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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 28939 | . . . 4 ⊢ +ℎ ∈ AbelOp | |
2 | ablogrpo 28326 | . . . 4 ⊢ ( +ℎ ∈ AbelOp → +ℎ ∈ GrpOp) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ +ℎ ∈ GrpOp |
4 | ax-hfvadd 28779 | . . . 4 ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ | |
5 | 4 | fdmi 6526 | . . 3 ⊢ dom +ℎ = ( ℋ × ℋ) |
6 | 3, 5 | grporn 28300 | . 2 ⊢ ℋ = ran +ℎ |
7 | hilid 28940 | . . 3 ⊢ (GId‘ +ℎ ) = 0ℎ | |
8 | 7 | eqcomi 2832 | . 2 ⊢ 0ℎ = (GId‘ +ℎ ) |
9 | hilvc 28941 | . 2 ⊢ 〈 +ℎ , ·ℎ 〉 ∈ CVecOLD | |
10 | normf 28902 | . 2 ⊢ normℎ: ℋ⟶ℝ | |
11 | norm-i 28908 | . . 3 ⊢ (𝑥 ∈ ℋ → ((normℎ‘𝑥) = 0 ↔ 𝑥 = 0ℎ)) | |
12 | 11 | biimpa 479 | . 2 ⊢ ((𝑥 ∈ ℋ ∧ (normℎ‘𝑥) = 0) → 𝑥 = 0ℎ) |
13 | norm-iii 28919 | . 2 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℋ) → (normℎ‘(𝑦 ·ℎ 𝑥)) = ((abs‘𝑦) · (normℎ‘𝑥))) | |
14 | norm-ii 28917 | . 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 28392 | 1 ⊢ 𝑈 ∈ NrmCVec |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 〈cop 4575 × cxp 5555 ‘cfv 6357 0cc0 10539 GrpOpcgr 28268 GIdcgi 28269 AbelOpcablo 28323 NrmCVeccnv 28363 ℋchba 28698 +ℎ cva 28699 ·ℎ csm 28700 normℎcno 28702 0ℎc0v 28703 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-hilex 28778 ax-hfvadd 28779 ax-hvcom 28780 ax-hvass 28781 ax-hv0cl 28782 ax-hvaddid 28783 ax-hfvmul 28784 ax-hvmulid 28785 ax-hvmulass 28786 ax-hvdistr1 28787 ax-hvdistr2 28788 ax-hvmul0 28789 ax-hfi 28858 ax-his1 28861 ax-his2 28862 ax-his3 28863 ax-his4 28864 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-grpo 28272 df-gid 28273 df-ablo 28324 df-vc 28338 df-nv 28371 df-hnorm 28747 df-hvsub 28750 |
This theorem is referenced by: hhva 28945 hh0v 28947 hhsm 28948 hhvs 28949 hhnm 28950 hhims 28951 hhmet 28953 hhmetdval 28955 hhip 28956 hhph 28957 hlimadd 28972 hhcau 28977 hhlm 28978 hhhl 28983 hhssabloilem 29040 hhsst 29045 hhshsslem1 29046 hhshsslem2 29047 hhsssh 29048 hhsssh2 29049 hhssvs 29051 occllem 29082 nmopsetretHIL 29643 hhlnoi 29679 hhnmoi 29680 hhbloi 29681 hh0oi 29682 nmopub2tHIL 29689 nmlnop0iHIL 29775 hmopidmchi 29930 |
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