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Mirrors > Home > MPE Home > Th. List > nmval | Structured version Visualization version GIF version |
Description: The value of the norm function. 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 6917 | . 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 22770 | . 2 ⊢ 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )) |
7 | ovex 6942 | . 2 ⊢ (𝐴𝐷 0 ) ∈ V | |
8 | 1, 6, 7 | fvmpt 6533 | 1 ⊢ (𝐴 ∈ 𝑋 → (𝑁‘𝐴) = (𝐴𝐷 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 ‘cfv 6127 (class class class)co 6910 Basecbs 16229 distcds 16321 0gc0g 16460 normcnm 22758 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-fv 6135 df-ov 6913 df-nm 22764 |
This theorem is referenced by: nmval2 22773 ngpds2 22787 isngp4 22793 nmge0 22798 nmeq0 22799 nminv 22802 nmmtri 22803 nmrtri 22805 |
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