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Mirrors > Home > HSE Home > Th. List > adjbdlnb | Structured version Visualization version GIF version |
Description: An operator is bounded and linear iff its adjoint is. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
adjbdlnb | ⊢ (𝑇 ∈ BndLinOp ↔ (adjℎ‘𝑇) ∈ BndLinOp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | adjbdln 29547 | . 2 ⊢ (𝑇 ∈ BndLinOp → (adjℎ‘𝑇) ∈ BndLinOp) | |
2 | bdopadj 29546 | . . . 4 ⊢ ((adjℎ‘𝑇) ∈ BndLinOp → (adjℎ‘𝑇) ∈ dom adjℎ) | |
3 | dmadjrnb 29370 | . . . 4 ⊢ (𝑇 ∈ dom adjℎ ↔ (adjℎ‘𝑇) ∈ dom adjℎ) | |
4 | 2, 3 | sylibr 235 | . . 3 ⊢ ((adjℎ‘𝑇) ∈ BndLinOp → 𝑇 ∈ dom adjℎ) |
5 | cnvadj 29356 | . . . . . 6 ⊢ ◡adjℎ = adjℎ | |
6 | 5 | fveq1i 6546 | . . . . 5 ⊢ (◡adjℎ‘(adjℎ‘𝑇)) = (adjℎ‘(adjℎ‘𝑇)) |
7 | adj1o 29358 | . . . . . 6 ⊢ adjℎ:dom adjℎ–1-1-onto→dom adjℎ | |
8 | simpl 483 | . . . . . 6 ⊢ ((𝑇 ∈ dom adjℎ ∧ (adjℎ‘𝑇) ∈ BndLinOp) → 𝑇 ∈ dom adjℎ) | |
9 | f1ocnvfv1 6905 | . . . . . 6 ⊢ ((adjℎ:dom adjℎ–1-1-onto→dom adjℎ ∧ 𝑇 ∈ dom adjℎ) → (◡adjℎ‘(adjℎ‘𝑇)) = 𝑇) | |
10 | 7, 8, 9 | sylancr 587 | . . . . 5 ⊢ ((𝑇 ∈ dom adjℎ ∧ (adjℎ‘𝑇) ∈ BndLinOp) → (◡adjℎ‘(adjℎ‘𝑇)) = 𝑇) |
11 | 6, 10 | syl5eqr 2847 | . . . 4 ⊢ ((𝑇 ∈ dom adjℎ ∧ (adjℎ‘𝑇) ∈ BndLinOp) → (adjℎ‘(adjℎ‘𝑇)) = 𝑇) |
12 | adjbdln 29547 | . . . . 5 ⊢ ((adjℎ‘𝑇) ∈ BndLinOp → (adjℎ‘(adjℎ‘𝑇)) ∈ BndLinOp) | |
13 | 12 | adantl 482 | . . . 4 ⊢ ((𝑇 ∈ dom adjℎ ∧ (adjℎ‘𝑇) ∈ BndLinOp) → (adjℎ‘(adjℎ‘𝑇)) ∈ BndLinOp) |
14 | 11, 13 | eqeltrrd 2886 | . . 3 ⊢ ((𝑇 ∈ dom adjℎ ∧ (adjℎ‘𝑇) ∈ BndLinOp) → 𝑇 ∈ BndLinOp) |
15 | 4, 14 | mpancom 684 | . 2 ⊢ ((adjℎ‘𝑇) ∈ BndLinOp → 𝑇 ∈ BndLinOp) |
16 | 1, 15 | impbii 210 | 1 ⊢ (𝑇 ∈ BndLinOp ↔ (adjℎ‘𝑇) ∈ BndLinOp) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1525 ∈ wcel 2083 ◡ccnv 5449 dom cdm 5450 –1-1-onto→wf1o 6231 ‘cfv 6232 BndLinOpcbo 28412 adjℎcado 28419 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-inf2 8957 ax-cc 9710 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 ax-pre-sup 10468 ax-addf 10469 ax-mulf 10470 ax-hilex 28463 ax-hfvadd 28464 ax-hvcom 28465 ax-hvass 28466 ax-hv0cl 28467 ax-hvaddid 28468 ax-hfvmul 28469 ax-hvmulid 28470 ax-hvmulass 28471 ax-hvdistr1 28472 ax-hvdistr2 28473 ax-hvmul0 28474 ax-hfi 28543 ax-his1 28546 ax-his2 28547 ax-his3 28548 ax-his4 28549 ax-hcompl 28666 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-fal 1538 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-iin 4834 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-se 5410 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-isom 