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Theorem dfhnorm2 28565
 Description: Alternate definition of the norm of a vector of Hilbert space. Definition of norm in [Beran] p. 96. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
dfhnorm2 norm = (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥)))

Proof of Theorem dfhnorm2
StepHypRef Expression
1 df-hnorm 28411 . 2 norm = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))
2 ax-hfi 28522 . . . . . 6 ·ih :( ℋ × ℋ)⟶ℂ
32fdmi 6301 . . . . 5 dom ·ih = ( ℋ × ℋ)
43dmeqi 5570 . . . 4 dom dom ·ih = dom ( ℋ × ℋ)
5 dmxpid 5590 . . . 4 dom ( ℋ × ℋ) = ℋ
64, 5eqtr2i 2802 . . 3 ℋ = dom dom ·ih
7 eqid 2777 . . 3 (√‘(𝑥 ·ih 𝑥)) = (√‘(𝑥 ·ih 𝑥))
86, 7mpteq12i 4977 . 2 (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥))) = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))
91, 8eqtr4i 2804 1 norm = (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥)))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1601   ↦ cmpt 4965   × cxp 5353  dom cdm 5355  ‘cfv 6135  (class class class)co 6922  ℂcc 10270  √csqrt 14380   ℋchba 28362   ·ih csp 28365  normℎcno 28366 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138  ax-hfi 28522 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4887  df-opab 4949  df-mpt 4966  df-xp 5361  df-dm 5365  df-fn 6138  df-f 6139  df-hnorm 28411 This theorem is referenced by:  normf  28566  normval  28567  hilnormi  28606
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