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Theorem normval 30954
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 7435 . . . 4 ((๐‘ฅ = ๐ด โˆง ๐‘ฅ = ๐ด) โ†’ (๐‘ฅ ยทih ๐‘ฅ) = (๐ด ยทih ๐ด))
21anidms 565 . . 3 (๐‘ฅ = ๐ด โ†’ (๐‘ฅ ยทih ๐‘ฅ) = (๐ด ยทih ๐ด))
32fveq2d 6906 . 2 (๐‘ฅ = ๐ด โ†’ (โˆšโ€˜(๐‘ฅ ยทih ๐‘ฅ)) = (โˆšโ€˜(๐ด ยทih ๐ด)))
4 dfhnorm2 30952 . 2 normโ„Ž = (๐‘ฅ โˆˆ โ„‹ โ†ฆ (โˆšโ€˜(๐‘ฅ ยทih ๐‘ฅ)))
5 fvex 6915 . 2 (โˆšโ€˜(๐ด ยทih ๐ด)) โˆˆ V
63, 4, 5fvmpt 7010 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 1533   โˆˆ wcel 2098  โ€˜cfv 6553  (class class class)co 7426  โˆšcsqrt 15220   โ„‹chba 30749   ยทih csp 30752  normโ„Žcno 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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