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Mirrors > Home > HSE Home > Th. List > nmcopexi | 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, 5-Feb-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmcopex.1 | ⊢ 𝑇 ∈ LinOp |
nmcopex.2 | ⊢ 𝑇 ∈ ContOp |
Ref | Expression |
---|---|
nmcopexi | ⊢ (normop‘𝑇) ∈ ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmcopex.2 | . . . 4 ⊢ 𝑇 ∈ ContOp | |
2 | ax-hv0cl 28774 | . . . 4 ⊢ 0ℎ ∈ ℋ | |
3 | 1rp 12387 | . . . 4 ⊢ 1 ∈ ℝ+ | |
4 | cnopc 29684 | . . . 4 ⊢ ((𝑇 ∈ ContOp ∧ 0ℎ ∈ ℋ ∧ 1 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℋ ((normℎ‘(𝑧 −ℎ 0ℎ)) < 𝑦 → (normℎ‘((𝑇‘𝑧) −ℎ (𝑇‘0ℎ))) < 1)) | |
5 | 1, 2, 3, 4 | mp3an 1457 | . . 3 ⊢ ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℋ ((normℎ‘(𝑧 −ℎ 0ℎ)) < 𝑦 → (normℎ‘((𝑇‘𝑧) −ℎ (𝑇‘0ℎ))) < 1) |
6 | hvsub0 28847 | . . . . . . . 8 ⊢ (𝑧 ∈ ℋ → (𝑧 −ℎ 0ℎ) = 𝑧) | |
7 | 6 | fveq2d 6669 | . . . . . . 7 ⊢ (𝑧 ∈ ℋ → (normℎ‘(𝑧 −ℎ 0ℎ)) = (normℎ‘𝑧)) |
8 | 7 | breq1d 5069 | . . . . . 6 ⊢ (𝑧 ∈ ℋ → ((normℎ‘(𝑧 −ℎ 0ℎ)) < 𝑦 ↔ (normℎ‘𝑧) < 𝑦)) |
9 | nmcopex.1 | . . . . . . . . . . 11 ⊢ 𝑇 ∈ LinOp | |
10 | 9 | lnop0i 29741 | . . . . . . . . . 10 ⊢ (𝑇‘0ℎ) = 0ℎ |
11 | 10 | oveq2i 7161 | . . . . . . . . 9 ⊢ ((𝑇‘𝑧) −ℎ (𝑇‘0ℎ)) = ((𝑇‘𝑧) −ℎ 0ℎ) |
12 | 9 | lnopfi 29740 | . . . . . . . . . . 11 ⊢ 𝑇: ℋ⟶ ℋ |
13 | 12 | ffvelrni 6845 | . . . . . . . . . 10 ⊢ (𝑧 ∈ ℋ → (𝑇‘𝑧) ∈ ℋ) |
14 | hvsub0 28847 | . . . . . . . . . 10 ⊢ ((𝑇‘𝑧) ∈ ℋ → ((𝑇‘𝑧) −ℎ 0ℎ) = (𝑇‘𝑧)) | |
15 | 13, 14 | syl 17 | . . . . . . . . 9 ⊢ (𝑧 ∈ ℋ → ((𝑇‘𝑧) −ℎ 0ℎ) = (𝑇‘𝑧)) |
16 | 11, 15 | syl5eq 2868 | . . . . . . . 8 ⊢ (𝑧 ∈ ℋ → ((𝑇‘𝑧) −ℎ (𝑇‘0ℎ)) = (𝑇‘𝑧)) |
17 | 16 | fveq2d 6669 | . . . . . . 7 ⊢ (𝑧 ∈ ℋ → (normℎ‘((𝑇‘𝑧) −ℎ (𝑇‘0ℎ))) = (normℎ‘(𝑇‘𝑧))) |
18 | 17 | breq1d 5069 | . . . . . 6 ⊢ (𝑧 ∈ ℋ → ((normℎ‘((𝑇‘𝑧) −ℎ (𝑇‘0ℎ))) < 1 ↔ (normℎ‘(𝑇‘𝑧)) < 1)) |
19 | 8, 18 | imbi12d 347 | . . . . 5 ⊢ (𝑧 ∈ ℋ → (((normℎ‘(𝑧 −ℎ 0ℎ)) < 𝑦 → (normℎ‘((𝑇‘𝑧) −ℎ (𝑇‘0ℎ))) < 1) ↔ ((normℎ‘𝑧) < 𝑦 → (normℎ‘(𝑇‘𝑧)) < 1))) |
20 | 19 | ralbiia 3164 | . . . 4 ⊢ (∀𝑧 ∈ ℋ ((normℎ‘(𝑧 −ℎ 0ℎ)) < 𝑦 → (normℎ‘((𝑇‘𝑧) −ℎ (𝑇‘0ℎ))) < 1) ↔ ∀𝑧 ∈ ℋ ((normℎ‘𝑧) < 𝑦 → (normℎ‘(𝑇‘𝑧)) < 1)) |
21 | 20 | rexbii 3247 | . . 3 ⊢ (∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℋ ((normℎ‘(𝑧 −ℎ 0ℎ)) < 𝑦 → (normℎ‘((𝑇‘𝑧) −ℎ (𝑇‘0ℎ))) < 1) ↔ ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℋ ((normℎ‘𝑧) < 𝑦 → (normℎ‘(𝑇‘𝑧)) < 1)) |
22 | 5, 21 | mpbi 232 | . 2 ⊢ ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℋ ((normℎ‘𝑧) < 𝑦 → (normℎ‘(𝑇‘𝑧)) < 1) |
23 | nmopval 29627 | . . 3 ⊢ (𝑇: ℋ⟶ ℋ → (normop‘𝑇) = sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (normℎ‘(𝑇‘𝑥)))}, ℝ*, < )) | |
24 | 12, 23 | ax-mp 5 | . 2 ⊢ (normop‘𝑇) = sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (normℎ‘(𝑇‘𝑥)))}, ℝ*, < ) |
