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Mirrors > Home > HSE Home > Th. List > hmopbdoptHIL | Structured version Visualization version GIF version |
Description: A Hermitian operator is a bounded linear operator (Hellinger-Toeplitz Theorem). (Contributed by NM, 18-Jan-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hmopbdoptHIL | ⊢ (𝑇 ∈ HrmOp → 𝑇 ∈ BndLinOp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hmoplin 30245 | . 2 ⊢ (𝑇 ∈ HrmOp → 𝑇 ∈ LinOp) | |
2 | hmop 30225 | . . . 4 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·ih 𝑦)) | |
3 | 2 | 3expib 1120 | . . 3 ⊢ (𝑇 ∈ HrmOp → ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·ih 𝑦))) |
4 | 3 | ralrimivv 3112 | . 2 ⊢ (𝑇 ∈ HrmOp → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·ih 𝑦)) |
5 | hilhl 29509 | . . 3 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 ∈ CHilOLD | |
6 | df-hba 29272 | . . . 4 ⊢ ℋ = (BaseSet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
7 | eqid 2737 | . . . . 5 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
8 | 7 | hhip 29480 | . . . 4 ⊢ ·ih = (·𝑖OLD‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
9 | eqid 2737 | . . . . 5 ⊢ (〈〈 +ℎ , ·ℎ 〉, normℎ〉 LnOp 〈〈 +ℎ , ·ℎ 〉, normℎ〉) = (〈〈 +ℎ , ·ℎ 〉, normℎ〉 LnOp 〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
10 | 7, 9 | hhlnoi 30203 | . . . 4 ⊢ LinOp = (〈〈 +ℎ , ·ℎ 〉, normℎ〉 LnOp 〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
11 | eqid 2737 | . . . . 5 ⊢ (〈〈 +ℎ , ·ℎ 〉, normℎ〉 BLnOp 〈〈 +ℎ , ·ℎ 〉, normℎ〉) = (〈〈 +ℎ , ·ℎ 〉, normℎ〉 BLnOp 〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
12 | 7, 11 | hhbloi 30205 | . . . 4 ⊢ BndLinOp = (〈〈 +ℎ , ·ℎ 〉, normℎ〉 BLnOp 〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
13 | 6, 8, 10, 12 | htth 29221 | . . 3 ⊢ ((〈〈 +ℎ , ·ℎ 〉, normℎ〉 ∈ CHilOLD ∧ 𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·ih 𝑦)) → 𝑇 ∈ BndLinOp) |
14 | 5, 13 | mp3an1 1446 | . 2 ⊢ ((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·ih 𝑦)) → 𝑇 ∈ BndLinOp) |
15 | 1, 4, 14 | syl2anc 583 | 1 ⊢ (𝑇 ∈ HrmOp → 𝑇 ∈ BndLinOp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ∀wral 3062 〈cop 4569 ‘cfv 6423 (class class class)co 7260 LnOp clno 29043 BLnOp cblo 29045 CHilOLDchlo 29188 ℋchba 29222 +ℎ cva 29223 ·ℎ csm 29224 ·ih csp 29225 normℎcno 29226 LinOpclo 29250 BndLinOpcbo 29251 HrmOpcho 29253 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 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 2708 ax-rep 5210 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7571 ax-inf2 9345 ax-cc 10138 ax-dc 10149 ax-cnex 10874 ax-resscn 10875 ax-1cn 10876 ax-icn 10877 ax-addcl 10878 ax-addrcl 10879 ax-mulcl 10880 ax-mulrcl 10881 ax-mulcom 10882 ax-addass 10883 ax-mulass 10884 ax-distr 10885 ax-i2m1 10886 ax-1ne0 10887 ax-1rid 10888 ax-rnegex 10889 ax-rrecex 10890 ax-cnre 10891 ax-pre-lttri 10892 ax-pre-lttrn 10893 ax-pre-ltadd 10894 ax-pre-mulgt0 10895 ax-pre-sup 10896 ax-addf 10897 ax-mulf 10898 ax-hilex 29302 ax-hfvadd 29303 ax-hvcom 29304 ax-hvass 29305 ax-hv0cl 29306 ax-hvaddid 29307 ax-hfvmul 29308 ax-hvmulid 