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Mirrors > Home > HSE Home > Th. List > cnlnadj | Structured version Visualization version GIF version |
Description: Every continuous linear operator has an adjoint. Theorem 3.10 of [Beran] p. 104. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cnlnadj | ⊢ (𝑇 ∈ (LinOp ∩ ContOp) → ∃𝑡 ∈ (LinOp ∩ ContOp)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑡‘𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnlnadjeu 29493 | . 2 ⊢ (𝑇 ∈ (LinOp ∩ ContOp) → ∃!𝑡 ∈ (LinOp ∩ ContOp)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑡‘𝑦))) | |
2 | reurex 3373 | . 2 ⊢ (∃!𝑡 ∈ (LinOp ∩ ContOp)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑡‘𝑦)) → ∃𝑡 ∈ (LinOp ∩ ContOp)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑡‘𝑦))) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝑇 ∈ (LinOp ∩ ContOp) → ∃𝑡 ∈ (LinOp ∩ ContOp)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑡‘𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 ∀wral 3118 ∃wrex 3119 ∃!wreu 3120 ∩ cin 3798 ‘cfv 6124 (class class class)co 6906 ℋchba 28332 ·ih csp 28335 ContOpccop 28359 LinOpclo 28360 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-inf2 8816 ax-cc 9573 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 ax-pre-sup 10331 ax-addf 10332 ax-mulf 10333 ax-hilex 28412 ax-hfvadd 28413 ax-hvcom 28414 ax-hvass 28415 ax-hv0cl 28416 ax-hvaddid 28417 ax-hfvmul 28418 ax-hvmulid 28419 ax-hvmulass 28420 ax-hvdistr1 28421 ax-hvdistr2 28422 ax-hvmul0 28423 ax-hfi 28492 ax-his1 28495 ax-his2 28496 ax-his3 28497 ax-his4 28498 ax-hcompl 28615 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-iun 4743 df-iin 4744 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-se 5303 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-isom 6133 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-of 7158 df-om 7328 df-1st 7429 df-2nd 7430 df-supp 7561 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-1o 7827 df-2o 7828 df-oadd 7831 df-omul 7832 df-er 8010 df-map 8125 df-pm 8126 df-ixp 8177 df-en 8224 df-dom 8225 df-sdom 8226 df-fin 8227 df-fsupp 8546 df-fi 8587 df-sup 8618 df-inf 8619 df-oi 8685 df-card 9079 df-acn 9082 df-cda 9306 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-div 11011 df-nn 11352 df-2 11415 df-3 11416 df-4 11417 df-5 11418 df-6 11419 df-7 11420 df-8 11421 df-9 11422 df-n0 11620 df-z 11706 df-dec 11823 df-uz 11970 df-q 12073 df-rp 12114 df-xneg 12233 df-xadd 12234 df-xmul 12235 df-ioo 12468 df-ico 12470 df-icc 12471 df-fz 12621 df-fzo 12762 df-fl 12889 df-seq 13097 df-exp 13156 df-hash 13412 df-cj 14217 df-re 14218 df-im 14219 df-sqrt 14353 df-abs 14354 df-clim 14597 df-rlim 14598 df-sum 14795 df-struct 16225 df-ndx 16226 df-slot 16227 df-base 16229 df-sets 16230 df-ress 16231 df-plusg 16319 df-mulr 16320 df-starv 16321 df-sca 16322 df-vsca 16323 df-ip 16324 df-tset 16325 df-ple 16326 df-ds 16328 df-unif 16329 df-hom 16330 df-cco 16331 df-rest 16437 df-topn 16438 df-0g 16456 df-gsum 16457 df-topgen 16458 df-pt 16459 df-prds 16462 df-xrs 16516 df-qtop 16521 df-imas 16522 df-xps 16524 df-mre 16600 df-mrc 16601 df-acs 16603 df-mgm 17596 df-sgrp 17638 df-mnd 17649 df-submnd 17690 df-mulg 17896 df-cntz 18101 df-cmn 18549 df-psmet 20099 df-xmet 20100 df-met 20101 df-bl 20102 df-mopn 20103 df-fbas 20104 df-fg 20105 df-cnfld 20108 df-top 21070 df-topon 21087 df-topsp 21109 df-bases 21122 df-cld 21195 df-ntr 21196 df-cls 21197 df-nei 21274 df-cn 21403 df-cnp 21404 df-lm 21405 df-t1 21490 df-haus 21491 df-tx 21737 df-hmeo 21930 df-fil 22021 df-fm 22113 df-flim 22114 df-flf 22115 df-xms 22496 df-ms 22497 df-tms 22498 df-cfil 23424 df-cau 23425 df-cmet 23426 df-grpo 27904 df-gid 27905 df-ginv 27906 df-gdiv 27907 df-ablo 27956 df-vc 27970 df-nv 28003 df-va 28006 df-ba 28007 df-sm 28008 df-0v 28009 df-vs 28010 df-nmcv 28011 df-ims 28012 df-dip 28112 df-ssp 28133 df-ph 28224 df-cbn 28275 df-hnorm 28381 df-hba 28382 df-hvsub 28384 df-hlim 28385 df-hcau 28386 df-sh 28620 df-ch 28634 df-oc 28665 df-ch0 28666 df-shs 28723 df-pjh 28810 df-h0op 29163 df-nmop 29254 df-cnop 29255 df-lnop 29256 df-unop 29258 df-hmop 29259 df-nmfn 29260 df-nlfn 29261 df-cnfn 29262 df-lnfn 29263 |
This theorem is referenced by: cnlnssadj 29495 adjbdln 29498 |
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