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Theorem dfhnorm2 27162
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 27008 . 2 norm = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))
2 ax-hfi 27119 . . . . . 6 ·ih :( ℋ × ℋ)⟶ℂ
32fdmi 5850 . . . . 5 dom ·ih = ( ℋ × ℋ)
43dmeqi 5138 . . . 4 dom dom ·ih = dom ( ℋ × ℋ)
5 dmxpid 5157 . . . 4 dom ( ℋ × ℋ) = ℋ
64, 5eqtr2i 2537 . . 3 ℋ = dom dom ·ih
7 eqid 2514 . . 3 (√‘(𝑥 ·ih 𝑥)) = (√‘(𝑥 ·ih 𝑥))
86, 7mpteq12i 4568 . 2 (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥))) = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))
91, 8eqtr4i 2539 1 norm = (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥)))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  cmpt 4541   × cxp 4930  dom cdm 4932  cfv 5689  (class class class)co 6425  cc 9687  csqrt 13675  chil 26959   ·ih csp 26962  normcno 26963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pr 4732  ax-hfi 27119
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rab 2809  df-v 3079  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-br 4482  df-opab 4542  df-mpt 4543  df-xp 4938  df-dm 4942  df-fn 5692  df-f 5693  df-hnorm 27008
This theorem is referenced by:  normf  27163  normval  27164  hilnormi  27203
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