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Theorem normval 27171
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 6536 . . . 4 ((𝑥 = 𝐴𝑥 = 𝐴) → (𝑥 ·ih 𝑥) = (𝐴 ·ih 𝐴))
21anidms 674 . . 3 (𝑥 = 𝐴 → (𝑥 ·ih 𝑥) = (𝐴 ·ih 𝐴))
32fveq2d 6092 . 2 (𝑥 = 𝐴 → (√‘(𝑥 ·ih 𝑥)) = (√‘(𝐴 ·ih 𝐴)))
4 dfhnorm2 27169 . 2 norm = (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥)))
5 fvex 6098 . 2 (√‘(𝐴 ·ih 𝐴)) ∈ V
63, 4, 5fvmpt 6176 1 (𝐴 ∈ ℋ → (norm𝐴) = (√‘(𝐴 ·ih 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  cfv 5790  (class class class)co 6527  csqrt 13767  chil 26966   ·ih csp 26969  normcno 26970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828  ax-hfi 27126
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-fv 5798  df-ov 6530  df-hnorm 27015
This theorem is referenced by:  normge0  27173  normgt0  27174  norm0  27175  normsqi  27179  norm-ii-i  27184  norm-iii-i  27186  bcsiALT  27226
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