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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 31189 | . . . 4 ⊢ +ℎ ∈ AbelOp | |
2 | ablogrpo 30576 | . . . 4 ⊢ ( +ℎ ∈ AbelOp → +ℎ ∈ GrpOp) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ +ℎ ∈ GrpOp |
4 | ax-hfvadd 31029 | . . . 4 ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ | |
5 | 4 | fdmi 6748 | . . 3 ⊢ dom +ℎ = ( ℋ × ℋ) |
6 | 3, 5 | grporn 30550 | . 2 ⊢ ℋ = ran +ℎ |
7 | hilid 31190 | . . 3 ⊢ (GId‘ +ℎ ) = 0ℎ | |
8 | 7 | eqcomi 2744 | . 2 ⊢ 0ℎ = (GId‘ +ℎ ) |
9 | hilvc 31191 | . 2 ⊢ 〈 +ℎ , ·ℎ 〉 ∈ CVecOLD | |
10 | normf 31152 | . 2 ⊢ normℎ: ℋ⟶ℝ | |
11 | norm-i 31158 | . . 3 ⊢ (𝑥 ∈ ℋ → ((normℎ‘𝑥) = 0 ↔ 𝑥 = 0ℎ)) | |
12 | 11 | biimpa 476 | . 2 ⊢ ((𝑥 ∈ ℋ ∧ (normℎ‘𝑥) = 0) → 𝑥 = 0ℎ) |
13 | norm-iii 31169 | . 2 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℋ) → (normℎ‘(𝑦 ·ℎ 𝑥)) = ((abs‘𝑦) · (normℎ‘𝑥))) | |
14 | norm-ii 31167 | . 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 30642 | 1 ⊢ 𝑈 ∈ NrmCVec |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 〈cop 4637 × cxp 5687 ‘cfv 6563 0cc0 11153 GrpOpcgr 30518 GIdcgi 30519 AbelOpcablo 30573 NrmCVeccnv 30613 ℋchba 30948 +ℎ cva 30949 ·ℎ csm 30950 normℎcno 30952 0ℎc0v 30953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-hilex 31028 ax-hfvadd 31029 ax-hvcom 31030 ax-hvass 31031 ax-hv0cl 31032 ax-hvaddid 31033 ax-hfvmul 31034 ax-hvmulid 31035 ax-hvmulass 31036 ax-hvdistr1 31037 ax-hvdistr2 31038 ax-hvmul0 31039 ax-hfi 31108 ax-his1 31111 ax-his2 31112 ax-his3 31113 ax-his4 31114 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-grpo 30522 df-gid 30523 df-ablo 30574 df-vc 30588 df-nv 30621 df-hnorm 30997 df-hvsub 31000 |
This theorem is referenced by: hhva 31195 hh0v 31197 hhsm 31198 hhvs 31199 hhnm 31200 hhims 31201 hhmet 31203 hhmetdval 31205 hhip 31206 hhph 31207 hlimadd 31222 hhcau 31227 hhlm 31228 hhhl 31233 hhssabloilem 31290 hhsst 31295 hhshsslem1 31296 hhshsslem2 31297 hhsssh 31298 hhsssh2 31299 hhssvs 31301 occllem 31332 nmopsetretHIL 31893 hhlnoi 31929 hhnmoi 31930 hhbloi 31931 hh0oi 31932 nmopub2tHIL 31939 nmlnop0iHIL 32025 hmopidmchi 32180 |
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