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| Mirrors > Home > HSE Home > Th. List > hilims | Structured version Visualization version GIF version | ||
| Description: Hilbert space distance metric. (Contributed by NM, 13-Sep-2007.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| hilims.1 | β’ β = (BaseSetβπ) |
| hilims.2 | β’ +β = ( +π£ βπ) |
| hilims.3 | β’ Β·β = ( Β·π OLD βπ) |
| hilims.5 | β’ Β·ih = (Β·πOLDβπ) |
| hilims.8 | β’ π· = (IndMetβπ) |
| hilims.9 | β’ π β NrmCVec |
| Ref | Expression |
|---|---|
| hilims | β’ π· = (normβ β ββ ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | hilims.1 | . . 3 β’ β = (BaseSetβπ) | |
| 2 | hilims.2 | . . 3 β’ +β = ( +π£ βπ) | |
| 3 | hilims.3 | . . 3 β’ Β·β = ( Β·π OLD βπ) | |
| 4 | hilims.5 | . . 3 β’ Β·ih = (Β·πOLDβπ) | |
| 5 | hilims.9 | . . 3 β’ π β NrmCVec | |
| 6 | 1, 2, 3, 4, 5 | hilhhi 30684 | . 2 β’ π = β¨β¨ +β , Β·β β©, normββ© |
| 7 | hilims.8 | . 2 β’ π· = (IndMetβπ) | |
| 8 | 6, 7 | hhims2 30693 | 1 β’ π· = (normβ β ββ ) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1539 β wcel 2104 β ccom 5679 βcfv 6542 NrmCVeccnv 30104 +π£ cpv 30105 BaseSetcba 30106 Β·π OLD cns 30107 IndMetcims 30111 Β·πOLDcdip 30220 βchba 30439 +β cva 30440 Β·β csm 30441 Β·ih csp 30442 normβcno 30443 ββ cmv 30445 |
| 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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-hilex 30519 ax-hfvadd 30520 ax-hvcom 30521 ax-hvass 30522 ax-hv0cl 30523 ax-hvaddid 30524 ax-hfvmul 30525 ax-hvmulid 30526 ax-hvmulass 30527 ax-hvdistr1 30528 ax-hvdistr2 30529 ax-hvmul0 30530 ax-hfi 30599 ax-his1 30602 ax-his2 30603 ax-his3 30604 ax-his4 30605 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12979 df-fz 13489 df-fzo 13632 df-seq 13971 df-exp 14032 df-hash 14295 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-clim 15436 df-sum 15637 df-grpo 30013 df-gid 30014 df-ginv 30015 df-gdiv 30016 df-ablo 30065 df-vc 30079 df-nv 30112 df-va 30115 df-ba 30116 df-sm 30117 df-0v 30118 df-vs 30119 df-nmcv 30120 df-ims 30121 df-dip 30221 df-hnorm 30488 df-hvsub 30491 |
| This theorem is referenced by: (None) |
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