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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 29653 | . . . 4 ⊢ 0ℎ ∈ ℋ | |
3 | 1rp 12835 | . . . 4 ⊢ 1 ∈ ℝ+ | |
4 | cnfnc 30580 | . . . 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 1460 | . . 3 ⊢ ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℋ ((normℎ‘(𝑧 −ℎ 0ℎ)) < 𝑦 → (abs‘((𝑇‘𝑧) − (𝑇‘0ℎ))) < 1) |
6 | hvsub0 29726 | . . . . . . . 8 ⊢ (𝑧 ∈ ℋ → (𝑧 −ℎ 0ℎ) = 𝑧) | |
7 | 6 | fveq2d 6829 | . . . . . . 7 ⊢ (𝑧 ∈ ℋ → (normℎ‘(𝑧 −ℎ 0ℎ)) = (normℎ‘𝑧)) |
8 | 7 | breq1d 5102 | . . . . . 6 ⊢ (𝑧 ∈ ℋ → ((normℎ‘(𝑧 −ℎ 0ℎ)) < 𝑦 ↔ (normℎ‘𝑧) < 𝑦)) |
9 | nmcfnex.1 | . . . . . . . . . . 11 ⊢ 𝑇 ∈ LinFn | |
10 | 9 | lnfn0i 30692 | . . . . . . . . . 10 ⊢ (𝑇‘0ℎ) = 0 |
11 | 10 | oveq2i 7348 | . . . . . . . . 9 ⊢ ((𝑇‘𝑧) − (𝑇‘0ℎ)) = ((𝑇‘𝑧) − 0) |
12 | 9 | lnfnfi 30691 | . . . . . . . . . . 11 ⊢ 𝑇: ℋ⟶ℂ |
13 | 12 | ffvelcdmi 7016 | . . . . . . . . . 10 ⊢ (𝑧 ∈ ℋ → (𝑇‘𝑧) ∈ ℂ) |
14 | 13 | subid1d 11422 | . . . . . . . . 9 ⊢ (𝑧 ∈ ℋ → ((𝑇‘𝑧) − 0) = (𝑇‘𝑧)) |
15 | 11, 14 | eqtrid 2788 | . . . . . . . 8 ⊢ (𝑧 ∈ ℋ → ((𝑇‘𝑧) − (𝑇‘0ℎ)) = (𝑇‘𝑧)) |
16 | 15 | fveq2d 6829 | . . . . . . 7 ⊢ (𝑧 ∈ ℋ → (abs‘((𝑇‘𝑧) − (𝑇‘0ℎ))) = (abs‘(𝑇‘𝑧))) |
17 | 16 | breq1d 5102 | . . . . . 6 ⊢ (𝑧 ∈ ℋ → ((abs‘((𝑇‘𝑧) − (𝑇‘0ℎ))) < 1 ↔ (abs‘(𝑇‘𝑧)) < 1)) |
18 | 8, 17 | imbi12d 344 | . . . . 5 ⊢ (𝑧 ∈ ℋ → (((normℎ‘(𝑧 −ℎ 0ℎ)) < 𝑦 → (abs‘((𝑇‘𝑧) − (𝑇‘0ℎ))) < 1) ↔ ((normℎ‘𝑧) < 𝑦 → (abs‘(𝑇‘𝑧)) < 1))) |
19 | 18 | ralbiia 3090 | . . . 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 3093 | . . 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 229 | . 2 ⊢ ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℋ ((normℎ‘𝑧) < 𝑦 → (abs‘(𝑇‘𝑧)) < 1) |
22 | nmfnval 30526 | . . 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 | ffvelcdmi 7016 | . . 3 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℂ) |
25 | 24 | abscld 15247 | . 2 ⊢ (𝑥 ∈ ℋ → (abs‘(𝑇‘𝑥)) ∈ ℝ) |
26 | 10 | fveq2i 6828 | . . 3 ⊢ (abs‘(𝑇‘0ℎ)) = (abs‘0) |
27 | abs0 15096 | . . 3 ⊢ (abs‘0) = 0 | |
28 | 26, 27 | eqtri 2764 | . 2 ⊢ (abs‘(𝑇‘0ℎ)) = 0 |
29 | rpcn 12841 | . . . . 5 ⊢ ((𝑦 / 2) ∈ ℝ+ → (𝑦 / 2) ∈ ℂ) | |
30 | 9 | lnfnmuli 30694 | . . . . 5 ⊢ (((𝑦 / 2) ∈ ℂ ∧ 𝑥 ∈ ℋ) → (𝑇‘((𝑦 / 2) ·ℎ 𝑥)) = ((𝑦 / 2) · (𝑇‘𝑥))) |
31 | 29, 30 | sylan 580 | . . . 4 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → (𝑇‘((𝑦 / 2) ·ℎ 𝑥)) = ((𝑦 / 2) · (𝑇‘𝑥))) |
32 | 31 | fveq2d 6829 | . . 3 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → (abs‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥))) = (abs‘((𝑦 / 2) · (𝑇‘𝑥)))) |
