Definition df-hosum 28589

Description: Define the sum of two Hilbert space operators. Definition of [Beran] p. 111. (Contributed by NM, 9-Nov-2000.) (New usage is discouraged.)

Assertion

Ref	Expression
df-hosum	⊢ +op = (𝑓 ∈ ( ℋ ↑𝑚 ℋ), 𝑔 ∈ ( ℋ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) +ℎ (𝑔‘𝑥))))

Distinct variable group:   𝑓,𝑔,𝑥

Detailed syntax breakdown of Definition df-hosum

Step	Hyp	Ref	Expression
1		chos 27795	. 2 class +op
2		vf	. . 3 setvar 𝑓
3		vg	. . 3 setvar 𝑔
4		chil 27776	. . . 4 class ℋ
5		cmap 7857	. . . 4 class ↑𝑚
6	4, 4, 5	co 6650	. . 3 class ( ℋ ↑𝑚 ℋ)
7		vx	. . . 4 setvar 𝑥
8	7	cv 1482	. . . . . 6 class 𝑥
9	2	cv 1482	. . . . . 6 class 𝑓
10	8, 9	cfv 5888	. . . . 5 class (𝑓‘𝑥)
11	3	cv 1482	. . . . . 6 class 𝑔
12	8, 11	cfv 5888	. . . . 5 class (𝑔‘𝑥)
13		cva 27777	. . . . 5 class +ℎ
14	10, 12, 13	co 6650	. . . 4 class ((𝑓‘𝑥) +ℎ (𝑔‘𝑥))
15	7, 4, 14	cmpt 4729	. . 3 class (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) +ℎ (𝑔‘𝑥)))
16	2, 3, 6, 6, 15	cmpt2 6652	. 2 class (𝑓 ∈ ( ℋ ↑𝑚 ℋ), 𝑔 ∈ ( ℋ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) +ℎ (𝑔‘𝑥))))
17	1, 16	wceq 1483	1 wff +op = (𝑓 ∈ ( ℋ ↑𝑚 ℋ), 𝑔 ∈ ( ℋ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) +ℎ (𝑔‘𝑥))))

This definition is referenced by:  hosmval  28594