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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 30689 | . . . 4 ⊢ 0ℎ ∈ ℋ | |
3 | 1rp 12985 | . . . 4 ⊢ 1 ∈ ℝ+ | |
4 | cnopc 31599 | . . . 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 1460 | . . 3 ⊢ ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℋ ((normℎ‘(𝑧 −ℎ 0ℎ)) < 𝑦 → (normℎ‘((𝑇‘𝑧) −ℎ (𝑇‘0ℎ))) < 1) |
6 | hvsub0 30762 | . . . . . . . 8 ⊢ (𝑧 ∈ ℋ → (𝑧 −ℎ 0ℎ) = 𝑧) | |
7 | 6 | fveq2d 6895 | . . . . . . 7 ⊢ (𝑧 ∈ ℋ → (normℎ‘(𝑧 −ℎ 0ℎ)) = (normℎ‘𝑧)) |
8 | 7 | breq1d 5158 | . . . . . 6 ⊢ (𝑧 ∈ ℋ → ((normℎ‘(𝑧 −ℎ 0ℎ)) < 𝑦 ↔ (normℎ‘𝑧) < 𝑦)) |
9 | nmcopex.1 | . . . . . . . . . . 11 ⊢ 𝑇 ∈ LinOp | |
10 | 9 | lnop0i 31656 | . . . . . . . . . 10 ⊢ (𝑇‘0ℎ) = 0ℎ |
11 | 10 | oveq2i 7423 | . . . . . . . . 9 ⊢ ((𝑇‘𝑧) −ℎ (𝑇‘0ℎ)) = ((𝑇‘𝑧) −ℎ 0ℎ) |
12 | 9 | lnopfi 31655 | . . . . . . . . . . 11 ⊢ 𝑇: ℋ⟶ ℋ |
13 | 12 | ffvelcdmi 7085 | . . . . . . . . . 10 ⊢ (𝑧 ∈ ℋ → (𝑇‘𝑧) ∈ ℋ) |
14 | hvsub0 30762 | . . . . . . . . . 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 | eqtrid 2783 | . . . . . . . 8 ⊢ (𝑧 ∈ ℋ → ((𝑇‘𝑧) −ℎ (𝑇‘0ℎ)) = (𝑇‘𝑧)) |
17 | 16 | fveq2d 6895 | . . . . . . 7 ⊢ (𝑧 ∈ ℋ → (normℎ‘((𝑇‘𝑧) −ℎ (𝑇‘0ℎ))) = (normℎ‘(𝑇‘𝑧))) |
18 | 17 | breq1d 5158 | . . . . . 6 ⊢ (𝑧 ∈ ℋ → ((normℎ‘((𝑇‘𝑧) −ℎ (𝑇‘0ℎ))) < 1 ↔ (normℎ‘(𝑇‘𝑧)) < 1)) |
19 | 8, 18 | imbi12d 344 | . . . . 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 3090 | . . . 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 3093 | . . 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 229 | . 2 ⊢ ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℋ ((normℎ‘𝑧) < 𝑦 → (normℎ‘(𝑇‘𝑧)) < 1) |
23 | nmopval 31542 | . . 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 | ffvelcdmi 7085 | . . 3 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ) |
26 | normcl 30811 | . . 3 ⊢ ((𝑇‘𝑥) ∈ ℋ → (normℎ‘(𝑇‘𝑥)) ∈ ℝ) | |
27 | 25, 26 | syl 17 | . 2 ⊢ (𝑥 ∈ ℋ → (normℎ‘(𝑇‘𝑥)) ∈ ℝ) |
28 | 10 | fveq2i 6894 | . . 3 ⊢ (normℎ‘(𝑇‘0ℎ)) = (normℎ‘0ℎ) |
29 | norm0 30814 | . . 3 ⊢ (normℎ‘0ℎ) = 0 | |
30 | 28, 29 | eqtri 2759 | . 2 ⊢ (normℎ‘(𝑇‘0ℎ)) = 0 |
31 | rpcn 12991 | . . . . 5 ⊢ ((𝑦 / 2) ∈ ℝ+ → (𝑦 / 2) ∈ ℂ) | |
32 | 9 | lnopmuli 31658 | . . . . 5 ⊢ (((𝑦 / 2) ∈ ℂ ∧ 𝑥 ∈ ℋ) → (𝑇‘((𝑦 / 2) ·ℎ 𝑥)) = ((𝑦 / 2) ·ℎ (𝑇‘𝑥))) |
33 | 31, 32 | sylan 579 | . . . 4 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → (𝑇‘((𝑦 / 2) ·ℎ 𝑥)) = ((𝑦 / 2) ·ℎ (𝑇‘𝑥))) |
