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Theorem normval 30881
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 7413 . . . 4 ((๐‘ฅ = ๐ด โˆง ๐‘ฅ = ๐ด) โ†’ (๐‘ฅ ยทih ๐‘ฅ) = (๐ด ยทih ๐ด))
21anidms 566 . . 3 (๐‘ฅ = ๐ด โ†’ (๐‘ฅ ยทih ๐‘ฅ) = (๐ด ยทih ๐ด))
32fveq2d 6888 . 2 (๐‘ฅ = ๐ด โ†’ (โˆšโ€˜(๐‘ฅ ยทih ๐‘ฅ)) = (โˆšโ€˜(๐ด ยทih ๐ด)))
4 dfhnorm2 30879 . 2 normโ„Ž = (๐‘ฅ โˆˆ โ„‹ โ†ฆ (โˆšโ€˜(๐‘ฅ ยทih ๐‘ฅ)))
5 fvex 6897 . 2 (โˆšโ€˜(๐ด ยทih ๐ด)) โˆˆ V
63, 4, 5fvmpt 6991 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 6536  (class class class)co 7404  โˆšcsqrt 15183   โ„‹chba 30676   ยทih csp 30679  normโ„Žcno 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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