29309 ax-hvmulass 29310 ax-hvdistr1 29311 ax-hvdistr2 29312 ax-hvmul0 29313 ax-hfi 29382 ax-his1 29385 ax-his2 29386 ax-his3 29387 ax-his4 29388 ax-hcompl 29505 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3429 df-sbc 3717 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-tp 4568 df-op 4570 df-uni 4842 df-int 4882 df-iun 4928 df-iin 4929 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-se 5541 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6259 df-on 6260 df-lim 6261 df-suc 6262 df-iota 6381 df-fun 6425 df-fn 6426 df-f 6427 df-f1 6428 df-fo 6429 df-f1o 6430 df-fv 6431 df-isom 6432 df-riota 7217 df-ov 7263 df-oprab 7264 df-mpo 7265 df-of 7516 df-om 7693 df-1st 7809 df-2nd 7810 df-supp 7954 df-frecs 8073 df-wrecs 8104 df-recs 8178 df-rdg 8217 df-1o 8272 df-2o 8273 df-oadd 8276 df-omul 8277 df-er 8461 df-map 8580 df-pm 8581 df-ixp 8649 df-en 8697 df-dom 8698 df-sdom 8699 df-fin 8700 df-fsupp 9075 df-fi 9116 df-sup 9147 df-inf 9148 df-oi 9215 df-card 9644 df-acn 9647 df-pnf 10958 df-mnf 10959 df-xr 10960 df-ltxr 10961 df-le 10962 df-sub 11153 df-neg 11154 df-div 11579 df-nn 11920 df-2 11982 df-3 11983 df-4 11984 df-5 11985 df-6 11986 df-7 11987 df-8 11988 df-9 11989 df-n0 12180 df-z 12266 df-dec 12383 df-uz 12528 df-q 12634 df-rp 12676 df-xneg 12793 df-xadd 12794 df-xmul 12795 df-ioo 13028 df-ico 13030 df-icc 13031 df-fz 13185 df-fzo 13328 df-fl 13456 df-seq 13666 df-exp 13727 df-hash 13989 df-cj 14754 df-re 14755 df-im 14756 df-sqrt 14890 df-abs 14891 df-clim 15141 df-rlim 15142 df-sum 15342 df-struct 16792 df-sets 16809 df-slot 16827 df-ndx 16839 df-base 16857 df-ress 16886 df-plusg 16919 df-mulr 16920 df-starv 16921 df-sca 16922 df-vsca 16923 df-ip 16924 df-tset 16925 df-ple 16926 df-ds 16928 df-unif 16929 df-hom 16930 df-cco 16931 df-rest 17077 df-topn 17078 df-0g 17096 df-gsum 17097 df-topgen 17098 df-pt 17099 df-prds 17102 df-xrs 17157 df-qtop 17162 df-imas 17163 df-xps 17165 df-mre 17239 df-mrc 17240 df-acs 17242 df-mgm 18270 df-sgrp 18319 df-mnd 18330 df-submnd 18375 df-mulg 18645 df-cntz 18867 df-cmn 19332 df-psmet 20533 df-xmet 20534 df-met 20535 df-bl 20536 df-mopn 20537 df-fbas 20538 df-fg 20539 df-cnfld 20542 df-top 21987 df-topon 22004 df-topsp 22026 df-bases 22040 df-cld 22114 df-ntr 22115 df-cls 22116 df-nei 22193 df-cn 22322 df-cnp 22323 df-lm 22324 df-t1 22409 df-haus 22410 df-cmp 22482 df-tx 22657 df-hmeo 22850 df-fil 22941 df-fm 23033 df-flim 23034 df-flf 23035 df-fcls 23036 df-xms 23417 df-ms 23418 df-tms 23419 df-cncf 23985 df-cfil 24362 df-cau 24363 df-cmet 24364 df-grpo 28796 df-gid 28797 df-ginv 28798 df-gdiv 28799 df-ablo 28848 df-vc 28862 df-nv 28895 df-va 28898 df-ba 28899 df-sm 28900 df-0v 28901 df-vs 28902 df-nmcv 28903 df-ims 28904 df-dip 29004 df-lno 29047 df-nmoo 29048 df-blo 29049 df-0o 29050 df-ph 29116 df-cbn 29166 df-hlo 29189 df-hnorm 29271 df-hba 29272 df-hvsub 29274 df-hlim 29275 df-hcau 29276 df-nmop 30142 df-lnop 30144 df-bdop 30145 df-hmop 30147 |
This theorem is referenced by: hmopidmchi 30454 |
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