![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 7437 | . 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 24616 | . 2 ⊢ 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )) |
7 | ovex 7463 | . 2 ⊢ (𝐴𝐷 0 ) ∈ V | |
8 | 1, 6, 7 | fvmpt 7015 | 1 ⊢ (𝐴 ∈ 𝑋 → (𝑁‘𝐴) = (𝐴𝐷 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 distcds 17306 0gc0g 17485 normcnm 24604 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-fv 6570 df-ov 7433 df-nm 24610 |
This theorem is referenced by: nmval2 24620 ngpds2 24634 isngp4 24640 nmge0 24645 nmeq0 24646 nminv 24649 nmmtri 24650 nmrtri 24652 |
Copyright terms: Public domain | W3C validator |