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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 |
---|---|
hhnv | ⊢ 𝑈 ∈ NrmCVec |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hilablo 28606 | . . . 4 ⊢ +ℎ ∈ AbelOp | |
2 | ablogrpo 27991 | . . . 4 ⊢ ( +ℎ ∈ AbelOp → +ℎ ∈ GrpOp) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ +ℎ ∈ GrpOp |
4 | ax-hfvadd 28446 | . . . 4 ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ | |
5 | 4 | fdmi 6303 | . . 3 ⊢ dom +ℎ = ( ℋ × ℋ) |
6 | 3, 5 | grporn 27965 | . 2 ⊢ ℋ = ran +ℎ |
7 | hilid 28607 | . . 3 ⊢ (GId‘ +ℎ ) = 0ℎ | |
8 | 7 | eqcomi 2787 | . 2 ⊢ 0ℎ = (GId‘ +ℎ ) |
9 | hilvc 28608 | . 2 ⊢ 〈 +ℎ , ·ℎ 〉 ∈ CVecOLD | |
10 | normf 28569 | . 2 ⊢ normℎ: ℋ⟶ℝ | |
11 | norm-i 28575 | . . 3 ⊢ (𝑥 ∈ ℋ → ((normℎ‘𝑥) = 0 ↔ 𝑥 = 0ℎ)) | |
12 | 11 | biimpa 470 | . 2 ⊢ ((𝑥 ∈ ℋ ∧ (normℎ‘𝑥) = 0) → 𝑥 = 0ℎ) |
13 | norm-iii 28586 | . 2 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℋ) → (normℎ‘(𝑦 ·ℎ 𝑥)) = ((abs‘𝑦) · (normℎ‘𝑥))) | |
14 | norm-ii 28584 | . 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 28057 | 1 ⊢ 𝑈 ∈ NrmCVec |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 ∈ wcel 2107 〈cop 4404 × cxp 5355 ‘cfv 6137 0cc0 10274 GrpOpcgr 27933 GIdcgi 27934 AbelOpcablo 27988 NrmCVeccnv 28028 ℋchba 28365 +ℎ cva 28366 ·ℎ csm 28367 normℎcno 28369 0ℎc0v 28370 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 ax-hilex 28445 ax-hfvadd 28446 ax-hvcom 28447 ax-hvass 28448 ax-hv0cl 28449 ax-hvaddid 28450 ax-hfvmul 28451 ax-hvmulid 28452 ax-hvmulass 28453 ax-hvdistr1 28454 ax-hvdistr2 28455 ax-hvmul0 28456 ax-hfi 28525 ax-his1 28528 ax-his2 28529 ax-his3 28530 ax-his4 28531 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-sup 8638 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11036 df-nn 11380 df-2 11443 df-3 11444 df-4 11445 df-n0 11648 df-z 11734 df-uz 11998 df-rp 12143 df-seq 13125 df-exp 13184 df-cj 14252 df-re 14253 df-im 14254 df-sqrt 14388 df-abs 14389 df-grpo 27937 df-gid 27938 df-ablo 27989 df-vc 28003 df-nv 28036 df-hnorm 28414 df-hvsub 28417 |
This theorem is referenced by: hhva 28612 hh0v 28614 hhsm 28615 hhvs 28616 hhnm 28617 hhims 28618 hhmet 28620 hhmetdval 28622 hhip 28623 hhph 28624 hlimadd 28639 hhcau 28644 hhlm 28645 hhhl 28650 hhssabloilem 28707 hhsst 28712 hhshsslem1 28713 hhshsslem2 28714 hhsssh 28715 hhsssh2 28716 hhssvs 28718 occllem 28751 nmopsetretHIL 29312 hhlnoi 29348 hhnmoi 29349 hhbloi 29350 hh0oi 29351 nmopub2tHIL 29358 nmlnop0iHIL 29444 hmopidmchi 29599 |
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