Definition df-homul 32345

Description: Define the scalar product with a Hilbert space operator. Definition of [Beran] p. 111. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)

Assertion

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

Distinct variable group:   𝑓,𝑔,𝑥

Detailed syntax breakdown of Definition df-homul

Step	Hyp	Ref	Expression
1		chot 31553	. 2 class ·op
2		vf	. . 3 setvar 𝑓
3		vg	. . 3 setvar 𝑔
4		cc 11208	. . 3 class ℂ
5		chba 31533	. . . 4 class ℋ
6		cmap 8858	. . . 4 class ↑m
7	5, 5, 6	co 7427	. . 3 class ( ℋ ↑m ℋ)
8		vx	. . . 4 setvar 𝑥
9	2	cv 1568	. . . . 5 class 𝑓
10	8	cv 1568	. . . . . 6 class 𝑥
11	3	cv 1568	. . . . . 6 class 𝑔
12	10, 11	cfv 6546	. . . . 5 class (𝑔‘𝑥)
13		csm 31535	. . . . 5 class ·ℎ
14	9, 12, 13	co 7427	. . . 4 class (𝑓 ·ℎ (𝑔‘𝑥))
15	8, 5, 14	cmpt 5188	. . 3 class (𝑥 ∈ ℋ ↦ (𝑓 ·ℎ (𝑔‘𝑥)))
16	2, 3, 4, 7, 15	cmpo 7429	. 2 class (𝑓 ∈ ℂ, 𝑔 ∈ ( ℋ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 ·ℎ (𝑔‘𝑥))))
17	1, 16	wceq 1569	1 wff ·op = (𝑓 ∈ ℂ, 𝑔 ∈ ( ℋ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 ·ℎ (𝑔‘𝑥))))

This definition is used by:  hommval  32350