HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  dfhnorm2 Structured version   Visualization version   GIF version

Theorem dfhnorm2 31141
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 30987 . 2 norm = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))
2 ax-hfi 31098 . . . . . 6 ·ih :( ℋ × ℋ)⟶ℂ
32fdmi 6747 . . . . 5 dom ·ih = ( ℋ × ℋ)
43dmeqi 5915 . . . 4 dom dom ·ih = dom ( ℋ × ℋ)
5 dmxpid 5941 . . . 4 dom ( ℋ × ℋ) = ℋ
64, 5eqtr2i 2766 . . 3 ℋ = dom dom ·ih
76mpteq1i 5238 . 2 (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥))) = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))
81, 7eqtr4i 2768 1 norm = (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥)))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  cmpt 5225   × cxp 5683  dom cdm 5685  cfv 6561  (class class class)co 7431  cc 11153  csqrt 15272  chba 30938   ·ih csp 30941  normcno 30942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-hfi 31098
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-mpt 5226  df-xp 5691  df-dm 5695  df-fn 6564  df-f 6565  df-hnorm 30987
This theorem is referenced by:  normf  31142  normval  31143  hilnormi  31182
  Copyright terms: Public domain W3C validator