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Theorem dfhnorm2 28826
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 28672 . 2 norm = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))
2 ax-hfi 28783 . . . . . 6 ·ih :( ℋ × ℋ)⟶ℂ
32fdmi 6517 . . . . 5 dom ·ih = ( ℋ × ℋ)
43dmeqi 5766 . . . 4 dom dom ·ih = dom ( ℋ × ℋ)
5 dmxpid 5793 . . . 4 dom ( ℋ × ℋ) = ℋ
64, 5eqtr2i 2842 . . 3 ℋ = dom dom ·ih
76mpteq1i 5147 . 2 (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥))) = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))
81, 7eqtr4i 2844 1 norm = (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥)))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  cmpt 5137   × cxp 5546  dom cdm 5548  cfv 6348  (class class class)co 7145  cc 10523  csqrt 14580  chba 28623   ·ih csp 28626  normcno 28627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-hfi 28783
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-mpt 5138  df-xp 5554  df-dm 5558  df-fn 6351  df-f 6352  df-hnorm 28672
This theorem is referenced by:  normf  28827  normval  28828  hilnormi  28867
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