Theorem dfhnorm2 31212

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 31058	. 2 ⊢ normℎ = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))
2		ax-hfi 31169	. . . . . 6 ⊢ ·ih :( ℋ × ℋ)⟶ℂ
3	2	fdmi 6667	. . . . 5 ⊢ dom ·ih = ( ℋ × ℋ)
4	3	dmeqi 5847	. . . 4 ⊢ dom dom ·ih = dom ( ℋ × ℋ)
5		dmxpid 5873	. . . 4 ⊢ dom ( ℋ × ℋ) = ℋ
6	4, 5	eqtr2i 2763	. . 3 ⊢ ℋ = dom dom ·ih
7	6	mpteq1i 5164	. 2 ⊢ (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥))) = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))
8	1, 7	eqtr4i 2765	1 ⊢ normℎ = (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥)))

Syntax hints:   = wceq 1547   ↦ cmpt 5154   × cxp 5617  dom cdm 5619  ‘cfv 6486  (class class class)co 7357  ℂcc 11028  √csqrt 15187   ℋchba 31009   ·_ih csp 31012  norm_ℎcno 31013

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5219  ax-pr 5363  ax-hfi 31169

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-sn 4557  df-pr 4559  df-op 4563  df-br 5074  df-opab 5136  df-mpt 5155  df-xp 5625  df-dm 5629  df-fn 6489  df-f 6490  df-hnorm 31058

This theorem is referenced by:  normf  31213  normval  31214  hilnormi  31253