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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 31023 | . . . 4 ⊢ 0ℎ ∈ ℋ | |
| 3 | 1rp 13039 | . . . 4 ⊢ 1 ∈ ℝ+ | |
| 4 | cnopc 31933 | . . . 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 1462 | . . 3 ⊢ ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℋ ((normℎ‘(𝑧 −ℎ 0ℎ)) < 𝑦 → (normℎ‘((𝑇‘𝑧) −ℎ (𝑇‘0ℎ))) < 1) | 
| 6 | hvsub0 31096 | . . . . . . . 8 ⊢ (𝑧 ∈ ℋ → (𝑧 −ℎ 0ℎ) = 𝑧) | |
| 7 | 6 | fveq2d 6909 | . . . . . . 7 ⊢ (𝑧 ∈ ℋ → (normℎ‘(𝑧 −ℎ 0ℎ)) = (normℎ‘𝑧)) | 
| 8 | 7 | breq1d 5152 | . . . . . 6 ⊢ (𝑧 ∈ ℋ → ((normℎ‘(𝑧 −ℎ 0ℎ)) < 𝑦 ↔ (normℎ‘𝑧) < 𝑦)) | 
| 9 | nmcopex.1 | . . . . . . . . . . 11 ⊢ 𝑇 ∈ LinOp | |
| 10 | 9 | lnop0i 31990 | . . . . . . . . . 10 ⊢ (𝑇‘0ℎ) = 0ℎ | 
| 11 | 10 | oveq2i 7443 | . . . . . . . . 9 ⊢ ((𝑇‘𝑧) −ℎ (𝑇‘0ℎ)) = ((𝑇‘𝑧) −ℎ 0ℎ) | 
| 12 | 9 | lnopfi 31989 | . . . . . . . . . . 11 ⊢ 𝑇: ℋ⟶ ℋ | 
| 13 | 12 | ffvelcdmi 7102 | . . . . . . . . . 10 ⊢ (𝑧 ∈ ℋ → (𝑇‘𝑧) ∈ ℋ) | 
| 14 | hvsub0 31096 | . . . . . . . . . 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 2788 | . . . . . . . 8 ⊢ (𝑧 ∈ ℋ → ((𝑇‘𝑧) −ℎ (𝑇‘0ℎ)) = (𝑇‘𝑧)) | 
| 17 | 16 | fveq2d 6909 | . . . . . . 7 ⊢ (𝑧 ∈ ℋ → (normℎ‘((𝑇‘𝑧) −ℎ (𝑇‘0ℎ))) = (normℎ‘(𝑇‘𝑧))) | 
| 18 | 17 | breq1d 5152 | . . . . . 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 230 | . 2 ⊢ ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℋ ((normℎ‘𝑧) < 𝑦 → (normℎ‘(𝑇‘𝑧)) < 1) | 
| 23 | nmopval 31876 | . . 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 7102 | . . 3 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ) | 
| 26 | normcl 31145 | . . 3 ⊢ ((𝑇‘𝑥) ∈ ℋ → (normℎ‘(𝑇‘𝑥)) ∈ ℝ) | |
| 27 | 25, 26 | syl 17 | . 2 ⊢ (𝑥 ∈ ℋ → (normℎ‘(𝑇‘𝑥)) ∈ ℝ) | 
| 28 | 10 | fveq2i 6908 | . . 3 ⊢ (normℎ‘(𝑇‘0ℎ)) = (normℎ‘0ℎ) | 
| 29 | norm0 31148 | . . 3 ⊢ (normℎ‘0ℎ) = 0 | |
| 30 | 28, 29 | eqtri 2764 | . 2 ⊢ (normℎ‘(𝑇‘0ℎ)) = 0 | 
| 31 | rpcn 13046 | . . . . 5 ⊢ ((𝑦 / 2) ∈ ℝ+ → (𝑦 / 2) ∈ ℂ) | |
| 32 | 9 | lnopmuli 31992 | . . . . 5 ⊢ (((𝑦 / 2) ∈ ℂ ∧ 𝑥 ∈ ℋ) → (𝑇‘((𝑦 / 2) ·ℎ 𝑥)) = ((𝑦 / 2) ·ℎ (𝑇‘𝑥))) | 
| 33 | 31, 32 | sylan 580 | . . . 4 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → (𝑇‘((𝑦 / 2) ·ℎ 𝑥)) = ((𝑦 / 2) ·ℎ (𝑇‘𝑥))) | 
| 34 | 33 | fveq2d 6909 | . . 3 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → (normℎ‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥))) = (normℎ‘((𝑦 / 2) ·ℎ (𝑇‘𝑥)))) | 
| 35 | norm-iii 31160 | . . . 4 ⊢ (((𝑦 / 2) ∈ ℂ ∧ (𝑇‘𝑥) ∈ ℋ) → (normℎ‘((𝑦 / 2) ·ℎ (𝑇‘𝑥))) = ((abs‘(𝑦 / 2)) · (normℎ‘(𝑇‘𝑥)))) | |
| 36 | 31, 25, 35 | syl2an 596 | . . 3 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → (normℎ‘((𝑦 / 2) ·ℎ (𝑇‘𝑥))) = ((abs‘(𝑦 / 2)) · (normℎ‘(𝑇‘𝑥)))) | 
| 37 | rpre 13044 | . . . . . 6 ⊢ ((𝑦 / 2) ∈ ℝ+ → (𝑦 / 2) ∈ ℝ) | |
| 38 | rpge0 13049 | . . . . . 6 ⊢ ((𝑦 / 2) ∈ ℝ+ → 0 ≤ (𝑦 / 2)) | |
| 39 | 37, 38 | absidd 15462 | . . . . 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 7447 | . . 3 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → ((abs‘(𝑦 / 2)) · (normℎ‘(𝑇‘𝑥))) = ((𝑦 / 2) · (normℎ‘(𝑇‘𝑥)))) | 
| 42 | 34, 36, 41 | 3eqtrrd 2781 | . 2 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → ((𝑦 / 2) · (normℎ‘(𝑇‘𝑥))) = (normℎ‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥)))) | 
| 43 | 22, 24, 27, 30, 42 | nmcexi 32046 | 1 ⊢ (normop‘𝑇) ∈ ℝ | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 {cab 2713 ∀wral 3060 ∃wrex 3069 class class class wbr 5142 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 supcsup 9481 ℂcc 11154 ℝcr 11155 0cc0 11156 1c1 11157 · cmul 11161 ℝ*cxr 11295 < clt 11296 ≤ cle 11297 / cdiv 11921 2c2 12322 ℝ+crp 13035 abscabs 15274 ℋchba 30939 ·ℎ csm 30941 normℎcno 30943 0ℎc0v 30944 −ℎ cmv 30945 normopcnop 30965 ContOpccop 30966 LinOpclo 30967 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-hilex 31019 ax-hfvadd 31020 ax-hvass 31022 ax-hv0cl 31023 ax-hvaddid 31024 ax-hfvmul 31025 ax-hvmulid 31026 ax-hvmulass 31027 ax-hvdistr2 31029 ax-hvmul0 31030 ax-hfi 31099 ax-his1 31102 ax-his3 31104 ax-his4 31105 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-sup 9483 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-n0 12529 df-z 12616 df-uz 12880 df-rp 13036 df-seq 14044 df-exp 14104 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-hnorm 30988 df-hvsub 30991 df-nmop 31859 df-cnop 31860 df-lnop 31861 | 
| This theorem is referenced by: nmcoplbi 32048 nmcopex 32049 cnlnadjlem2 32088 cnlnadjlem7 32093 cnlnadjlem8 32094 | 
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