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| Mirrors > Home > MPE Home > Th. List > nmval | Structured version Visualization version GIF version | ||
| Description: The value of the norm as the distance to zero. Problem 1 of [Kreyszig] p. 63. (Contributed by NM, 4-Dec-2006.) (Revised by Mario Carneiro, 2-Oct-2015.) |
| Ref | Expression |
|---|---|
| nmfval.n | ⊢ 𝑁 = (norm‘𝑊) |
| nmfval.x | ⊢ 𝑋 = (Base‘𝑊) |
| nmfval.z | ⊢ 0 = (0g‘𝑊) |
| nmfval.d | ⊢ 𝐷 = (dist‘𝑊) |
| Ref | Expression |
|---|---|
| nmval | ⊢ (𝐴 ∈ 𝑋 → (𝑁‘𝐴) = (𝐴𝐷 0 )) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | oveq1 7438 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥𝐷 0 ) = (𝐴𝐷 0 )) | |
| 2 | nmfval.n | . . 3 ⊢ 𝑁 = (norm‘𝑊) | |
| 3 | nmfval.x | . . 3 ⊢ 𝑋 = (Base‘𝑊) | |
| 4 | nmfval.z | . . 3 ⊢ 0 = (0g‘𝑊) | |
| 5 | nmfval.d | . . 3 ⊢ 𝐷 = (dist‘𝑊) | |
| 6 | 2, 3, 4, 5 | nmfval 24601 | . 2 ⊢ 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )) |
| 7 | ovex 7464 | . 2 ⊢ (𝐴𝐷 0 ) ∈ V | |
| 8 | 1, 6, 7 | fvmpt 7016 | 1 ⊢ (𝐴 ∈ 𝑋 → (𝑁‘𝐴) = (𝐴𝐷 0 )) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 distcds 17306 0gc0g 17484 normcnm 24589 |
| 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-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 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-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 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-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 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-fv 6569 df-ov 7434 df-nm 24595 |
| This theorem is referenced by: nmval2 24605 ngpds2 24619 isngp4 24625 nmge0 24630 nmeq0 24631 nminv 24634 nmmtri 24635 nmrtri 24637 |
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