25 | 12 | ffvelrni 6845 | . . 3 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ) |
26 | normcl 28896 | . . 3 ⊢ ((𝑇‘𝑥) ∈ ℋ → (normℎ‘(𝑇‘𝑥)) ∈ ℝ) | |
27 | 25, 26 | syl 17 | . 2 ⊢ (𝑥 ∈ ℋ → (normℎ‘(𝑇‘𝑥)) ∈ ℝ) |
28 | 10 | fveq2i 6668 | . . 3 ⊢ (normℎ‘(𝑇‘0ℎ)) = (normℎ‘0ℎ) |
29 | norm0 28899 | . . 3 ⊢ (normℎ‘0ℎ) = 0 | |
30 | 28, 29 | eqtri 2844 | . 2 ⊢ (normℎ‘(𝑇‘0ℎ)) = 0 |
31 | rpcn 12393 | . . . . 5 ⊢ ((𝑦 / 2) ∈ ℝ+ → (𝑦 / 2) ∈ ℂ) | |
32 | 9 | lnopmuli 29743 | . . . . 5 ⊢ (((𝑦 / 2) ∈ ℂ ∧ 𝑥 ∈ ℋ) → (𝑇‘((𝑦 / 2) ·ℎ 𝑥)) = ((𝑦 / 2) ·ℎ (𝑇‘𝑥))) |
33 | 31, 32 | sylan 582 | . . . 4 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → (𝑇‘((𝑦 / 2) ·ℎ 𝑥)) = ((𝑦 / 2) ·ℎ (𝑇‘𝑥))) |
34 | 33 | fveq2d 6669 | . . 3 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → (normℎ‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥))) = (normℎ‘((𝑦 / 2) ·ℎ (𝑇‘𝑥)))) |
35 | norm-iii 28911 | . . . 4 ⊢ (((𝑦 / 2) ∈ ℂ ∧ (𝑇‘𝑥) ∈ ℋ) → (normℎ‘((𝑦 / 2) ·ℎ (𝑇‘𝑥))) = ((abs‘(𝑦 / 2)) · (normℎ‘(𝑇‘𝑥)))) | |
36 | 31, 25, 35 | syl2an 597 | . . 3 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → (normℎ‘((𝑦 / 2) ·ℎ (𝑇‘𝑥))) = ((abs‘(𝑦 / 2)) · (normℎ‘(𝑇‘𝑥)))) |
37 | rpre 12391 | . . . . . 6 ⊢ ((𝑦 / 2) ∈ ℝ+ → (𝑦 / 2) ∈ ℝ) | |
38 | rpge0 12396 | . . . . . 6 ⊢ ((𝑦 / 2) ∈ ℝ+ → 0 ≤ (𝑦 / 2)) | |
39 | 37, 38 | absidd 14776 | . . . . 5 ⊢ ((𝑦 / 2) ∈ ℝ+ → (abs‘(𝑦 / 2)) = (𝑦 / 2)) |
40 | 39 | adantr 483 | . . . 4 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → (abs‘(𝑦 / 2)) = (𝑦 / 2)) |
41 | 40 | oveq1d 7165 | . . 3 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → ((abs‘(𝑦 / 2)) · (normℎ‘(𝑇‘𝑥))) = ((𝑦 / 2) · (normℎ‘(𝑇‘𝑥)))) |
42 | 34, 36, 41 | 3eqtrrd 2861 | . 2 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → ((𝑦 / 2) · (normℎ‘(𝑇‘𝑥))) = (normℎ‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥)))) |
43 | 22, 24, 27, 30, 42 | nmcexi 29797 | 1 ⊢ (normop‘𝑇) ∈ ℝ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {cab 2799 ∀wral 3138 ∃wrex 3139 class class class wbr 5059 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 supcsup 8898 ℂcc 10529 ℝcr 10530 0cc0 10531 1c1 10532 · cmul 10536 ℝ*cxr 10668 < clt 10669 ≤ cle 10670 / cdiv 11291 2c2 11686 ℝ+crp 12383 abscabs 14587 ℋchba 28690 ·ℎ csm 28692 normℎcno 28694 0ℎc0v 28695 −ℎ cmv 28696 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 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 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-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-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 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 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 |
This theorem is referenced by: nmcoplbi 29799 nmcopex 29800 cnlnadjlem2 29839 cnlnadjlem7 29844 cnlnadjlem8 29845 |
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