊) ∈ CMetSp ∧ 𝑈 ∈ 𝑆) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → 𝑋 ∈ Ban) |
14 | 2, 7, 8, 9, 13 | syl31anc 1374 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑈 ∈ 𝑆 ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → 𝑋 ∈ Ban) |
15 | 10, 11 | cphssphl 24123 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑈 ∈ 𝑆 ∧ 𝑋 ∈ Ban) → 𝑋 ∈ ℂHil) |
16 | 14, 15 | syld3an3 1410 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑈 ∈ 𝑆 ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → 𝑋 ∈ ℂHil) |
17 | eqid 2738 | . . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
18 | 10, 17 | resssca 16753 | . . . 4 ⊢ (𝑈 ∈ 𝑆 → (Scalar‘𝑊) = (Scalar‘𝑋)) |
19 | 3, 18 | syl5reqr 2788 | . . 3 ⊢ (𝑈 ∈ 𝑆 → (Scalar‘𝑋) = ℂfld) |
20 | 19 | 3ad2ant2 1135 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑈 ∈ 𝑆 ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡‘(MetOpen‘𝐷))) → (Scalar‘𝑋) = ℂfld) |
21 | 16, 20 | jca 515 | 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 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ⊆ wss 3843 × cxp 5523 dom cdm 5525 ↾ cres 5527 ‘cfv 6339 (class class class)co 7170 ↾s cress 16587 Scalarcsca 16671 distcds 16677 LSubSpclss 19822 MetOpencmopn 20207 ℂfldccnfld 20217 ⇝𝑡clm 21977 NrmVeccnvc 23334 ℂPreHilccph 23918 Cauccau 24005 CMetSpccms 24084 Bancbn 24085 ℂHilchl 24086 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-inf2 9177 ax-cc 9935 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-pre-sup 10693 ax-addf 10694 ax-mulf 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-of 7425 df-om 7600 df-1st 7714 df-2nd 7715 df-supp 7857 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-2o 8132 df-oadd 8135 df-omul 8136 df-er 8320 df-map 8439 df-pm 8440 df-ixp 8508 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-fsupp 8907 df-fi 8948 df-sup 8979 df-inf 8980 df-oi 9047 df-card 9441 df-acn 9444 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-z 12063 df-dec 12180 df-uz 12325 df-q 12431 df-rp 12473 df-xneg 12590 df-xadd 12591 df-xmul 12592 df-ioo 12825 df-ico 12827 df-icc 12828 df-fz 12982 df-fzo 13125 df-fl 13253 df-seq 13461 df-exp 13522 df-hash 13783 df-cj 14548 df-re 14549 df-im 14550 df-sqrt 14684 df-abs 14685 df-clim 14935 df-rlim 14936 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-mulr 16682 df-starv 16683 df-sca 16684 df-vsca 16685 df-ip 16686 df-tset 16687 df-ple 16688 df-ds 16690 df-unif 16691 df-hom 16692 df-cco 16693 df-rest 16799 df-topn 16800 df-0g 16818 df-gsum 16819 df-topgen 16820 df-pt 16821 df-prds 16824 df-xrs 16878 df-qtop 16883 df-imas 16884 df-xps 16886 df-mre 16960 df-mrc 16961 df-acs 16963 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-submnd 18073 df-grp 18222 df-minusg 18223 df-sbg 18224 df-mulg 18343 df-subg 18394 df-ghm 18474 df-cntz 18565 df-cmn 19026 df-mgp 19359 df-ur 19371 df-ring 19418 df-subrg 19652 df-lmod 19755 df-lss 19823 df-lsp 19863 df-lmhm 19913 df-lvec 19994 df-sra 20063 df-rgmod 20064 df-psmet 20209 df-xmet 20210 df-met 20211 df-bl 20212 df-mopn 20213 df-fbas 20214 df-fg 20215 df-cnfld 20218 df-phl 20442 df-top 21645 df-topon 21662 df-topsp 21684 df-bases 21697 df-cld 21770 df-ntr 21771 df-cls 21772 df-nei 21849 df-cn 21978 df-cnp 21979 df-lm 21980 df-haus 22066 df-cmp 22138 df-tx 22313 df-hmeo 22506 df-fil 22597 df-fm 22689 df-flim 22690 df-flf 22691 df-fcls 22692 df-xms 23073 df-ms 23074 df-tms 23075 df-nm 23335 df-ngp 23336 df-nlm 23339 df-nvc 23340 df-cncf 23630 df-cph 23920 df-cfil 24007 df-cau 24008 df-cmet 24009 df-cms 24087 df-bn 24088 df-hl 24089 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |