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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 7367 | . . . 4 โข ((๐ฅ = ๐ด โง ๐ฅ = ๐ด) โ (๐ฅ ยทih ๐ฅ) = (๐ด ยทih ๐ด)) | |
2 | 1 | anidms 568 | . . 3 โข (๐ฅ = ๐ด โ (๐ฅ ยทih ๐ฅ) = (๐ด ยทih ๐ด)) |
3 | 2 | fveq2d 6847 | . 2 โข (๐ฅ = ๐ด โ (โโ(๐ฅ ยทih ๐ฅ)) = (โโ(๐ด ยทih ๐ด))) |
4 | dfhnorm2 30106 | . 2 โข normโ = (๐ฅ โ โ โฆ (โโ(๐ฅ ยทih ๐ฅ))) | |
5 | fvex 6856 | . 2 โข (โโ(๐ด ยทih ๐ด)) โ V | |
6 | 3, 4, 5 | fvmpt 6949 | 1 โข (๐ด โ โ โ (normโโ๐ด) = (โโ(๐ด ยทih ๐ด))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1542 โ wcel 2107 โcfv 6497 (class class class)co 7358 โcsqrt 15124 โchba 29903 ยทih csp 29906 normโcno 29907 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-hfi 30063 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-ov 7361 df-hnorm 29952 |
This theorem is referenced by: normge0 30110 normgt0 30111 norm0 30112 normsqi 30116 norm-ii-i 30121 norm-iii-i 30123 bcsiALT 30163 |
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