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Mirrors > Home > HSE Home > Th. List > nmcoplb | Structured version Visualization version GIF version |
Description: A lower bound for the norm of a continuous linear Hilbert space operator. Theorem 3.5(ii) of [Beran] p. 99. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmcoplb | ⊢ ((𝑇 ∈ LinOp ∧ 𝑇 ∈ ContOp ∧ 𝐴 ∈ ℋ) → (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 4172 | . . 3 ⊢ (𝑇 ∈ (LinOp ∩ ContOp) ↔ (𝑇 ∈ LinOp ∧ 𝑇 ∈ ContOp)) | |
2 | fveq1 6672 | . . . . . . . 8 ⊢ (𝑇 = if(𝑇 ∈ (LinOp ∩ ContOp), 𝑇, ( I ↾ ℋ)) → (𝑇‘𝐴) = (if(𝑇 ∈ (LinOp ∩ ContOp), 𝑇, ( I ↾ ℋ))‘𝐴)) | |
3 | 2 | fveq2d 6677 | . . . . . . 7 ⊢ (𝑇 = if(𝑇 ∈ (LinOp ∩ ContOp), 𝑇, ( I ↾ ℋ)) → (normℎ‘(𝑇‘𝐴)) = (normℎ‘(if(𝑇 ∈ (LinOp ∩ ContOp), 𝑇, ( I ↾ ℋ))‘𝐴))) |
4 | fveq2 6673 | . . . . . . . 8 ⊢ (𝑇 = if(𝑇 ∈ (LinOp ∩ ContOp), 𝑇, ( I ↾ ℋ)) → (normop‘𝑇) = (normop‘if(𝑇 ∈ (LinOp ∩ ContOp), 𝑇, ( I ↾ ℋ)))) | |
5 | 4 | oveq1d 7174 | . . . . . . 7 ⊢ (𝑇 = if(𝑇 ∈ (LinOp ∩ ContOp), 𝑇, ( I ↾ ℋ)) → ((normop‘𝑇) · (normℎ‘𝐴)) = ((normop‘if(𝑇 ∈ (LinOp ∩ ContOp), 𝑇, ( I ↾ ℋ))) · (normℎ‘𝐴))) |
6 | 3, 5 | breq12d 5082 | . . . . . 6 ⊢ (𝑇 = if(𝑇 ∈ (LinOp ∩ ContOp), 𝑇, ( I ↾ ℋ)) → ((normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴)) ↔ (normℎ‘(if(𝑇 ∈ (LinOp ∩ ContOp), 𝑇, ( I ↾ ℋ))‘𝐴)) ≤ ((normop‘if(𝑇 ∈ (LinOp ∩ ContOp), 𝑇, ( I ↾ ℋ))) · (normℎ‘𝐴)))) |
7 | 6 | imbi2d 343 | . . . . 5 ⊢ (𝑇 = if(𝑇 ∈ (LinOp ∩ ContOp), 𝑇, ( I ↾ ℋ)) → ((𝐴 ∈ ℋ → (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴))) ↔ (𝐴 ∈ ℋ → (normℎ‘(if(𝑇 ∈ (LinOp ∩ ContOp), 𝑇, ( I ↾ ℋ))‘𝐴)) ≤ ((normop‘if(𝑇 ∈ (LinOp ∩ ContOp), 𝑇, ( I ↾ ℋ))) · (normℎ‘𝐴))))) |
8 | idlnop 29772 | . . . . . . . . . 10 ⊢ ( I ↾ ℋ) ∈ LinOp | |
9 | idcnop 29761 | . . . . . . . . . 10 ⊢ ( I ↾ ℋ) ∈ ContOp | |
10 | elin 4172 | . . . . . . . . . 10 ⊢ (( I ↾ ℋ) ∈ (LinOp ∩ ContOp) ↔ (( I ↾ ℋ) ∈ LinOp ∧ ( I ↾ ℋ) ∈ ContOp)) | |
11 | 8, 9, 10 | mpbir2an 709 | . . . . . . . . 9 ⊢ ( I ↾ ℋ) ∈ (LinOp ∩ ContOp) |
12 | 11 | elimel 4537 | . . . . . . . 8 ⊢ if(𝑇 ∈ (LinOp ∩ ContOp), 𝑇, ( I ↾ ℋ)) ∈ (LinOp ∩ ContOp) |
