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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 7435 | . . . 4 โข ((๐ฅ = ๐ด โง ๐ฅ = ๐ด) โ (๐ฅ ยทih ๐ฅ) = (๐ด ยทih ๐ด)) | |
2 | 1 | anidms 565 | . . 3 โข (๐ฅ = ๐ด โ (๐ฅ ยทih ๐ฅ) = (๐ด ยทih ๐ด)) |
3 | 2 | fveq2d 6906 | . 2 โข (๐ฅ = ๐ด โ (โโ(๐ฅ ยทih ๐ฅ)) = (โโ(๐ด ยทih ๐ด))) |
4 | dfhnorm2 30952 | . 2 โข normโ = (๐ฅ โ โ โฆ (โโ(๐ฅ ยทih ๐ฅ))) | |
5 | fvex 6915 | . 2 โข (โโ(๐ด ยทih ๐ด)) โ V | |
6 | 3, 4, 5 | fvmpt 7010 | 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 6553 (class class class)co 7426 โcsqrt 15220 โchba 30749 ยทih csp 30752 normโcno 30753 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-hfi 30909 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-ov 7429 df-hnorm 30798 |
This theorem is referenced by: normge0 30956 normgt0 30957 norm0 30958 normsqi 30962 norm-ii-i 30967 norm-iii-i 30969 bcsiALT 31009 |
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