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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 30934 | . . . 4 ⊢ 0ℎ ∈ ℋ | |
| 3 | 1rp 12885 | . . . 4 ⊢ 1 ∈ ℝ+ | |
| 4 | cnfnc 31861 | . . . 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 1463 | . . 3 ⊢ ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℋ ((normℎ‘(𝑧 −ℎ 0ℎ)) < 𝑦 → (abs‘((𝑇‘𝑧) − (𝑇‘0ℎ))) < 1) |
| 6 | hvsub0 31007 | . . . . . . . 8 ⊢ (𝑧 ∈ ℋ → (𝑧 −ℎ 0ℎ) = 𝑧) | |
| 7 | 6 | fveq2d 6820 | . . . . . . 7 ⊢ (𝑧 ∈ ℋ → (normℎ‘(𝑧 −ℎ 0ℎ)) = (normℎ‘𝑧)) |
| 8 | 7 | breq1d 5098 | . . . . . 6 ⊢ (𝑧 ∈ ℋ → ((normℎ‘(𝑧 −ℎ 0ℎ)) < 𝑦 ↔ (normℎ‘𝑧) < 𝑦)) |
| 9 | nmcfnex.1 | . . . . . . . . . . 11 ⊢ 𝑇 ∈ LinFn | |
| 10 | 9 | lnfn0i 31973 | . . . . . . . . . 10 ⊢ (𝑇‘0ℎ) = 0 |
| 11 | 10 | oveq2i 7351 | . . . . . . . . 9 ⊢ ((𝑇‘𝑧) − (𝑇‘0ℎ)) = ((𝑇‘𝑧) − 0) |
| 12 | 9 | lnfnfi 31972 | . . . . . . . . . . 11 ⊢ 𝑇: ℋ⟶ℂ |
| 13 | 12 | ffvelcdmi 7010 | . . . . . . . . . 10 ⊢ (𝑧 ∈ ℋ → (𝑇‘𝑧) ∈ ℂ) |
| 14 | 13 | subid1d 11452 | . . . . . . . . 9 ⊢ (𝑧 ∈ ℋ → ((𝑇‘𝑧) − 0) = (𝑇‘𝑧)) |
| 15 | 11, 14 | eqtrid 2776 | . . . . . . . 8 ⊢ (𝑧 ∈ ℋ → ((𝑇‘𝑧) − (𝑇‘0ℎ)) = (𝑇‘𝑧)) |
| 16 | 15 | fveq2d 6820 | . . . . . . 7 ⊢ (𝑧 ∈ ℋ → (abs‘((𝑇‘𝑧) − (𝑇‘0ℎ))) = (abs‘(𝑇‘𝑧))) |
| 17 | 16 | breq1d 5098 | . . . . . 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 3073 | . . . 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 3076 | . . 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 230 | . 2 ⊢ ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℋ ((normℎ‘𝑧) < 𝑦 → (abs‘(𝑇‘𝑧)) < 1) |
| 22 | nmfnval 31807 | . . 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 7010 | . . 3 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℂ) |
| 25 | 24 | abscld 15333 | . 2 ⊢ (𝑥 ∈ ℋ → (abs‘(𝑇‘𝑥)) ∈ ℝ) |
| 26 | 10 | fveq2i 6819 | . . 3 ⊢ (abs‘(𝑇‘0ℎ)) = (abs‘0) |
| 27 | abs0 15179 | . . 3 ⊢ (abs‘0) = 0 | |
| 28 | 26, 27 | eqtri 2752 | . 2 ⊢ (abs‘(𝑇‘0ℎ)) = 0 |
| 29 | rpcn 12892 | . . . . 5 ⊢ ((𝑦 / 2) ∈ ℝ+ → (𝑦 / 2) ∈ ℂ) | |
| 30 | 9 | lnfnmuli 31975 | . . . . 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 6820 | . . 3 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → (abs‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥))) = (abs‘((𝑦 / 2) · (𝑇‘𝑥)))) |
| 33 | absmul 15188 | . . . 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 12890 | . . . . . 6 ⊢ ((𝑦 / 2) ∈ ℝ+ → (𝑦 / 2) ∈ ℝ) | |
| 36 | rpge0 12895 | . . . . . 6 ⊢ ((𝑦 / 2) ∈ ℝ+ → 0 ≤ (𝑦 / 2)) | |
