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Mirrors > Home > MPE Home > Th. List > nmooval | Structured version Visualization version GIF version |
Description: The operator norm function. (Contributed by NM, 27-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmoofval.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
nmoofval.2 | ⊢ 𝑌 = (BaseSet‘𝑊) |
nmoofval.3 | ⊢ 𝐿 = (normCV‘𝑈) |
nmoofval.4 | ⊢ 𝑀 = (normCV‘𝑊) |
nmoofval.6 | ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) |
Ref | Expression |
---|---|
nmooval | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝑁‘𝑇) = sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑧)))}, ℝ*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmoofval.2 | . . . . 5 ⊢ 𝑌 = (BaseSet‘𝑊) | |
2 | 1 | fvexi 6907 | . . . 4 ⊢ 𝑌 ∈ V |
3 | nmoofval.1 | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
4 | 3 | fvexi 6907 | . . . 4 ⊢ 𝑋 ∈ V |
5 | 2, 4 | elmap 8892 | . . 3 ⊢ (𝑇 ∈ (𝑌 ↑m 𝑋) ↔ 𝑇:𝑋⟶𝑌) |
6 | nmoofval.3 | . . . . . 6 ⊢ 𝐿 = (normCV‘𝑈) | |
7 | nmoofval.4 | . . . . . 6 ⊢ 𝑀 = (normCV‘𝑊) | |
8 | nmoofval.6 | . . . . . 6 ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) | |
9 | 3, 1, 6, 7, 8 | nmoofval 30692 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑁 = (𝑡 ∈ (𝑌 ↑m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧)))}, ℝ*, < ))) |
10 | 9 | fveq1d 6895 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑁‘𝑇) = ((𝑡 ∈ (𝑌 ↑m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧)))}, ℝ*, < ))‘𝑇)) |
11 | fveq1 6892 | . . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑧) = (𝑇‘𝑧)) | |
12 | 11 | fveq2d 6897 | . . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (𝑀‘(𝑡‘𝑧)) = (𝑀‘(𝑇‘𝑧))) |
13 | 12 | eqeq2d 2737 | . . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (𝑥 = (𝑀‘(𝑡‘𝑧)) ↔ 𝑥 = (𝑀‘(𝑇‘𝑧)))) |
14 | 13 | anbi2d 628 | . . . . . . . 8 ⊢ (𝑡 = 𝑇 → (((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧))) ↔ ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑧))))) |
15 | 14 | rexbidv 3169 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧))) ↔ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑧))))) |
16 | 15 | abbidv 2795 | . . . . . 6 ⊢ (𝑡 = 𝑇 → {𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧)))} = {𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑧)))}) |
17 | 16 | supeq1d 9482 | . . . . 5 ⊢ (𝑡 = 𝑇 → sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧)))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑧)))}, ℝ*, < )) |
18 | eqid 2726 | . . . . 5 ⊢ (𝑡 ∈ (𝑌 ↑m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧)))}, ℝ*, < )) = (𝑡 ∈ (𝑌 ↑m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧)))}, ℝ*, < )) | |
19 | xrltso 13168 | . . . . . 6 ⊢ < Or ℝ* | |
20 | 19 | supex 9499 | . . . . 5 ⊢ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑧)))}, ℝ*, < ) ∈ V |
21 | 17, 18, 20 | fvmpt 7001 | . . . 4 ⊢ (𝑇 ∈ (𝑌 ↑m 𝑋) → ((𝑡 ∈ (𝑌 ↑m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧)))}, ℝ*, < ))‘𝑇) = sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑧)))}, ℝ*, < )) |
22 | 10, 21 | sylan9eq 2786 | . . 3 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑇 ∈ (𝑌 ↑m 𝑋)) → (𝑁‘𝑇) = sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑧)))}, ℝ*, < )) |
23 | 5, 22 | sylan2br 593 | . 2 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑇:𝑋⟶𝑌) → (𝑁‘𝑇) = sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑧)))}, ℝ*, < )) |
24 | 23 | 3impa 1107 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝑁‘𝑇) = sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑧)))}, ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 {cab 2703 ∃wrex 3060 class class class wbr 5145 ↦ cmpt 5228 ⟶wf 6542 ‘cfv 6546 (class class class)co 7416 ↑m cmap 8847 supcsup 9476 1c1 11150 ℝ*cxr 11288 < clt 11289 ≤ cle 11290 NrmCVeccnv 30514 BaseSetcba 30516 normCVcnmcv 30520 normOpOLD cnmoo 30671 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-pre-lttri 11223 ax-pre-lttrn 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-po 5586 df-so 5587 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-ov 7419 df-oprab 7420 df-mpo 7421 df-er 8726 df-map 8849 df-en 8967 df-dom 8968 df-sdom 8969 df-sup 9478 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-nmoo 30675 |
This theorem is referenced by: nmoxr 30696 nmooge0 30697 nmorepnf 30698 nmoolb 30701 nmoubi 30702 nmoo0 30721 nmlno0lem 30723 |
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