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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 13180 | . . 3 ⊢ < Or ℝ* | |
2 | 1 | supex 9501 | . 2 ⊢ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}, ℝ*, < ) ∈ V |
3 | ax-hilex 31028 | . 2 ⊢ ℋ ∈ V | |
4 | cnex 11234 | . 2 ⊢ ℂ ∈ V | |
5 | fveq1 6906 | . . . . . . . 8 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑦) = (𝑇‘𝑦)) | |
6 | 5 | fveq2d 6911 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (abs‘(𝑡‘𝑦)) = (abs‘(𝑇‘𝑦))) |
7 | 6 | eqeq2d 2746 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (𝑥 = (abs‘(𝑡‘𝑦)) ↔ 𝑥 = (abs‘(𝑇‘𝑦)))) |
8 | 7 | anbi2d 630 | . . . . 5 ⊢ (𝑡 = 𝑇 → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑡‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦))))) |
9 | 8 | rexbidv 3177 | . . . 4 ⊢ (𝑡 = 𝑇 → (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑡‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦))))) |
10 | 9 | abbidv 2806 | . . 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 9484 | . 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 31874 | . 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 8920 | 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 395 = wceq 1537 {cab 2712 ∃wrex 3068 class class class wbr 5148 ⟶wf 6559 ‘cfv 6563 supcsup 9478 ℂcc 11151 1c1 11154 ℝ*cxr 11292 < clt 11293 ≤ cle 11294 abscabs 15270 ℋchba 30948 normℎcno 30952 normfncnmf 30980 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-hilex 31028 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-nmfn 31874 |
This theorem is referenced by: nmfnxr 31908 nmfnrepnf 31909 nmfnlb 31953 nmfnleub 31954 nmfn0 32016 nmcfnexi 32080 branmfn 32134 |
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