34 | 33 | fveq2d 6895 | . . 3 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → (normℎ‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥))) = (normℎ‘((𝑦 / 2) ·ℎ (𝑇‘𝑥)))) |
35 | norm-iii 30826 | . . . 4 ⊢ (((𝑦 / 2) ∈ ℂ ∧ (𝑇‘𝑥) ∈ ℋ) → (normℎ‘((𝑦 / 2) ·ℎ (𝑇‘𝑥))) = ((abs‘(𝑦 / 2)) · (normℎ‘(𝑇‘𝑥)))) | |
36 | 31, 25, 35 | syl2an 595 | . . 3 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → (normℎ‘((𝑦 / 2) ·ℎ (𝑇‘𝑥))) = ((abs‘(𝑦 / 2)) · (normℎ‘(𝑇‘𝑥)))) |
37 | rpre 12989 | . . . . . 6 ⊢ ((𝑦 / 2) ∈ ℝ+ → (𝑦 / 2) ∈ ℝ) | |
38 | rpge0 12994 | . . . . . 6 ⊢ ((𝑦 / 2) ∈ ℝ+ → 0 ≤ (𝑦 / 2)) | |
39 | 37, 38 | absidd 15376 | . . . . 5 ⊢ ((𝑦 / 2) ∈ ℝ+ → (abs‘(𝑦 / 2)) = (𝑦 / 2)) |
40 | 39 | adantr 480 | . . . 4 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → (abs‘(𝑦 / 2)) = (𝑦 / 2)) |
41 | 40 | oveq1d 7427 | . . 3 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → ((abs‘(𝑦 / 2)) · (normℎ‘(𝑇‘𝑥))) = ((𝑦 / 2) · (normℎ‘(𝑇‘𝑥)))) |
42 | 34, 36, 41 | 3eqtrrd 2776 | . 2 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → ((𝑦 / 2) · (normℎ‘(𝑇‘𝑥))) = (normℎ‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥)))) |
43 | 22, 24, 27, 30, 42 | nmcexi 31712 | 1 ⊢ (normop‘𝑇) ∈ ℝ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 {cab 2708 ∀wral 3060 ∃wrex 3069 class class class wbr 5148 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 supcsup 9441 ℂcc 11114 ℝcr 11115 0cc0 11116 1c1 11117 · cmul 11121 ℝ*cxr 11254 < clt 11255 ≤ cle 11256 / cdiv 11878 2c2 12274 ℝ+crp 12981 abscabs 15188 ℋchba 30605 ·ℎ csm 30607 normℎcno 30609 0ℎc0v 30610 −ℎ cmv 30611 normopcnop 30631 ContOpccop 30632 LinOpclo 30633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 ax-hilex 30685 ax-hfvadd 30686 ax-hvass 30688 ax-hv0cl 30689 ax-hvaddid 30690 ax-hfvmul 30691 ax-hvmulid 30692 ax-hvmulass 30693 ax-hvdistr2 30695 ax-hvmul0 30696 ax-hfi 30765 ax-his1 30768 ax-his3 30770 ax-his4 30771 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-sup 9443 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-n0 12480 df-z 12566 df-uz 12830 df-rp 12982 df-seq 13974 df-exp 14035 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-hnorm 30654 df-hvsub 30657 df-nmop 31525 df-cnop 31526 df-lnop 31527 |
This theorem is referenced by: nmcoplbi 31714 nmcopex 31715 cnlnadjlem2 31754 cnlnadjlem7 31759 cnlnadjlem8 31760 |
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