13 | elin 4172 | . . . . . . . 8 ⊢ (if(𝑇 ∈ (LinOp ∩ ContOp), 𝑇, ( I ↾ ℋ)) ∈ (LinOp ∩ ContOp) ↔ (if(𝑇 ∈ (LinOp ∩ ContOp), 𝑇, ( I ↾ ℋ)) ∈ LinOp ∧ if(𝑇 ∈ (LinOp ∩ ContOp), 𝑇, ( I ↾ ℋ)) ∈ ContOp)) | |
14 | 12, 13 | mpbi 232 | . . . . . . 7 ⊢ (if(𝑇 ∈ (LinOp ∩ ContOp), 𝑇, ( I ↾ ℋ)) ∈ LinOp ∧ if(𝑇 ∈ (LinOp ∩ ContOp), 𝑇, ( I ↾ ℋ)) ∈ ContOp) |
15 | 14 | simpli 486 | . . . . . 6 ⊢ if(𝑇 ∈ (LinOp ∩ ContOp), 𝑇, ( I ↾ ℋ)) ∈ LinOp |
16 | 14 | simpri 488 | . . . . . 6 ⊢ if(𝑇 ∈ (LinOp ∩ ContOp), 𝑇, ( I ↾ ℋ)) ∈ ContOp |
17 | 15, 16 | nmcoplbi 29808 | . . . . 5 ⊢ (𝐴 ∈ ℋ → (normℎ‘(if(𝑇 ∈ (LinOp ∩ ContOp), 𝑇, ( I ↾ ℋ))‘𝐴)) ≤ ((normop‘if(𝑇 ∈ (LinOp ∩ ContOp), 𝑇, ( I ↾ ℋ))) · (normℎ‘𝐴))) |
18 | 7, 17 | dedth 4526 | . . . 4 ⊢ (𝑇 ∈ (LinOp ∩ ContOp) → (𝐴 ∈ ℋ → (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴)))) |
19 | 18 | imp 409 | . . 3 ⊢ ((𝑇 ∈ (LinOp ∩ ContOp) ∧ 𝐴 ∈ ℋ) → (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴))) |
20 | 1, 19 | sylanbr 584 | . 2 ⊢ (((𝑇 ∈ LinOp ∧ 𝑇 ∈ ContOp) ∧ 𝐴 ∈ ℋ) → (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴))) |
21 | 20 | 3impa 1106 | 1 ⊢ ((𝑇 ∈ LinOp ∧ 𝑇 ∈ ContOp ∧ 𝐴 ∈ ℋ) → (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ∩ cin 3938 ifcif 4470 class class class wbr 5069 I cid 5462 ↾ cres 5560 ‘cfv 6358 (class class class)co 7159 · cmul 10545 ≤ cle 10679 ℋchba 28699 normℎcno 28703 normopcnop 28725 ContOpccop 28726 LinOpclo 28727 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 ax-hilex 28779 ax-hfvadd 28780 ax-hvcom 28781 ax-hvass 28782 ax-hv0cl 28783 ax-hvaddid 28784 ax-hfvmul 28785 ax-hvmulid 28786 ax-hvmulass 28787 ax-hvdistr1 28788 ax-hvdistr2 28789 ax-hvmul0 28790 ax-hfi 28859 ax-his1 28862 ax-his2 28863 ax-his3 28864 ax-his4 28865 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-sup 8909 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-seq 13373 df-exp 13433 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-grpo 28273 df-gid 28274 df-ablo 28325 df-vc 28339 df-nv 28372 df-va 28375 df-ba 28376 df-sm 28377 df-0v 28378 df-nmcv 28380 df-hnorm 28748 df-hba 28749 df-hvsub 28751 df-nmop 29619 df-cnop 29620 df-lnop 29621 df-unop 29623 |
This theorem is referenced by: lnopconi 29814 |
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