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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 7278 | . 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 23742 | . 2 ⊢ 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )) |
7 | ovex 7304 | . 2 ⊢ (𝐴𝐷 0 ) ∈ V | |
8 | 1, 6, 7 | fvmpt 6872 | 1 ⊢ (𝐴 ∈ 𝑋 → (𝑁‘𝐴) = (𝐴𝐷 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 ‘cfv 6432 (class class class)co 7271 Basecbs 16910 distcds 16969 0gc0g 17148 normcnm 23730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-fv 6440 df-ov 7274 df-nm 23736 |
This theorem is referenced by: nmval2 23746 ngpds2 23760 isngp4 23766 nmge0 23771 nmeq0 23772 nminv 23775 nmmtri 23776 nmrtri 23778 |
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