normval - Hilbert Space Explorer

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Theorem normval 31102

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 7355	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑥 ·_ih 𝑥) = (𝐴 ·_ih 𝐴))
2	1	anidms 566	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ·_ih 𝑥) = (𝐴 ·_ih 𝐴))
3	2	fveq2d 6826	. 2 ⊢ (𝑥 = 𝐴 → (√‘(𝑥 ·_ih 𝑥)) = (√‘(𝐴 ·_ih 𝐴)))
4		dfhnorm2 31100	. 2 ⊢ norm_ℎ = (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·_ih 𝑥)))
5		fvex 6835	. 2 ⊢ (√‘(𝐴 ·_ih 𝐴)) ∈ V
6	3, 4, 5	fvmpt 6929	1 ⊢ (𝐴 ∈ ℋ → (norm_ℎ‘𝐴) = (√‘(𝐴 ·_ih 𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2111 ‘cfv 6481 (class class class)co 7346 √csqrt 15140 ℋchba 30897 ·_ih csp 30900 norm_ℎcno 30901

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5234 ax-nul 5244 ax-pr 5370 ax-hfi 31057

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-dif 3905 df-un 3907 df-ss 3919 df-nul 4284 df-if 4476 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-br 5092 df-opab 5154 df-mpt 5173 df-id 5511 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-fv 6489 df-ov 7349 df-hnorm 30946

This theorem is referenced by: normge0 31104 normgt0 31105 norm0 31106 normsqi 31110 norm-ii-i 31115 norm-iii-i 31117 bcsiALT 31157

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