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Mirrors > Home > HSE Home > Th. List > hhnv | Structured version Visualization version GIF version |
Description: Hilbert space is a normed complex vector space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hhnv.1 | ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 |
Ref | Expression |
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hhnv | ⊢ 𝑈 ∈ NrmCVec |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hilablo 29195 | . . . 4 ⊢ +ℎ ∈ AbelOp | |
2 | ablogrpo 28582 | . . . 4 ⊢ ( +ℎ ∈ AbelOp → +ℎ ∈ GrpOp) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ +ℎ ∈ GrpOp |
4 | ax-hfvadd 29035 | . . . 4 ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ | |
5 | 4 | fdmi 6535 | . . 3 ⊢ dom +ℎ = ( ℋ × ℋ) |
6 | 3, 5 | grporn 28556 | . 2 ⊢ ℋ = ran +ℎ |
7 | hilid 29196 | . . 3 ⊢ (GId‘ +ℎ ) = 0ℎ | |
8 | 7 | eqcomi 2745 | . 2 ⊢ 0ℎ = (GId‘ +ℎ ) |
9 | hilvc 29197 | . 2 ⊢ 〈 +ℎ , ·ℎ 〉 ∈ CVecOLD | |
10 | normf 29158 | . 2 ⊢ normℎ: ℋ⟶ℝ | |
11 | norm-i 29164 | . . 3 ⊢ (𝑥 ∈ ℋ → ((normℎ‘𝑥) = 0 ↔ 𝑥 = 0ℎ)) | |
12 | 11 | biimpa 480 | . 2 ⊢ ((𝑥 ∈ ℋ ∧ (normℎ‘𝑥) = 0) → 𝑥 = 0ℎ) |
13 | norm-iii 29175 | . 2 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℋ) → (normℎ‘(𝑦 ·ℎ 𝑥)) = ((abs‘𝑦) · (normℎ‘𝑥))) | |
14 | norm-ii 29173 | . 2 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (normℎ‘(𝑥 +ℎ 𝑦)) ≤ ((normℎ‘𝑥) + (normℎ‘𝑦))) | |
15 | hhnv.1 | . 2 ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
16 | 6, 8, 9, 10, 12, 13, 14, 15 | isnvi 28648 | 1 ⊢ 𝑈 ∈ NrmCVec |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2112 〈cop 4533 × cxp 5534 ‘cfv 6358 0cc0 10694 GrpOpcgr 28524 GIdcgi 28525 AbelOpcablo 28579 NrmCVeccnv 28619 ℋchba 28954 +ℎ cva 28955 ·ℎ csm 28956 normℎcno 28958 0ℎc0v 28959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 ax-hilex 29034 ax-hfvadd 29035 ax-hvcom 29036 ax-hvass 29037 ax-hv0cl 29038 ax-hvaddid 29039 ax-hfvmul 29040 ax-hvmulid 29041 ax-hvmulass 29042 ax-hvdistr1 29043 ax-hvdistr2 29044 ax-hvmul0 29045 ax-hfi 29114 ax-his1 29117 ax-his2 29118 ax-his3 29119 ax-his4 29120 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-sup 9036 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-n0 12056 df-z 12142 df-uz 12404 df-rp 12552 df-seq 13540 df-exp 13601 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-grpo 28528 df-gid 28529 df-ablo 28580 df-vc 28594 df-nv 28627 df-hnorm 29003 df-hvsub 29006 |
This theorem is referenced by: hhva 29201 hh0v 29203 hhsm 29204 hhvs 29205 hhnm 29206 hhims 29207 hhmet 29209 hhmetdval 29211 hhip 29212 hhph 29213 hlimadd 29228 hhcau 29233 hhlm 29234 hhhl 29239 hhssabloilem 29296 hhsst 29301 hhshsslem1 29302 hhshsslem2 29303 hhsssh 29304 hhsssh2 29305 hhssvs 29307 occllem 29338 nmopsetretHIL 29899 hhlnoi 29935 hhnmoi 29936 hhbloi 29937 hh0oi 29938 nmopub2tHIL 29945 nmlnop0iHIL 30031 hmopidmchi 30186 |
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