Definition df-homul 29162

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 28368	. 2 class ·op
2		vf	. . 3 setvar 𝑓
3		vg	. . 3 setvar 𝑔
4		cc 10270	. . 3 class ℂ
5		chba 28348	. . . 4 class ℋ
6		cmap 8140	. . . 4 class ↑𝑚
7	5, 5, 6	co 6922	. . 3 class ( ℋ ↑𝑚 ℋ)
8		vx	. . . 4 setvar 𝑥
9	2	cv 1600	. . . . 5 class 𝑓
10	8	cv 1600	. . . . . 6 class 𝑥
11	3	cv 1600	. . . . . 6 class 𝑔
12	10, 11	cfv 6135	. . . . 5 class (𝑔‘𝑥)
13		csm 28350	. . . . 5 class ·ℎ
14	9, 12, 13	co 6922	. . . 4 class (𝑓 ·ℎ (𝑔‘𝑥))
15	8, 5, 14	cmpt 4965	. . 3 class (𝑥 ∈ ℋ ↦ (𝑓 ·ℎ (𝑔‘𝑥)))
16	2, 3, 4, 7, 15	cmpt2 6924	. 2 class (𝑓 ∈ ℂ, 𝑔 ∈ ( ℋ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 ·ℎ (𝑔‘𝑥))))
17	1, 16	wceq 1601	1 wff ·op = (𝑓 ∈ ℂ, 𝑔 ∈ ( ℋ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 ·ℎ (𝑔‘𝑥))))

This definition is referenced by:  hommval  29167