Definition df-hfsum 31753

Description: Define the sum of two Hilbert space functionals. Definition of [Beran] p. 111. Note that unlike some authors, we define a functional as any function from ℋ to ℂ, not just linear (or bounded linear) ones. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
df-hfsum	⊢ +fn = (𝑓 ∈ (ℂ ↑m ℋ), 𝑔 ∈ (ℂ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) + (𝑔‘𝑥))))

Distinct variable group:   𝑓,𝑔,𝑥

Detailed syntax breakdown of Definition df-hfsum

Step	Hyp	Ref	Expression
1		chfs 30961	. 2 class +fn
2		vf	. . 3 setvar 𝑓
3		vg	. . 3 setvar 𝑔
4		cc 11154	. . . 4 class ℂ
5		chba 30939	. . . 4 class ℋ
6		cmap 8867	. . . 4 class ↑m
7	4, 5, 6	co 7432	. . 3 class (ℂ ↑m ℋ)
8		vx	. . . 4 setvar 𝑥
9	8	cv 1538	. . . . . 6 class 𝑥
10	2	cv 1538	. . . . . 6 class 𝑓
11	9, 10	cfv 6560	. . . . 5 class (𝑓‘𝑥)
12	3	cv 1538	. . . . . 6 class 𝑔
13	9, 12	cfv 6560	. . . . 5 class (𝑔‘𝑥)
14		caddc 11159	. . . . 5 class +
15	11, 13, 14	co 7432	. . . 4 class ((𝑓‘𝑥) + (𝑔‘𝑥))
16	8, 5, 15	cmpt 5224	. . 3 class (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) + (𝑔‘𝑥)))
17	2, 3, 7, 7, 16	cmpo 7434	. 2 class (𝑓 ∈ (ℂ ↑m ℋ), 𝑔 ∈ (ℂ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) + (𝑔‘𝑥))))
18	1, 17	wceq 1539	1 wff +fn = (𝑓 ∈ (ℂ ↑m ℋ), 𝑔 ∈ (ℂ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) + (𝑔‘𝑥))))

This definition is referenced by:  hfsmval  31758