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Theorem dfhnorm2 30302
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 30148 . 2 norm = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))
2 ax-hfi 30259 . . . . . 6 ·ih :( ℋ × ℋ)⟶ℂ
32fdmi 6717 . . . . 5 dom ·ih = ( ℋ × ℋ)
43dmeqi 5897 . . . 4 dom dom ·ih = dom ( ℋ × ℋ)
5 dmxpid 5922 . . . 4 dom ( ℋ × ℋ) = ℋ
64, 5eqtr2i 2761 . . 3 ℋ = dom dom ·ih
76mpteq1i 5238 . 2 (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥))) = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))
81, 7eqtr4i 2763 1 norm = (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥)))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  cmpt 5225   × cxp 5668  dom cdm 5670  cfv 6533  (class class class)co 7394  cc 11092  csqrt 15164  chba 30099   ·ih csp 30102  normcno 30103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5293  ax-nul 5300  ax-pr 5421  ax-hfi 30259
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rab 3433  df-v 3476  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5143  df-opab 5205  df-mpt 5226  df-xp 5676  df-dm 5680  df-fn 6536  df-f 6537  df-hnorm 30148
This theorem is referenced by:  normf  30303  normval  30304  hilnormi  30343
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