Theorem dfhnorm2 31193

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 31039	. 2 ⊢ normℎ = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))
2		ax-hfi 31150	. . . . . 6 ⊢ ·ih :( ℋ × ℋ)⟶ℂ
3	2	fdmi 6679	. . . . 5 ⊢ dom ·ih = ( ℋ × ℋ)
4	3	dmeqi 5859	. . . 4 ⊢ dom dom ·ih = dom ( ℋ × ℋ)
5		dmxpid 5885	. . . 4 ⊢ dom ( ℋ × ℋ) = ℋ
6	4, 5	eqtr2i 2760	. . 3 ⊢ ℋ = dom dom ·ih
7	6	mpteq1i 5176	. 2 ⊢ (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥))) = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))
8	1, 7	eqtr4i 2762	1 ⊢ normℎ = (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥)))

Syntax hints:   = wceq 1542   ↦ cmpt 5166   × cxp 5629  dom cdm 5631  ‘cfv 6498  (class class class)co 7367  ℂcc 11036  √csqrt 15195   ℋchba 30990   ·_ih csp 30993  norm_ℎcno 30994

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375  ax-hfi 31150

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-mpt 5167  df-xp 5637  df-dm 5641  df-fn 6501  df-f 6502  df-hnorm 31039

This theorem is referenced by:  normf  31194  normval  31195  hilnormi  31234