33 | absmul 15105 | . . . 4 ⊢ (((𝑦 / 2) ∈ ℂ ∧ (𝑇‘𝑥) ∈ ℂ) → (abs‘((𝑦 / 2) · (𝑇‘𝑥))) = ((abs‘(𝑦 / 2)) · (abs‘(𝑇‘𝑥)))) | |
34 | 29, 24, 33 | syl2an 596 | . . 3 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → (abs‘((𝑦 / 2) · (𝑇‘𝑥))) = ((abs‘(𝑦 / 2)) · (abs‘(𝑇‘𝑥)))) |
35 | rpre 12839 | . . . . . 6 ⊢ ((𝑦 / 2) ∈ ℝ+ → (𝑦 / 2) ∈ ℝ) | |
36 | rpge0 12844 | . . . . . 6 ⊢ ((𝑦 / 2) ∈ ℝ+ → 0 ≤ (𝑦 / 2)) | |
37 | 35, 36 | absidd 15233 | . . . . 5 ⊢ ((𝑦 / 2) ∈ ℝ+ → (abs‘(𝑦 / 2)) = (𝑦 / 2)) |
38 | 37 | adantr 481 | . . . 4 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → (abs‘(𝑦 / 2)) = (𝑦 / 2)) |
39 | 38 | oveq1d 7352 | . . 3 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → ((abs‘(𝑦 / 2)) · (abs‘(𝑇‘𝑥))) = ((𝑦 / 2) · (abs‘(𝑇‘𝑥)))) |
40 | 32, 34, 39 | 3eqtrrd 2781 | . 2 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → ((𝑦 / 2) · (abs‘(𝑇‘𝑥))) = (abs‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥)))) |
41 | 21, 23, 25, 28, 40 | nmcexi 30676 | 1 ⊢ (normfn‘𝑇) ∈ ℝ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1540 ∈ wcel 2105 {cab 2713 ∀wral 3061 ∃wrex 3070 class class class wbr 5092 ⟶wf 6475 ‘cfv 6479 (class class class)co 7337 supcsup 9297 ℂcc 10970 ℝcr 10971 0cc0 10972 1c1 10973 · cmul 10977 ℝ*cxr 11109 < clt 11110 ≤ cle 11111 − cmin 11306 / cdiv 11733 2c2 12129 ℝ+crp 12831 abscabs 15044 ℋchba 29569 ·ℎ csm 29571 normℎcno 29573 0ℎc0v 29574 −ℎ cmv 29575 normfncnmf 29601 ContFnccnfn 29603 LinFnclf 29604 |
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 2707 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 ax-pre-sup 11050 ax-hilex 29649 ax-hv0cl 29653 ax-hvaddid 29654 ax-hfvmul 29655 ax-hvmulid 29656 ax-hvmulass 29657 ax-hvmul0 29660 ax-hfi 29729 ax-his1 29732 ax-his3 29734 ax-his4 29735 |
This theorem depends on definitions: df-bi 206 df-an 397 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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-2nd 7900 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-er 8569 df-map 8688 df-en 8805 df-dom 8806 df-sdom 8807 df-sup 9299 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-div 11734 df-nn 12075 df-2 12137 df-3 12138 df-n0 12335 df-z 12421 df-uz 12684 df-rp 12832 df-seq 13823 df-exp 13884 df-cj 14909 df-re 14910 df-im 14911 df-sqrt 15045 df-abs 15046 df-hnorm 29618 df-hvsub 29621 df-nmfn 30495 df-cnfn 30497 df-lnfn 30498 |
This theorem is referenced by: nmcfnlbi 30702 nmcfnex 30703 |
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