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Mirrors > Home > HSE Home > Th. List > hhnmoi | Structured version Visualization version GIF version |
Description: The norm of an operator in Hilbert space. (Contributed by NM, 19-Nov-2007.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hhnmo.1 | ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 |
hhnmo.2 | ⊢ 𝑁 = (𝑈 normOpOLD 𝑈) |
Ref | Expression |
---|---|
hhnmoi | ⊢ normop = 𝑁 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nmop 29610 | . 2 ⊢ normop = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑡‘𝑦)))}, ℝ*, < )) | |
2 | hhnmo.1 | . . . 4 ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
3 | 2 | hhnv 28936 | . . 3 ⊢ 𝑈 ∈ NrmCVec |
4 | 2 | hhba 28938 | . . . 4 ⊢ ℋ = (BaseSet‘𝑈) |
5 | 2 | hhnm 28942 | . . . 4 ⊢ normℎ = (normCV‘𝑈) |
6 | hhnmo.2 | . . . 4 ⊢ 𝑁 = (𝑈 normOpOLD 𝑈) | |
7 | 4, 4, 5, 5, 6 | nmoofval 28533 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑈 ∈ NrmCVec) → 𝑁 = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑡‘𝑦)))}, ℝ*, < ))) |
8 | 3, 3, 7 | mp2an 690 | . 2 ⊢ 𝑁 = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑡‘𝑦)))}, ℝ*, < )) |
9 | 1, 8 | eqtr4i 2847 | 1 ⊢ normop = 𝑁 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1533 ∈ wcel 2110 {cab 2799 ∃wrex 3139 〈cop 4567 class class class wbr 5059 ↦ cmpt 5139 ‘cfv 6350 (class class class)co 7150 ↑m cmap 8400 supcsup 8898 1c1 10532 ℝ*cxr 10668 < clt 10669 ≤ cle 10670 NrmCVeccnv 28355 normOpOLD cnmoo 28512 ℋchba 28690 +ℎ cva 28691 ·ℎ csm 28692 normℎcno 28694 normopcnop 28716 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-hilex 28770 ax-hfvadd 28771 ax-hvcom 28772 ax-hvass 28773 ax-hv0cl 28774 ax-hvaddid 28775 ax-hfvmul 28776 ax-hvmulid 28777 ax-hvmulass 28778 ax-hvdistr1 28779 ax-hvdistr2 28780 ax-hvmul0 28781 ax-hfi 28850 ax-his1 28853 ax-his2 28854 ax-his3 28855 ax-his4 28856 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-grpo 28264 df-gid 28265 df-ablo 28316 df-vc 28330 df-nv 28363 df-va 28366 df-ba 28367 df-nmcv 28371 df-nmoo 28516 df-hnorm 28739 df-hvsub 28742 df-nmop 29610 |
This theorem is referenced by: hhbloi 29673 nmopub2tHIL 29681 nmlnop0iHIL 29767 |
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