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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 28432 | . . . 4 ⊢ 0ℎ ∈ ℋ | |
3 | 1rp 12141 | . . . 4 ⊢ 1 ∈ ℝ+ | |
4 | cnopc 29344 | . . . 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 1534 | . . 3 ⊢ ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℋ ((normℎ‘(𝑧 −ℎ 0ℎ)) < 𝑦 → (normℎ‘((𝑇‘𝑧) −ℎ (𝑇‘0ℎ))) < 1) |
6 | hvsub0 28505 | . . . . . . . 8 ⊢ (𝑧 ∈ ℋ → (𝑧 −ℎ 0ℎ) = 𝑧) | |
7 | 6 | fveq2d 6450 | . . . . . . 7 ⊢ (𝑧 ∈ ℋ → (normℎ‘(𝑧 −ℎ 0ℎ)) = (normℎ‘𝑧)) |
8 | 7 | breq1d 4896 | . . . . . 6 ⊢ (𝑧 ∈ ℋ → ((normℎ‘(𝑧 −ℎ 0ℎ)) < 𝑦 ↔ (normℎ‘𝑧) < 𝑦)) |
9 | nmcopex.1 | . . . . . . . . . . 11 ⊢ 𝑇 ∈ LinOp | |
10 | 9 | lnop0i 29401 | . . . . . . . . . 10 ⊢ (𝑇‘0ℎ) = 0ℎ |
11 | 10 | oveq2i 6933 | . . . . . . . . 9 ⊢ ((𝑇‘𝑧) −ℎ (𝑇‘0ℎ)) = ((𝑇‘𝑧) −ℎ 0ℎ) |
12 | 9 | lnopfi 29400 | . . . . . . . . . . 11 ⊢ 𝑇: ℋ⟶ ℋ |
13 | 12 | ffvelrni 6622 | . . . . . . . . . 10 ⊢ (𝑧 ∈ ℋ → (𝑇‘𝑧) ∈ ℋ) |
14 | hvsub0 28505 | . . . . . . . . . 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 2825 | . . . . . . . 8 ⊢ (𝑧 ∈ ℋ → ((𝑇‘𝑧) −ℎ (𝑇‘0ℎ)) = (𝑇‘𝑧)) |
17 | 16 | fveq2d 6450 | . . . . . . 7 ⊢ (𝑧 ∈ ℋ → (normℎ‘((𝑇‘𝑧) −ℎ (𝑇‘0ℎ))) = (normℎ‘(𝑇‘𝑧))) |
18 | 17 | breq1d 4896 | . . . . . 6 ⊢ (𝑧 ∈ ℋ → ((normℎ‘((𝑇‘𝑧) −ℎ (𝑇‘0ℎ))) < 1 ↔ (normℎ‘(𝑇‘𝑧)) < 1)) |
19 | 8, 18 | imbi12d 336 | . . . . 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 3160 | . . . 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 3223 | . . 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 222 | . 2 ⊢ ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℋ ((normℎ‘𝑧) < 𝑦 → (normℎ‘(𝑇‘𝑧)) < 1) |
23 | nmopval 29287 | . . 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 6622 | . . 3 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ) |
26 | normcl 28554 | . . 3 ⊢ ((𝑇‘𝑥) ∈ ℋ → (normℎ‘(𝑇‘𝑥)) ∈ ℝ) | |
27 | 25, 26 | syl 17 | . 2 ⊢ (𝑥 ∈ ℋ → (normℎ‘(𝑇‘𝑥)) ∈ ℝ) |
28 | 10 | fveq2i 6449 | . . 3 ⊢ (normℎ‘(𝑇‘0ℎ)) = (normℎ‘0ℎ) |
29 | norm0 28557 | . . 3 ⊢ (normℎ‘0ℎ) = 0 | |
30 | 28, 29 | eqtri 2801 | . 2 ⊢ (normℎ‘(𝑇‘0ℎ)) = 0 |
31 | rpcn 12149 | . . . . 5 ⊢ ((𝑦 / 2) ∈ ℝ+ → (𝑦 / 2) ∈ ℂ) | |
32 | 9 | lnopmuli 29403 | . . . . 5 ⊢ (((𝑦 / 2) ∈ ℂ ∧ 𝑥 ∈ ℋ) → (𝑇‘((𝑦 / 2) ·ℎ 𝑥)) = ((𝑦 / 2) ·ℎ (𝑇‘𝑥))) |
33 | 31, 32 | sylan 575 | . . . 4 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → (𝑇‘((𝑦 / 2) ·ℎ 𝑥)) = ((𝑦 / 2) ·ℎ (𝑇‘𝑥))) |
