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Theorem dfhnorm2 28908
 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 28754 . 2 norm = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))
2 ax-hfi 28865 . . . . . 6 ·ih :( ℋ × ℋ)⟶ℂ
32fdmi 6502 . . . . 5 dom ·ih = ( ℋ × ℋ)
43dmeqi 5741 . . . 4 dom dom ·ih = dom ( ℋ × ℋ)
5 dmxpid 5768 . . . 4 dom ( ℋ × ℋ) = ℋ
64, 5eqtr2i 2825 . . 3 ℋ = dom dom ·ih
76mpteq1i 5123 . 2 (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥))) = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))
81, 7eqtr4i 2827 1 norm = (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥)))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ↦ cmpt 5113   × cxp 5521  dom cdm 5523  ‘cfv 6328  (class class class)co 7139  ℂcc 10528  √csqrt 14587   ℋchba 28705   ·ih csp 28708  normℎcno 28709 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-hfi 28865 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-mpt 5114  df-xp 5529  df-dm 5533  df-fn 6331  df-f 6332  df-hnorm 28754 This theorem is referenced by:  normf  28909  normval  28910  hilnormi  28949
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