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Mirrors > Home > HSE Home > Th. List > nmfnval | Structured version Visualization version GIF version |
Description: Value of the norm of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmfnval | ⊢ (𝑇: ℋ⟶ℂ → (normfn‘𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}, ℝ*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrltso 11974 | . . 3 ⊢ < Or ℝ* | |
2 | 1 | supex 8369 | . 2 ⊢ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}, ℝ*, < ) ∈ V |
3 | ax-hilex 27856 | . 2 ⊢ ℋ ∈ V | |
4 | cnex 10017 | . 2 ⊢ ℂ ∈ V | |
5 | fveq1 6190 | . . . . . . . 8 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑦) = (𝑇‘𝑦)) | |
6 | 5 | fveq2d 6195 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (abs‘(𝑡‘𝑦)) = (abs‘(𝑇‘𝑦))) |
7 | 6 | eqeq2d 2632 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (𝑥 = (abs‘(𝑡‘𝑦)) ↔ 𝑥 = (abs‘(𝑇‘𝑦)))) |
8 | 7 | anbi2d 740 | . . . . 5 ⊢ (𝑡 = 𝑇 → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑡‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦))))) |
9 | 8 | rexbidv 3052 | . . . 4 ⊢ (𝑡 = 𝑇 → (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑡‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦))))) |
10 | 9 | abbidv 2741 | . . 3 ⊢ (𝑡 = 𝑇 → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑡‘𝑦)))} = {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}) |
11 | 10 | supeq1d 8352 | . 2 ⊢ (𝑡 = 𝑇 → sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑡‘𝑦)))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}, ℝ*, < )) |
12 | df-nmfn 28704 | . 2 ⊢ normfn = (𝑡 ∈ (ℂ ↑𝑚 ℋ) ↦ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑡‘𝑦)))}, ℝ*, < )) | |
13 | 2, 3, 4, 11, 12 | fvmptmap 7894 | 1 ⊢ (𝑇: ℋ⟶ℂ → (normfn‘𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}, ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 {cab 2608 ∃wrex 2913 class class class wbr 4653 ⟶wf 5884 ‘cfv 5888 supcsup 8346 ℂcc 9934 1c1 9937 ℝ*cxr 10073 < clt 10074 ≤ cle 10075 abscabs 13974 ℋchil 27776 normℎcno 27780 normfncnmf 27808 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-hilex 27856 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-nmfn 28704 |
This theorem is referenced by: nmfnxr 28738 nmfnrepnf 28739 nmfnlb 28783 nmfnleub 28784 nmfn0 28846 nmcfnexi 28910 branmfn 28964 |
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