Theorem dfhnorm2 31101

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 30947	. 2 ⊢ normℎ = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))
2		ax-hfi 31058	. . . . . 6 ⊢ ·ih :( ℋ × ℋ)⟶ℂ
3	2	fdmi 6681	. . . . 5 ⊢ dom ·ih = ( ℋ × ℋ)
4	3	dmeqi 5858	. . . 4 ⊢ dom dom ·ih = dom ( ℋ × ℋ)
5		dmxpid 5883	. . . 4 ⊢ dom ( ℋ × ℋ) = ℋ
6	4, 5	eqtr2i 2753	. . 3 ⊢ ℋ = dom dom ·ih
7	6	mpteq1i 5193	. 2 ⊢ (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥))) = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))
8	1, 7	eqtr4i 2755	1 ⊢ normℎ = (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥)))

Syntax hints:   = wceq 1540   ↦ cmpt 5183   × cxp 5629  dom cdm 5631  ‘cfv 6499  (class class class)co 7369  ℂcc 11042  √csqrt 15175   ℋchba 30898   ·_ih csp 30901  norm_ℎcno 30902

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-hfi 31058

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-mpt 5184  df-xp 5637  df-dm 5641  df-fn 6502  df-f 6503  df-hnorm 30947

This theorem is referenced by:  normf  31102  normval  31103  hilnormi  31142