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Mirrors > Home > HSE Home > Th. List > normval | Structured version Visualization version GIF version |
Description: The value of the norm of a vector in Hilbert space. Definition of norm in [Beran] p. 96. In the literature, the norm of ๐ด is usually written as "|| ๐ด ||", but we use function value notation to take advantage of our existing theorems about functions. (Contributed by NM, 29-May-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
normval | โข (๐ด โ โ โ (normโโ๐ด) = (โโ(๐ด ยทih ๐ด))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq12 7413 | . . . 4 โข ((๐ฅ = ๐ด โง ๐ฅ = ๐ด) โ (๐ฅ ยทih ๐ฅ) = (๐ด ยทih ๐ด)) | |
2 | 1 | anidms 566 | . . 3 โข (๐ฅ = ๐ด โ (๐ฅ ยทih ๐ฅ) = (๐ด ยทih ๐ด)) |
3 | 2 | fveq2d 6888 | . 2 โข (๐ฅ = ๐ด โ (โโ(๐ฅ ยทih ๐ฅ)) = (โโ(๐ด ยทih ๐ด))) |
4 | dfhnorm2 30879 | . 2 โข normโ = (๐ฅ โ โ โฆ (โโ(๐ฅ ยทih ๐ฅ))) | |
5 | fvex 6897 | . 2 โข (โโ(๐ด ยทih ๐ด)) โ V | |
6 | 3, 4, 5 | fvmpt 6991 | 1 โข (๐ด โ โ โ (normโโ๐ด) = (โโ(๐ด ยทih ๐ด))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1533 โ wcel 2098 โcfv 6536 (class class class)co 7404 โcsqrt 15183 โchba 30676 ยทih csp 30679 normโcno 30680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-hfi 30836 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-ov 7407 df-hnorm 30725 |
This theorem is referenced by: normge0 30883 normgt0 30884 norm0 30885 normsqi 30889 norm-ii-i 30894 norm-iii-i 30896 bcsiALT 30936 |
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