normval - Hilbert Space Explorer

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Theorem normval 30954

Description: The value of the norm of a vector in Hilbert space. Definition of norm in [Beran] p. 96. In the literature, the norm of 𝐴 is usually written as "|| 𝐴 ||", but we use function value notation to take advantage of our existing theorems about functions. (Contributed by NM, 29-May-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

**Assertion**
Ref	Expression
normval	⊢ (𝐴 ∈ ℋ → (norm_ℎ‘𝐴) = (√‘(𝐴 ·_ih 𝐴)))

Proof of Theorem normval Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq12 7435	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑥 ·_ih 𝑥) = (𝐴 ·_ih 𝐴))
2	1	anidms 565	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ·_ih 𝑥) = (𝐴 ·_ih 𝐴))
3	2	fveq2d 6906	. 2 ⊢ (𝑥 = 𝐴 → (√‘(𝑥 ·_ih 𝑥)) = (√‘(𝐴 ·_ih 𝐴)))
4		dfhnorm2 30952	. 2 ⊢ norm_ℎ = (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·_ih 𝑥)))
5		fvex 6915	. 2 ⊢ (√‘(𝐴 ·_ih 𝐴)) ∈ V
6	3, 4, 5	fvmpt 7010	1 ⊢ (𝐴 ∈ ℋ → (norm_ℎ‘𝐴) = (√‘(𝐴 ·_ih 𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ‘cfv 6553 (class class class)co 7426 √csqrt 15220 ℋchba 30749 ·_ih csp 30752 norm_ℎcno 30753

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-hfi 30909

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-ov 7429 df-hnorm 30798

This theorem is referenced by: normge0 30956 normgt0 30957 norm0 30958 normsqi 30962 norm-ii-i 30967 norm-iii-i 30969 bcsiALT 31009

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