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Theorem dfhnorm2 28319
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 28165 . 2 norm = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))
2 ax-hfi 28276 . . . . . 6 ·ih :( ℋ × ℋ)⟶ℂ
32fdmi 6194 . . . . 5 dom ·ih = ( ℋ × ℋ)
43dmeqi 5462 . . . 4 dom dom ·ih = dom ( ℋ × ℋ)
5 dmxpid 5482 . . . 4 dom ( ℋ × ℋ) = ℋ
64, 5eqtr2i 2794 . . 3 ℋ = dom dom ·ih
7 eqid 2771 . . 3 (√‘(𝑥 ·ih 𝑥)) = (√‘(𝑥 ·ih 𝑥))
86, 7mpteq12i 4877 . 2 (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥))) = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))
91, 8eqtr4i 2796 1 norm = (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥)))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  cmpt 4864   × cxp 5248  dom cdm 5250  cfv 6030  (class class class)co 6796  cc 10140  csqrt 14181  chil 28116   ·ih csp 28119  normcno 28120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035  ax-hfi 28276
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-mpt 4865  df-xp 5256  df-dm 5260  df-fn 6033  df-f 6034  df-hnorm 28165
This theorem is referenced by:  normf  28320  normval  28321  hilnormi  28360
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