34 | 33 | fveq2d 6450 | . . 3 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → (normℎ‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥))) = (normℎ‘((𝑦 / 2) ·ℎ (𝑇‘𝑥)))) |
35 | norm-iii 28569 | . . . 4 ⊢ (((𝑦 / 2) ∈ ℂ ∧ (𝑇‘𝑥) ∈ ℋ) → (normℎ‘((𝑦 / 2) ·ℎ (𝑇‘𝑥))) = ((abs‘(𝑦 / 2)) · (normℎ‘(𝑇‘𝑥)))) | |
36 | 31, 25, 35 | syl2an 589 | . . 3 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → (normℎ‘((𝑦 / 2) ·ℎ (𝑇‘𝑥))) = ((abs‘(𝑦 / 2)) · (normℎ‘(𝑇‘𝑥)))) |
37 | rpre 12145 | . . . . . 6 ⊢ ((𝑦 / 2) ∈ ℝ+ → (𝑦 / 2) ∈ ℝ) | |
38 | rpge0 12152 | . . . . . 6 ⊢ ((𝑦 / 2) ∈ ℝ+ → 0 ≤ (𝑦 / 2)) | |
39 | 37, 38 | absidd 14569 | . . . . 5 ⊢ ((𝑦 / 2) ∈ ℝ+ → (abs‘(𝑦 / 2)) = (𝑦 / 2)) |
40 | 39 | adantr 474 | . . . 4 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → (abs‘(𝑦 / 2)) = (𝑦 / 2)) |
41 | 40 | oveq1d 6937 | . . 3 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → ((abs‘(𝑦 / 2)) · (normℎ‘(𝑇‘𝑥))) = ((𝑦 / 2) · (normℎ‘(𝑇‘𝑥)))) |
42 | 34, 36, 41 | 3eqtrrd 2818 | . 2 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → ((𝑦 / 2) · (normℎ‘(𝑇‘𝑥))) = (normℎ‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥)))) |
43 | 22, 24, 27, 30, 42 | nmcexi 29457 | 1 ⊢ (normop‘𝑇) ∈ ℝ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 {cab 2762 ∀wral 3089 ∃wrex 3090 class class class wbr 4886 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 supcsup 8634 ℂcc 10270 ℝcr 10271 0cc0 10272 1c1 10273 · cmul 10277 ℝ*cxr 10410 < clt 10411 ≤ cle 10412 / cdiv 11032 2c2 11430 ℝ+crp 12137 abscabs 14381 ℋchba 28348 ·ℎ csm 28350 normℎcno 28352 0ℎc0v 28353 −ℎ cmv 28354 normopcnop 28374 ContOpccop 28375 LinOpclo 28376 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-hilex 28428 ax-hfvadd 28429 ax-hvass 28431 ax-hv0cl 28432 ax-hvaddid 28433 ax-hfvmul 28434 ax-hvmulid 28435 ax-hvmulass 28436 ax-hvdistr2 28438 ax-hvmul0 28439 ax-hfi 28508 ax-his1 28511 ax-his3 28513 ax-his4 28514 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-sup 8636 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-n0 11643 df-z 11729 df-uz 11993 df-rp 12138 df-seq 13120 df-exp 13179 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-hnorm 28397 df-hvsub 28400 df-nmop 29270 df-cnop 29271 df-lnop 29272 |
This theorem is referenced by: nmcoplbi 29459 nmcopex 29460 cnlnadjlem2 29499 cnlnadjlem7 29504 cnlnadjlem8 29505 |
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