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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 | β’ π = β¨β¨ +β , Β·β β©, normββ© |
hhnmo.2 | β’ π = (π normOpOLD π) |
Ref | Expression |
---|---|
hhnmoi | β’ normop = π |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nmop 31597 | . 2 β’ normop = (π‘ β ( β βm β) β¦ sup({π₯ β£ βπ¦ β β ((normββπ¦) β€ 1 β§ π₯ = (normββ(π‘βπ¦)))}, β*, < )) | |
2 | hhnmo.1 | . . . 4 β’ π = β¨β¨ +β , Β·β β©, normββ© | |
3 | 2 | hhnv 30923 | . . 3 β’ π β NrmCVec |
4 | 2 | hhba 30925 | . . . 4 β’ β = (BaseSetβπ) |
5 | 2 | hhnm 30929 | . . . 4 β’ normβ = (normCVβπ) |
6 | hhnmo.2 | . . . 4 β’ π = (π normOpOLD π) | |
7 | 4, 4, 5, 5, 6 | nmoofval 30520 | . . 3 β’ ((π β NrmCVec β§ π β NrmCVec) β π = (π‘ β ( β βm β) β¦ sup({π₯ β£ βπ¦ β β ((normββπ¦) β€ 1 β§ π₯ = (normββ(π‘βπ¦)))}, β*, < ))) |
8 | 3, 3, 7 | mp2an 689 | . 2 β’ π = (π‘ β ( β βm β) β¦ sup({π₯ β£ βπ¦ β β ((normββπ¦) β€ 1 β§ π₯ = (normββ(π‘βπ¦)))}, β*, < )) |
9 | 1, 8 | eqtr4i 2757 | 1 β’ normop = π |
Colors of variables: wff setvar class |
Syntax hints: β§ wa 395 = wceq 1533 β wcel 2098 {cab 2703 βwrex 3064 β¨cop 4629 class class class wbr 5141 β¦ cmpt 5224 βcfv 6536 (class class class)co 7404 βm cmap 8819 supcsup 9434 1c1 11110 β*cxr 11248 < clt 11249 β€ cle 11250 NrmCVeccnv 30342 normOpOLD cnmoo 30499 βchba 30677 +β cva 30678 Β·β csm 30679 normβcno 30681 normopcnop 30703 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-hilex 30757 ax-hfvadd 30758 ax-hvcom 30759 ax-hvass 30760 ax-hv0cl 30761 ax-hvaddid 30762 ax-hfvmul 30763 ax-hvmulid 30764 ax-hvmulass 30765 ax-hvdistr1 30766 ax-hvdistr2 30767 ax-hvmul0 30768 ax-hfi 30837 ax-his1 30840 ax-his2 30841 ax-his3 30842 ax-his4 30843 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-n0 12474 df-z 12560 df-uz 12824 df-rp 12978 df-seq 13970 df-exp 14031 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-grpo 30251 df-gid 30252 df-ablo 30303 df-vc 30317 df-nv 30350 df-va 30353 df-ba 30354 df-nmcv 30358 df-nmoo 30503 df-hnorm 30726 df-hvsub 30729 df-nmop 31597 |
This theorem is referenced by: hhbloi 31660 nmopub2tHIL 31668 nmlnop0iHIL 31754 |
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