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Mirrors > Home > HSE Home > Th. List > nmcfnexi | Structured version Visualization version GIF version |
Description: The norm of a continuous linear Hilbert space functional exists. Theorem 3.5(i) of [Beran] p. 99. (Contributed by NM, 14-Feb-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmcfnex.1 | ⊢ 𝑇 ∈ LinFn |
nmcfnex.2 | ⊢ 𝑇 ∈ ContFn |
Ref | Expression |
---|---|
nmcfnexi | ⊢ (normfn‘𝑇) ∈ ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmcfnex.2 | . . . 4 ⊢ 𝑇 ∈ ContFn | |
2 | ax-hv0cl 27988 | . . . 4 ⊢ 0ℎ ∈ ℋ | |
3 | 1rp 11874 | . . . 4 ⊢ 1 ∈ ℝ+ | |
4 | cnfnc 28917 | . . . 4 ⊢ ((𝑇 ∈ ContFn ∧ 0ℎ ∈ ℋ ∧ 1 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℋ ((normℎ‘(𝑧 −ℎ 0ℎ)) < 𝑦 → (abs‘((𝑇‘𝑧) − (𝑇‘0ℎ))) < 1)) | |
5 | 1, 2, 3, 4 | mp3an 1464 | . . 3 ⊢ ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℋ ((normℎ‘(𝑧 −ℎ 0ℎ)) < 𝑦 → (abs‘((𝑇‘𝑧) − (𝑇‘0ℎ))) < 1) |
6 | hvsub0 28061 | . . . . . . . 8 ⊢ (𝑧 ∈ ℋ → (𝑧 −ℎ 0ℎ) = 𝑧) | |
7 | 6 | fveq2d 6233 | . . . . . . 7 ⊢ (𝑧 ∈ ℋ → (normℎ‘(𝑧 −ℎ 0ℎ)) = (normℎ‘𝑧)) |
8 | 7 | breq1d 4695 | . . . . . 6 ⊢ (𝑧 ∈ ℋ → ((normℎ‘(𝑧 −ℎ 0ℎ)) < 𝑦 ↔ (normℎ‘𝑧) < 𝑦)) |
9 | nmcfnex.1 | . . . . . . . . . . 11 ⊢ 𝑇 ∈ LinFn | |
10 | 9 | lnfn0i 29029 | . . . . . . . . . 10 ⊢ (𝑇‘0ℎ) = 0 |
11 | 10 | oveq2i 6701 | . . . . . . . . 9 ⊢ ((𝑇‘𝑧) − (𝑇‘0ℎ)) = ((𝑇‘𝑧) − 0) |
12 | 9 | lnfnfi 29028 | . . . . . . . . . . 11 ⊢ 𝑇: ℋ⟶ℂ |
13 | 12 | ffvelrni 6398 | . . . . . . . . . 10 ⊢ (𝑧 ∈ ℋ → (𝑇‘𝑧) ∈ ℂ) |
14 | 13 | subid1d 10419 | . . . . . . . . 9 ⊢ (𝑧 ∈ ℋ → ((𝑇‘𝑧) − 0) = (𝑇‘𝑧)) |
15 | 11, 14 | syl5eq 2697 | . . . . . . . 8 ⊢ (𝑧 ∈ ℋ → ((𝑇‘𝑧) − (𝑇‘0ℎ)) = (𝑇‘𝑧)) |
16 | 15 | fveq2d 6233 | . . . . . . 7 ⊢ (𝑧 ∈ ℋ → (abs‘((𝑇‘𝑧) − (𝑇‘0ℎ))) = (abs‘(𝑇‘𝑧))) |
17 | 16 | breq1d 4695 | . . . . . 6 ⊢ (𝑧 ∈ ℋ → ((abs‘((𝑇‘𝑧) − (𝑇‘0ℎ))) < 1 ↔ (abs‘(𝑇‘𝑧)) < 1)) |
18 | 8, 17 | imbi12d 333 | . . . . 5 ⊢ (𝑧 ∈ ℋ → (((normℎ‘(𝑧 −ℎ 0ℎ)) < 𝑦 → (abs‘((𝑇‘𝑧) − (𝑇‘0ℎ))) < 1) ↔ ((normℎ‘𝑧) < 𝑦 → (abs‘(𝑇‘𝑧)) < 1))) |
19 | 18 | ralbiia 3008 | . . . 4 ⊢ (∀𝑧 ∈ ℋ ((normℎ‘(𝑧 −ℎ 0ℎ)) < 𝑦 → (abs‘((𝑇‘𝑧) − (𝑇‘0ℎ))) < 1) ↔ ∀𝑧 ∈ ℋ ((normℎ‘𝑧) < 𝑦 → (abs‘(𝑇‘𝑧)) < 1)) |
20 | 19 | rexbii 3070 | . . 3 ⊢ (∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℋ ((normℎ‘(𝑧 −ℎ 0ℎ)) < 𝑦 → (abs‘((𝑇‘𝑧) − (𝑇‘0ℎ))) < 1) ↔ ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℋ ((normℎ‘𝑧) < 𝑦 → (abs‘(𝑇‘𝑧)) < 1)) |
21 | 5, 20 | mpbi 220 | . 2 ⊢ ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℋ ((normℎ‘𝑧) < 𝑦 → (abs‘(𝑇‘𝑧)) < 1) |
22 | nmfnval 28863 | . . 3 ⊢ (𝑇: ℋ⟶ℂ → (normfn‘𝑇) = sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (abs‘(𝑇‘𝑥)))}, ℝ*, < )) | |
23 | 12, 22 | ax-mp 5 | . 2 ⊢ (normfn‘𝑇) = sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (abs‘(𝑇‘𝑥)))}, ℝ*, < ) |
24 | 12 | ffvelrni 6398 | . . 3 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℂ) |
