normval - Hilbert Space Explorer

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Theorem normval 29387

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 7264	. . . 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 6760	. 2 ⊢ (𝑥 = 𝐴 → (√‘(𝑥 ·_ih 𝑥)) = (√‘(𝐴 ·_ih 𝐴)))
4		dfhnorm2 29385	. 2 ⊢ norm_ℎ = (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·_ih 𝑥)))
5		fvex 6769	. 2 ⊢ (√‘(𝐴 ·_ih 𝐴)) ∈ V
6	3, 4, 5	fvmpt 6857	1 ⊢ (𝐴 ∈ ℋ → (norm_ℎ‘𝐴) = (√‘(𝐴 ·_ih 𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2108 ‘cfv 6418 (class class class)co 7255 √csqrt 14872 ℋchba 29182 ·_ih csp 29185 norm_ℎcno 29186

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pr 5347 ax-hfi 29342

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-fv 6426 df-ov 7258 df-hnorm 29231

This theorem is referenced by: normge0 29389 normgt0 29390 norm0 29391 normsqi 29395 norm-ii-i 29400 norm-iii-i 29402 bcsiALT 29442

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