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| Mirrors > Home > HSE Home > Th. List > hhhl | Structured version Visualization version GIF version | ||
| Description: The Hilbert space structure is a complex Hilbert space. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| hhhl.1 | β’ π = β¨β¨ +β , Β·β β©, normββ© |
| Ref | Expression |
|---|---|
| hhhl | β’ π β CHilOLD |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | hhhl.1 | . . . 4 β’ π = β¨β¨ +β , Β·β β©, normββ© | |
| 2 | 1 | hhnv 30418 | . . 3 β’ π β NrmCVec |
| 3 | eqid 2733 | . . . 4 β’ (IndMetβπ) = (IndMetβπ) | |
| 4 | 1, 3 | hhcms 30456 | . . 3 β’ (IndMetβπ) β (CMetβ β) |
| 5 | 1 | hhba 30420 | . . . 4 β’ β = (BaseSetβπ) |
| 6 | 5, 3 | iscbn 30117 | . . 3 β’ (π β CBan β (π β NrmCVec β§ (IndMetβπ) β (CMetβ β))) |
| 7 | 2, 4, 6 | mpbir2an 710 | . 2 β’ π β CBan |
| 8 | 1 | hhph 30431 | . 2 β’ π β CPreHilOLD |
| 9 | ishlo 30140 | . 2 β’ (π β CHilOLD β (π β CBan β§ π β CPreHilOLD)) | |
| 10 | 7, 8, 9 | mpbir2an 710 | 1 β’ π β CHilOLD |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1542 β wcel 2107 β¨cop 4635 βcfv 6544 CMetccmet 24771 NrmCVeccnv 29837 IndMetcims 29844 CPreHilOLDccphlo 30065 CBanccbn 30115 CHilOLDchlo 30138 βchba 30172 +β cva 30173 Β·β csm 30174 normβcno 30176 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cc 10430 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190 ax-hilex 30252 ax-hfvadd 30253 ax-hvcom 30254 ax-hvass 30255 ax-hv0cl 30256 ax-hvaddid 30257 ax-hfvmul 30258 ax-hvmulid 30259 ax-hvmulass 30260 ax-hvdistr1 30261 ax-hvdistr2 30262 ax-hvmul0 30263 ax-hfi 30332 ax-his1 30335 ax-his2 30336 ax-his3 30337 ax-his4 30338 ax-hcompl 30455 |
| This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-oadd 8470 df-omul 8471 df-er 8703 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-acn 9937 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-n0 12473 df-z 12559 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ico 13330 df-fz 13485 df-fl 13757 df-seq 13967 df-exp 14028 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-rlim 15433 df-rest 17368 df-topgen 17389 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-fbas 20941 df-fg 20942 df-top 22396 df-topon 22413 df-bases 22449 df-ntr 22524 df-nei 22602 df-lm 22733 df-fil 23350 df-fm 23442 df-flim 23443 df-flf 23444 df-cfil 24772 df-cau 24773 df-cmet 24774 df-grpo 29746 df-gid 29747 df-ginv 29748 df-gdiv 29749 df-ablo 29798 df-vc 29812 df-nv 29845 df-va 29848 df-ba 29849 df-sm 29850 df-0v 29851 df-vs 29852 df-nmcv 29853 df-ims 29854 df-ph 30066 df-cbn 30116 df-hlo 30139 df-hnorm 30221 df-hvsub 30224 df-hlim 30225 df-hcau 30226 |
| This theorem is referenced by: hilhl 30459 |
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