25 | 24 | abscld 14219 | . 2 ⊢ (𝑥 ∈ ℋ → (abs‘(𝑇‘𝑥)) ∈ ℝ) |
26 | 10 | fveq2i 6232 | . . 3 ⊢ (abs‘(𝑇‘0ℎ)) = (abs‘0) |
27 | abs0 14069 | . . 3 ⊢ (abs‘0) = 0 | |
28 | 26, 27 | eqtri 2673 | . 2 ⊢ (abs‘(𝑇‘0ℎ)) = 0 |
29 | rpcn 11879 | . . . . 5 ⊢ ((𝑦 / 2) ∈ ℝ+ → (𝑦 / 2) ∈ ℂ) | |
30 | 9 | lnfnmuli 29031 | . . . . 5 ⊢ (((𝑦 / 2) ∈ ℂ ∧ 𝑥 ∈ ℋ) → (𝑇‘((𝑦 / 2) ·ℎ 𝑥)) = ((𝑦 / 2) · (𝑇‘𝑥))) |
31 | 29, 30 | sylan 487 | . . . 4 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → (𝑇‘((𝑦 / 2) ·ℎ 𝑥)) = ((𝑦 / 2) · (𝑇‘𝑥))) |
32 | 31 | fveq2d 6233 | . . 3 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → (abs‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥))) = (abs‘((𝑦 / 2) · (𝑇‘𝑥)))) |
33 | absmul 14078 | . . . 4 ⊢ (((𝑦 / 2) ∈ ℂ ∧ (𝑇‘𝑥) ∈ ℂ) → (abs‘((𝑦 / 2) · (𝑇‘𝑥))) = ((abs‘(𝑦 / 2)) · (abs‘(𝑇‘𝑥)))) | |
34 | 29, 24, 33 | syl2an 493 | . . 3 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → (abs‘((𝑦 / 2) · (𝑇‘𝑥))) = ((abs‘(𝑦 / 2)) · (abs‘(𝑇‘𝑥)))) |
35 | rpre 11877 | . . . . . 6 ⊢ ((𝑦 / 2) ∈ ℝ+ → (𝑦 / 2) ∈ ℝ) | |
36 | rpge0 11883 | . . . . . 6 ⊢ ((𝑦 / 2) ∈ ℝ+ → 0 ≤ (𝑦 / 2)) | |
37 | 35, 36 | absidd 14205 | . . . . 5 ⊢ ((𝑦 / 2) ∈ ℝ+ → (abs‘(𝑦 / 2)) = (𝑦 / 2)) |
38 | 37 | adantr 480 | . . . 4 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → (abs‘(𝑦 / 2)) = (𝑦 / 2)) |
39 | 38 | oveq1d 6705 | . . 3 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → ((abs‘(𝑦 / 2)) · (abs‘(𝑇‘𝑥))) = ((𝑦 / 2) · (abs‘(𝑇‘𝑥)))) |
40 | 32, 34, 39 | 3eqtrrd 2690 | . 2 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → ((𝑦 / 2) · (abs‘(𝑇‘𝑥))) = (abs‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥)))) |
41 | 21, 23, 25, 28, 40 | nmcexi 29013 | 1 ⊢ (normfn‘𝑇) ∈ ℝ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 {cab 2637 ∀wral 2941 ∃wrex 2942 class class class wbr 4685 ⟶wf 5922 ‘cfv 5926 (class class class)co 6690 supcsup 8387 ℂcc 9972 ℝcr 9973 0cc0 9974 1c1 9975 · cmul 9979 ℝ*cxr 10111 < clt 10112 ≤ cle 10113 − cmin 10304 / cdiv 10722 2c2 11108 ℝ+crp 11870 abscabs 14018 ℋchil 27904 ·ℎ csm 27906 normℎcno 27908 0ℎc0v 27909 −ℎ cmv 27910 normfncnmf 27936 ContFnccnfn 27938 LinFnclf 27939 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 ax-hilex 27984 ax-hv0cl 27988 ax-hvaddid 27989 ax-hfvmul 27990 ax-hvmulid 27991 ax-hvmulass 27992 ax-hvmul0 27995 ax-hfi 28064 ax-his1 28067 ax-his3 28069 ax-his4 28070 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-map 7901 df-en 7998 df-dom 7999 df-sdom 8000 df-sup 8389 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-n0 11331 df-z 11416 df-uz 11726 df-rp 11871 df-seq 12842 df-exp 12901 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-hnorm 27953 df-hvsub 27956 df-nmfn 28832 df-cnfn 28834 df-lnfn 28835 |
This theorem is referenced by: nmcfnlbi 29039 nmcfnex 29040 |
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