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Theorem normval 31006
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 𝐴 is usually written as "|| 𝐴 ||", 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 (𝐴 ∈ ℋ → (norm𝐴) = (√‘(𝐴 ·ih 𝐴)))

Proof of Theorem normval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 oveq12 7428 . . . 4 ((𝑥 = 𝐴𝑥 = 𝐴) → (𝑥 ·ih 𝑥) = (𝐴 ·ih 𝐴))
21anidms 565 . . 3 (𝑥 = 𝐴 → (𝑥 ·ih 𝑥) = (𝐴 ·ih 𝐴))
32fveq2d 6900 . 2 (𝑥 = 𝐴 → (√‘(𝑥 ·ih 𝑥)) = (√‘(𝐴 ·ih 𝐴)))
4 dfhnorm2 31004 . 2 norm = (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥)))
5 fvex 6909 . 2 (√‘(𝐴 ·ih 𝐴)) ∈ V
63, 4, 5fvmpt 7004 1 (𝐴 ∈ ℋ → (norm𝐴) = (√‘(𝐴 ·ih 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cfv 6549  (class class class)co 7419  csqrt 15216  chba 30801   ·ih csp 30804  normcno 30805
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 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-hfi 30961
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 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557  df-ov 7422  df-hnorm 30850
This theorem is referenced by:  normge0  31008  normgt0  31009  norm0  31010  normsqi  31014  norm-ii-i  31019  norm-iii-i  31021  bcsiALT  31061
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