normval - Hilbert Space Explorer

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Theorem normval 31220

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 7372	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑥 ·_ih 𝑥) = (𝐴 ·_ih 𝐴))
2	1	anidms 571	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ·_ih 𝑥) = (𝐴 ·_ih 𝐴))
3	2	fveq2d 6838	. 2 ⊢ (𝑥 = 𝐴 → (√‘(𝑥 ·_ih 𝑥)) = (√‘(𝐴 ·_ih 𝐴)))
4		dfhnorm2 31218	. 2 ⊢ norm_ℎ = (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·_ih 𝑥)))
5		fvex 6847	. 2 ⊢ (√‘(𝐴 ·_ih 𝐴)) ∈ V
6	3, 4, 5	fvmpt 6942	1 ⊢ (𝐴 ∈ ℋ → (norm_ℎ‘𝐴) = (√‘(𝐴 ·_ih 𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1547 ∈ wcel 2119 ‘cfv 6492 (class class class)co 7363 √csqrt 15193 ℋchba 31015 ·_ih csp 31018 norm_ℎcno 31019

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-sep 5225 ax-nul 5235 ax-pr 5369 ax-hfi 31175

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-ral 3055 df-rex 3065 df-rab 3393 df-v 3434 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4269 df-if 4462 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-br 5080 df-opab 5142 df-mpt 5161 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-fv 6500 df-ov 7366 df-hnorm 31064

This theorem is referenced by: normge0 31222 normgt0 31223 norm0 31224 normsqi 31228 norm-ii-i 31233 norm-iii-i 31235 bcsiALT 31275

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