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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 | ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 |
Ref | Expression |
---|---|
hhhl | ⊢ 𝑈 ∈ CHilOLD |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hhhl.1 | . . . 4 ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
2 | 1 | hhnv 30406 | . . 3 ⊢ 𝑈 ∈ NrmCVec |
3 | eqid 2733 | . . . 4 ⊢ (IndMet‘𝑈) = (IndMet‘𝑈) | |
4 | 1, 3 | hhcms 30444 | . . 3 ⊢ (IndMet‘𝑈) ∈ (CMet‘ ℋ) |
5 | 1 | hhba 30408 | . . . 4 ⊢ ℋ = (BaseSet‘𝑈) |
6 | 5, 3 | iscbn 30105 | . . 3 ⊢ (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ (IndMet‘𝑈) ∈ (CMet‘ ℋ))) |
7 | 2, 4, 6 | mpbir2an 710 | . 2 ⊢ 𝑈 ∈ CBan |
8 | 1 | hhph 30419 | . 2 ⊢ 𝑈 ∈ CPreHilOLD |
9 | ishlo 30128 | . 2 ⊢ (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD)) | |
10 | 7, 8, 9 | mpbir2an 710 | 1 ⊢ 𝑈 ∈ CHilOLD |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 〈cop 4634 ‘cfv 6541 CMetccmet 24763 NrmCVeccnv 29825 IndMetcims 29832 CPreHilOLDccphlo 30053 CBanccbn 30103 CHilOLDchlo 30126 ℋchba 30160 +ℎ cva 30161 ·ℎ csm 30162 normℎcno 30164 |
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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-inf2 9633 ax-cc 10427 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 ax-addf 11186 ax-mulf 11187 ax-hilex 30240 ax-hfvadd 30241 ax-hvcom 30242 ax-hvass 30243 ax-hv0cl 30244 ax-hvaddid 30245 ax-hfvmul 30246 ax-hvmulid 30247 ax-hvmulass 30248 ax-hvdistr1 30249 ax-hvdistr2 30250 ax-hvmul0 30251 ax-hfi 30320 ax-his1 30323 ax-his2 30324 ax-his3 30325 ax-his4 30326 ax-hcompl 30443 |
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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-oadd 8467 df-omul 8468 df-er 8700 df-map 8819 df-pm 8820 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9502 df-card 9931 df-acn 9934 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-n0 12470 df-z 12556 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-ico 13327 df-fz 13482 df-fl 13754 df-seq 13964 df-exp 14025 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-clim 15429 df-rlim 15430 df-rest 17365 df-topgen 17386 df-psmet 20929 df-xmet 20930 df-met 20931 df-bl 20932 df-mopn 20933 df-fbas 20934 df-fg 20935 df-top 22388 df-topon 22405 df-bases 22441 df-ntr 22516 df-nei 22594 df-lm 22725 df-fil 23342 df-fm 23434 df-flim 23435 df-flf 23436 df-cfil 24764 df-cau 24765 df-cmet 24766 df-grpo 29734 df-gid 29735 df-ginv 29736 df-gdiv 29737 df-ablo 29786 df-vc 29800 df-nv 29833 df-va 29836 df-ba 29837 df-sm 29838 df-0v 29839 df-vs 29840 df-nmcv 29841 df-ims 29842 df-ph 30054 df-cbn 30104 df-hlo 30127 df-hnorm 30209 df-hvsub 30212 df-hlim 30213 df-hcau 30214 |
This theorem is referenced by: hilhl 30447 |
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