Theorem dfhnorm2 27979

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 27825	. 2 ⊢ normℎ = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))
2		ax-hfi 27936	. . . . . 6 ⊢ ·ih :( ℋ × ℋ)⟶ℂ
3	2	fdmi 6052	. . . . 5 ⊢ dom ·ih = ( ℋ × ℋ)
4	3	dmeqi 5325	. . . 4 ⊢ dom dom ·ih = dom ( ℋ × ℋ)
5		dmxpid 5345	. . . 4 ⊢ dom ( ℋ × ℋ) = ℋ
6	4, 5	eqtr2i 2645	. . 3 ⊢ ℋ = dom dom ·ih
7		eqid 2622	. . 3 ⊢ (√‘(𝑥 ·ih 𝑥)) = (√‘(𝑥 ·ih 𝑥))
8	6, 7	mpteq12i 4742	. 2 ⊢ (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥))) = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))
9	1, 8	eqtr4i 2647	1 ⊢ normℎ = (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥)))

Syntax hints:   = wceq 1483   ↦ cmpt 4729   × cxp 5112  dom cdm 5114  ‘cfv 5888  (class class class)co 6650  ℂcc 9934  √csqrt 13973   ℋchil 27776   ·_ih csp 27779  norm_ℎcno 27780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-hfi 27936

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-mpt 4730  df-xp 5120  df-dm 5124  df-fn 5891  df-f 5892  df-hnorm 27825

This theorem is referenced by:  normf  27980  normval  27981  hilnormi  28020