normval - Hilbert Space Explorer

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Theorem normval 31106

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 7361	. . . 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 6832	. 2 ⊢ (𝑥 = 𝐴 → (√‘(𝑥 ·_ih 𝑥)) = (√‘(𝐴 ·_ih 𝐴)))
4		dfhnorm2 31104	. 2 ⊢ norm_ℎ = (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·_ih 𝑥)))
5		fvex 6841	. 2 ⊢ (√‘(𝐴 ·_ih 𝐴)) ∈ V
6	3, 4, 5	fvmpt 6935	1 ⊢ (𝐴 ∈ ℋ → (norm_ℎ‘𝐴) = (√‘(𝐴 ·_ih 𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2113 ‘cfv 6486 (class class class)co 7352 √csqrt 15142 ℋchba 30901 ·_ih csp 30904 norm_ℎcno 30905

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 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5236 ax-nul 5246 ax-pr 5372 ax-hfi 31061

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 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-ral 3049 df-rex 3058 df-rab 3397 df-v 3439 df-dif 3901 df-un 3903 df-ss 3915 df-nul 4283 df-if 4475 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-br 5094 df-opab 5156 df-mpt 5175 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-fv 6494 df-ov 7355 df-hnorm 30950

This theorem is referenced by: normge0 31108 normgt0 31109 norm0 31110 normsqi 31114 norm-ii-i 31119 norm-iii-i 31121 bcsiALT 31161

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