| 37 | 35, 36 | absidd 15317 | . . . . 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 7355 | . . 3 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → ((abs‘(𝑦 / 2)) · (abs‘(𝑇‘𝑥))) = ((𝑦 / 2) · (abs‘(𝑇‘𝑥)))) |
| 40 | 32, 34, 39 | 3eqtrrd 2769 | . 2 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → ((𝑦 / 2) · (abs‘(𝑇‘𝑥))) = (abs‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥)))) |
| 41 | 21, 23, 25, 28, 40 | nmcexi 31957 | 1 ⊢ (normfn‘𝑇) ∈ ℝ |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 {cab 2707 ∀wral 3044 ∃wrex 3053 class class class wbr 5088 ⟶wf 6472 ‘cfv 6476 (class class class)co 7340 supcsup 9318 ℂcc 10995 ℝcr 10996 0cc0 10997 1c1 10998 · cmul 11002 ℝ*cxr 11136 < clt 11137 ≤ cle 11138 − cmin 11335 / cdiv 11765 2c2 12171 ℝ+crp 12881 abscabs 15128 ℋchba 30850 ·ℎ csm 30852 normℎcno 30854 0ℎc0v 30855 −ℎ cmv 30856 normfncnmf 30882 ContFnccnfn 30884 LinFnclf 30885 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5367 ax-un 7662 ax-cnex 11053 ax-resscn 11054 ax-1cn 11055 ax-icn 11056 ax-addcl 11057 ax-addrcl 11058 ax-mulcl 11059 ax-mulrcl 11060 ax-mulcom 11061 ax-addass 11062 ax-mulass 11063 ax-distr 11064 ax-i2m1 11065 ax-1ne0 11066 ax-1rid 11067 ax-rnegex 11068 ax-rrecex 11069 ax-cnre 11070 ax-pre-lttri 11071 ax-pre-lttrn 11072 ax-pre-ltadd 11073 ax-pre-mulgt0 11074 ax-pre-sup 11075 ax-hilex 30930 ax-hv0cl 30934 ax-hvaddid 30935 ax-hfvmul 30936 ax-hvmulid 30937 ax-hvmulass 30938 ax-hvmul0 30941 ax-hfi 31010 ax-his1 31013 ax-his3 31015 ax-his4 31016 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3393 df-v 3435 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-iun 4940 df-br 5089 df-opab 5151 df-mpt 5170 df-tr 5196 df-id 5508 df-eprel 5513 df-po 5521 df-so 5522 df-fr 5566 df-we 5568 df-xp 5619 df-rel 5620 df-cnv 5621 df-co 5622 df-dm 5623 df-rn 5624 df-res 5625 df-ima 5626 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-riota 7297 df-ov 7343 df-oprab 7344 df-mpo 7345 df-om 7791 df-2nd 7916 df-frecs 8205 df-wrecs 8236 df-recs 8285 df-rdg 8323 df-er 8616 df-map 8746 df-en 8864 df-dom 8865 df-sdom 8866 df-sup 9320 df-pnf 11139 df-mnf 11140 df-xr 11141 df-ltxr 11142 df-le 11143 df-sub 11337 df-neg 11338 df-div 11766 df-nn 12117 df-2 12179 df-3 12180 df-n0 12373 df-z 12460 df-uz 12724 df-rp 12882 df-seq 13897 df-exp 13957 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-hnorm 30899 df-hvsub 30902 df-nmfn 31776 df-cnfn 31778 df-lnfn 31779 |
| This theorem is referenced by: nmcfnlbi 31983 nmcfnex 31984 |
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