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Theorem normval 31144
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 7441 . . . 4 ((𝑥 = 𝐴𝑥 = 𝐴) → (𝑥 ·ih 𝑥) = (𝐴 ·ih 𝐴))
21anidms 566 . . 3 (𝑥 = 𝐴 → (𝑥 ·ih 𝑥) = (𝐴 ·ih 𝐴))
32fveq2d 6909 . 2 (𝑥 = 𝐴 → (√‘(𝑥 ·ih 𝑥)) = (√‘(𝐴 ·ih 𝐴)))
4 dfhnorm2 31142 . 2 norm = (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥)))
5 fvex 6918 . 2 (√‘(𝐴 ·ih 𝐴)) ∈ V
63, 4, 5fvmpt 7015 1 (𝐴 ∈ ℋ → (norm𝐴) = (√‘(𝐴 ·ih 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  cfv 6560  (class class class)co 7432  csqrt 15273  chba 30939   ·ih csp 30942  normcno 30943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-hfi 31099
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-ov 7435  df-hnorm 30988
This theorem is referenced by:  normge0  31146  normgt0  31147  norm0  31148  normsqi  31152  norm-ii-i  31157  norm-iii-i  31159  bcsiALT  31199
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