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Theorem dfhnorm2 28107
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 27953 . 2 norm = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))
2 ax-hfi 28064 . . . . . 6 ·ih :( ℋ × ℋ)⟶ℂ
32fdmi 6090 . . . . 5 dom ·ih = ( ℋ × ℋ)
43dmeqi 5357 . . . 4 dom dom ·ih = dom ( ℋ × ℋ)
5 dmxpid 5377 . . . 4 dom ( ℋ × ℋ) = ℋ
64, 5eqtr2i 2674 . . 3 ℋ = dom dom ·ih
7 eqid 2651 . . 3 (√‘(𝑥 ·ih 𝑥)) = (√‘(𝑥 ·ih 𝑥))
86, 7mpteq12i 4775 . 2 (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥))) = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))
91, 8eqtr4i 2676 1 norm = (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥)))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  cmpt 4762   × cxp 5141  dom cdm 5143  cfv 5926  (class class class)co 6690  cc 9972  csqrt 14017  chil 27904   ·ih csp 27907  normcno 27908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-hfi 28064
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-mpt 4763  df-xp 5149  df-dm 5153  df-fn 5929  df-f 5930  df-hnorm 27953
This theorem is referenced by:  normf  28108  normval  28109  hilnormi  28148
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