Definition df-hodif 30094

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

Assertion

Ref	Expression
df-hodif	⊢ −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-hodif

Step	Hyp	Ref	Expression
1		chod 29302	. 2 class −op
2		vf	. . 3 setvar 𝑓
3		vg	. . 3 setvar 𝑔
4		chba 29281	. . . 4 class ℋ
5		cmap 8615	. . . 4 class ↑m
6	4, 4, 5	co 7275	. . 3 class ( ℋ ↑m ℋ)
7		vx	. . . 4 setvar 𝑥
8	7	cv 1538	. . . . . 6 class 𝑥
9	2	cv 1538	. . . . . 6 class 𝑓
10	8, 9	cfv 6433	. . . . 5 class (𝑓‘𝑥)
11	3	cv 1538	. . . . . 6 class 𝑔
12	8, 11	cfv 6433	. . . . 5 class (𝑔‘𝑥)
13		cmv 29287	. . . . 5 class −ℎ
14	10, 12, 13	co 7275	. . . 4 class ((𝑓‘𝑥) −ℎ (𝑔‘𝑥))
15	7, 4, 14	cmpt 5157	. . 3 class (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) −ℎ (𝑔‘𝑥)))
16	2, 3, 6, 6, 15	cmpo 7277	. 2 class (𝑓 ∈ ( ℋ ↑m ℋ), 𝑔 ∈ ( ℋ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) −ℎ (𝑔‘𝑥))))
17	1, 16	wceq 1539	1 wff −op = (𝑓 ∈ ( ℋ ↑m ℋ), 𝑔 ∈ ( ℋ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) −ℎ (𝑔‘𝑥))))

This definition is referenced by:  hodmval  30099