6241 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-of 7274 df-om 7444 df-1st 7552 df-2nd 7553 df-supp 7689 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-2o 7961 df-oadd 7964 df-omul 7965 df-er 8146 df-map 8265 df-pm 8266 df-ixp 8318 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-fsupp 8687 df-fi 8728 df-sup 8759 df-inf 8760 df-oi 8827 df-card 9221 df-acn 9224 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-div 11152 df-nn 11493 df-2 11554 df-3 11555 df-4 11556 df-5 11557 df-6 11558 df-7 11559 df-8 11560 df-9 11561 df-n0 11752 df-z 11836 df-dec 11953 df-uz 12098 df-q 12202 df-rp 12244 df-xneg 12361 df-xadd 12362 df-xmul 12363 df-ioo 12596 df-ico 12598 df-icc 12599 df-fz 12747 df-fzo 12888 df-fl 13016 df-seq 13224 df-exp 13284 df-hash 13545 df-cj 14296 df-re 14297 df-im 14298 df-sqrt 14432 df-abs 14433 df-clim 14683 df-rlim 14684 df-sum 14881 df-struct 16318 df-ndx 16319 df-slot 16320 df-base 16322 df-sets 16323 df-ress 16324 df-plusg 16411 df-mulr 16412 df-starv 16413 df-sca 16414 df-vsca 16415 df-ip 16416 df-tset 16417 df-ple 16418 df-ds 16420 df-unif 16421 df-hom 16422 df-cco 16423 df-rest 16529 df-topn 16530 df-0g 16548 df-gsum 16549 df-topgen 16550 df-pt 16551 df-prds 16554 df-xrs 16608 df-qtop 16613 df-imas 16614 df-xps 16616 df-mre 16690 df-mrc 16691 df-acs 16693 df-mgm 17685 df-sgrp 17727 df-mnd 17738 df-submnd 17779 df-mulg 17986 df-cntz 18192 df-cmn 18639 df-psmet 20223 df-xmet 20224 df-met 20225 df-bl 20226 df-mopn 20227 df-fbas 20228 df-fg 20229 df-cnfld 20232 df-top 21190 df-topon 21207 df-topsp 21229 df-bases 21242 df-cld 21315 df-ntr 21316 df-cls 21317 df-nei 21394 df-cn 21523 df-cnp 21524 df-lm 21525 df-t1 21610 df-haus 21611 df-tx 21858 df-hmeo 22051 df-fil 22142 df-fm 22234 df-flim 22235 df-flf 22236 df-xms 22617 df-ms 22618 df-tms 22619 df-cfil 23545 df-cau 23546 df-cmet 23547 df-grpo 27957 df-gid 27958 df-ginv 27959 df-gdiv 27960 df-ablo 28009 df-vc 28023 df-nv 28056 df-va 28059 df-ba 28060 df-sm 28061 df-0v 28062 df-vs 28063 df-nmcv 28064 df-ims 28065 df-dip 28165 df-ssp 28186 df-ph 28277 df-cbn 28327 df-hnorm 28432 df-hba 28433 df-hvsub 28435 df-hlim 28436 df-hcau 28437 df-sh 28671 df-ch 28685 df-oc 28716 df-ch0 28717 df-shs 28772 df-pjh 28859 df-h0op 29212 df-nmop 29303 df-cnop 29304 df-lnop 29305 df-bdop 29306 df-unop 29307 df-hmop 29308 df-nmfn 29309 df-nlfn 29310 df-cnfn 29311 df-lnfn 29312 df-adjh 29313 |
This theorem is referenced by: adjbd1o 29549 nmopcoadji 29565 |
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