Theorem dfhnorm2 28565

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	⊢ normℎ = (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥)))

Proof of Theorem dfhnorm2

Step	Hyp	Ref	Expression
1		df-hnorm 28411	. 2 ⊢ normℎ = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))
2		ax-hfi 28522	. . . . . 6 ⊢ ·ih :( ℋ × ℋ)⟶ℂ
3	2	fdmi 6301	. . . . 5 ⊢ dom ·ih = ( ℋ × ℋ)
4	3	dmeqi 5570	. . . 4 ⊢ dom dom ·ih = dom ( ℋ × ℋ)
5		dmxpid 5590	. . . 4 ⊢ dom ( ℋ × ℋ) = ℋ
6	4, 5	eqtr2i 2802	. . 3 ⊢ ℋ = dom dom ·ih
7		eqid 2777	. . 3 ⊢ (√‘(𝑥 ·ih 𝑥)) = (√‘(𝑥 ·ih 𝑥))
8	6, 7	mpteq12i 4977	. 2 ⊢ (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥))) = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))
9	1, 8	eqtr4i 2804	1 ⊢ normℎ = (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥)))

Syntax hints:   = wceq 1601   ↦ cmpt 4965   × cxp 5353  dom cdm 5355  ‘cfv 6135  (class class class)co 6922  ℂcc 10270  √csqrt 14380   ℋchba 28362   ·_ih csp 28365  norm_ℎcno 28366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138  ax-hfi 28522

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4887  df-opab 4949  df-mpt 4966  df-xp 5361  df-dm 5365  df-fn 6138  df-f 6139  df-hnorm 28411

This theorem is referenced by:  normf  28566  normval  28567  hilnormi  28606