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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 6920 | . . . 4 ⊢ 𝑌 ∈ V |
| 3 | nmoofval.1 | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
| 4 | 3 | fvexi 6920 | . . . 4 ⊢ 𝑋 ∈ V |
| 5 | 2, 4 | elmap 8911 | . . 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 30781 | . . . . 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 6908 | . . . 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 6905 | . . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑧) = (𝑇‘𝑧)) | |
| 12 | 11 | fveq2d 6910 | . . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (𝑀‘(𝑡‘𝑧)) = (𝑀‘(𝑇‘𝑧))) |
| 13 | 12 | eqeq2d 2748 | . . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (𝑥 = (𝑀‘(𝑡‘𝑧)) ↔ 𝑥 = (𝑀‘(𝑇‘𝑧)))) |
| 14 | 13 | anbi2d 630 | . . . . . . . 8 ⊢ (𝑡 = 𝑇 → (((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧))) ↔ ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑧))))) |
| 15 | 14 | rexbidv 3179 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧))) ↔ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑧))))) |
| 16 | 15 | abbidv 2808 | . . . . . 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 9486 | . . . . 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 2737 | . . . . 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 13183 | . . . . . 6 ⊢ < Or ℝ* | |
| 20 | 19 | supex 9503 | . . . . 5 ⊢ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑧)))}, ℝ*, < ) ∈ V |
| 21 | 17, 18, 20 | fvmpt 7016 | . . . 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 2797 | . . 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 595 | . 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 1110 | 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 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 {cab 2714 ∃wrex 3070 class class class wbr 5143 ↦ cmpt 5225 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 supcsup 9480 1c1 11156 ℝ*cxr 11294 < clt 11295 ≤ cle 11296 NrmCVeccnv 30603 BaseSetcba 30605 normCVcnmcv 30609 normOpOLD cnmoo 30760 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-nmoo 30764 |
| This theorem is referenced by: nmoxr 30785 nmooge0 30786 nmorepnf 30787 nmoolb 30790 nmoubi 30791 nmoo0 30810 nmlno0lem 30812 |
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