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Mirrors > Home > HSE Home > Th. List > nmcopex | Structured version Visualization version GIF version |
Description: The norm of a continuous linear Hilbert space operator exists. Theorem 3.5(i) of [Beran] p. 99. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmcopex | ⊢ ((𝑇 ∈ LinOp ∧ 𝑇 ∈ ContOp) → (normop‘𝑇) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 4168 | . 2 ⊢ (𝑇 ∈ (LinOp ∩ ContOp) ↔ (𝑇 ∈ LinOp ∧ 𝑇 ∈ ContOp)) | |
2 | fveq2 6664 | . . . 4 ⊢ (𝑇 = if(𝑇 ∈ (LinOp ∩ ContOp), 𝑇, ( I ↾ ℋ)) → (normop‘𝑇) = (normop‘if(𝑇 ∈ (LinOp ∩ ContOp), 𝑇, ( I ↾ ℋ)))) | |
3 | 2 | eleq1d 2897 | . . 3 ⊢ (𝑇 = if(𝑇 ∈ (LinOp ∩ ContOp), 𝑇, ( I ↾ ℋ)) → ((normop‘𝑇) ∈ ℝ ↔ (normop‘if(𝑇 ∈ (LinOp ∩ ContOp), 𝑇, ( I ↾ ℋ))) ∈ ℝ)) |
4 | idlnop 29763 | . . . . . . . 8 ⊢ ( I ↾ ℋ) ∈ LinOp | |
5 | idcnop 29752 | . . . . . . . 8 ⊢ ( I ↾ ℋ) ∈ ContOp | |
6 | elin 4168 | . . . . . . . 8 ⊢ (( I ↾ ℋ) ∈ (LinOp ∩ ContOp) ↔ (( I ↾ ℋ) ∈ LinOp ∧ ( I ↾ ℋ) ∈ ContOp)) | |
7 | 4, 5, 6 | mpbir2an 709 | . . . . . . 7 ⊢ ( I ↾ ℋ) ∈ (LinOp ∩ ContOp) |
8 | 7 | elimel 4533 | . . . . . 6 ⊢ if(𝑇 ∈ (LinOp ∩ ContOp), 𝑇, ( I ↾ ℋ)) ∈ (LinOp ∩ ContOp) |
9 | elin 4168 | . . . . . 6 ⊢ (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)) | |
10 | 8, 9 | mpbi 232 | . . . . 5 ⊢ (if(𝑇 ∈ (LinOp ∩ ContOp), 𝑇, ( I ↾ ℋ)) ∈ LinOp ∧ if(𝑇 ∈ (LinOp ∩ ContOp), 𝑇, ( I ↾ ℋ)) ∈ ContOp) |
11 | 10 | simpli 486 | . . . 4 ⊢ if(𝑇 ∈ (LinOp ∩ ContOp), 𝑇, ( I ↾ ℋ)) ∈ LinOp |
12 | 10 | simpri 488 | . . . 4 ⊢ if(𝑇 ∈ (LinOp ∩ ContOp), 𝑇, ( I ↾ ℋ)) ∈ ContOp |
13 | 11, 12 | nmcopexi 29798 | . . 3 ⊢ (normop‘if(𝑇 ∈ (LinOp ∩ ContOp), 𝑇, ( I ↾ ℋ))) ∈ ℝ |
14 | 3, 13 | dedth 4522 | . 2 ⊢ (𝑇 ∈ (LinOp ∩ ContOp) → (normop‘𝑇) ∈ ℝ) |
15 | 1, 14 | sylbir 237 | 1 ⊢ ((𝑇 ∈ LinOp ∧ 𝑇 ∈ ContOp) → (normop‘𝑇) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∩ cin 3934 ifcif 4466 I cid 5453 ↾ cres 5551 ‘cfv 6349 ℝcr 10530 ℋchba 28690 normopcnop 28716 ContOpccop 28717 LinOpclo 28718 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-hilex 28770 ax-hfvadd 28771 ax-hvcom 28772 ax-hvass 28773 ax-hv0cl 28774 ax-hvaddid 28775 ax-hfvmul 28776 ax-hvmulid 28777 ax-hvmulass 28778 ax-hvdistr2 28780 ax-hvmul0 28781 ax-hfi 28850 ax-his1 28853 ax-his2 28854 ax-his3 28855 ax-his4 28856 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-hnorm 28739 df-hvsub 28742 df-nmop 29610 df-cnop 29611 df-lnop 29612 df-unop 29614 |
This theorem is referenced by: lnopconi 29805 lnopcnbd 29807 |
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