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Theorem normval 30108
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 ๐ด is usually written as "|| ๐ด ||", 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.)
Assertion
Ref Expression
normval (๐ด โˆˆ โ„‹ โ†’ (normโ„Žโ€˜๐ด) = (โˆšโ€˜(๐ด ยทih ๐ด)))

Proof of Theorem normval
Dummy variable ๐‘ฅ is distinct from all other variables.
StepHypRef Expression
1 oveq12 7367 . . . 4 ((๐‘ฅ = ๐ด โˆง ๐‘ฅ = ๐ด) โ†’ (๐‘ฅ ยทih ๐‘ฅ) = (๐ด ยทih ๐ด))
21anidms 568 . . 3 (๐‘ฅ = ๐ด โ†’ (๐‘ฅ ยทih ๐‘ฅ) = (๐ด ยทih ๐ด))
32fveq2d 6847 . 2 (๐‘ฅ = ๐ด โ†’ (โˆšโ€˜(๐‘ฅ ยทih ๐‘ฅ)) = (โˆšโ€˜(๐ด ยทih ๐ด)))
4 dfhnorm2 30106 . 2 normโ„Ž = (๐‘ฅ โˆˆ โ„‹ โ†ฆ (โˆšโ€˜(๐‘ฅ ยทih ๐‘ฅ)))
5 fvex 6856 . 2 (โˆšโ€˜(๐ด ยทih ๐ด)) โˆˆ V
63, 4, 5fvmpt 6949 1 (๐ด โˆˆ โ„‹ โ†’ (normโ„Žโ€˜๐ด) = (โˆšโ€˜(๐ด ยทih ๐ด)))
Colors of variables: wff setvar class
Syntax hints:   โ†’ wi 4   = wceq 1542   โˆˆ wcel 2107  โ€˜cfv 6497  (class class class)co 7358  โˆšcsqrt 15124   โ„‹chba 29903   ยทih csp 29906  normโ„Žcno 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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