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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 31179 | . . . 4 ⊢ +ℎ ∈ AbelOp | |
| 2 | ablogrpo 30566 | . . . 4 ⊢ ( +ℎ ∈ AbelOp → +ℎ ∈ GrpOp) | |
| 3 | 1, 2 | ax-mp 5 | . . 3 ⊢ +ℎ ∈ GrpOp |
| 4 | ax-hfvadd 31019 | . . . 4 ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ | |
| 5 | 4 | fdmi 6747 | . . 3 ⊢ dom +ℎ = ( ℋ × ℋ) |
| 6 | 3, 5 | grporn 30540 | . 2 ⊢ ℋ = ran +ℎ |
| 7 | hilid 31180 | . . 3 ⊢ (GId‘ +ℎ ) = 0ℎ | |
| 8 | 7 | eqcomi 2746 | . 2 ⊢ 0ℎ = (GId‘ +ℎ ) |
| 9 | hilvc 31181 | . 2 ⊢ 〈 +ℎ , ·ℎ 〉 ∈ CVecOLD | |
| 10 | normf 31142 | . 2 ⊢ normℎ: ℋ⟶ℝ | |
| 11 | norm-i 31148 | . . 3 ⊢ (𝑥 ∈ ℋ → ((normℎ‘𝑥) = 0 ↔ 𝑥 = 0ℎ)) | |
| 12 | 11 | biimpa 476 | . 2 ⊢ ((𝑥 ∈ ℋ ∧ (normℎ‘𝑥) = 0) → 𝑥 = 0ℎ) |
| 13 | norm-iii 31159 | . 2 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℋ) → (normℎ‘(𝑦 ·ℎ 𝑥)) = ((abs‘𝑦) · (normℎ‘𝑥))) | |
| 14 | norm-ii 31157 | . 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 30632 | 1 ⊢ 𝑈 ∈ NrmCVec |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 〈cop 4632 × cxp 5683 ‘cfv 6561 0cc0 11155 GrpOpcgr 30508 GIdcgi 30509 AbelOpcablo 30563 NrmCVeccnv 30603 ℋchba 30938 +ℎ cva 30939 ·ℎ csm 30940 normℎcno 30942 0ℎc0v 30943 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-hilex 31018 ax-hfvadd 31019 ax-hvcom 31020 ax-hvass 31021 ax-hv0cl 31022 ax-hvaddid 31023 ax-hfvmul 31024 ax-hvmulid 31025 ax-hvmulass 31026 ax-hvdistr1 31027 ax-hvdistr2 31028 ax-hvmul0 31029 ax-hfi 31098 ax-his1 31101 ax-his2 31102 ax-his3 31103 ax-his4 31104 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-grpo 30512 df-gid 30513 df-ablo 30564 df-vc 30578 df-nv 30611 df-hnorm 30987 df-hvsub 30990 |
| This theorem is referenced by: hhva 31185 hh0v 31187 hhsm 31188 hhvs 31189 hhnm 31190 hhims 31191 hhmet 31193 hhmetdval 31195 hhip 31196 hhph 31197 hlimadd 31212 hhcau 31217 hhlm 31218 hhhl 31223 hhssabloilem 31280 hhsst 31285 hhshsslem1 31286 hhshsslem2 31287 hhsssh 31288 hhsssh2 31289 hhssvs 31291 occllem 31322 nmopsetretHIL 31883 hhlnoi 31919 hhnmoi 31920 hhbloi 31921 hh0oi 31922 nmopub2tHIL 31929 nmlnop0iHIL 32015 hmopidmchi 32170 |
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