Definition df-hfmul 31715

Description: Define the scalar product with a Hilbert space functional. Definition of [Beran] p. 111. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)

Assertion

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

Distinct variable group:   𝑓,𝑔,𝑥

Detailed syntax breakdown of Definition df-hfmul

Step	Hyp	Ref	Expression
1		chft 30923	. 2 class ·fn
2		vf	. . 3 setvar 𝑓
3		vg	. . 3 setvar 𝑔
4		cc 11127	. . 3 class ℂ
5		chba 30900	. . . 4 class ℋ
6		cmap 8840	. . . 4 class ↑m
7	4, 5, 6	co 7405	. . 3 class (ℂ ↑m ℋ)
8		vx	. . . 4 setvar 𝑥
9	2	cv 1539	. . . . 5 class 𝑓
10	8	cv 1539	. . . . . 6 class 𝑥
11	3	cv 1539	. . . . . 6 class 𝑔
12	10, 11	cfv 6531	. . . . 5 class (𝑔‘𝑥)
13		cmul 11134	. . . . 5 class ·
14	9, 12, 13	co 7405	. . . 4 class (𝑓 · (𝑔‘𝑥))
15	8, 5, 14	cmpt 5201	. . 3 class (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔‘𝑥)))
16	2, 3, 4, 7, 15	cmpo 7407	. 2 class (𝑓 ∈ ℂ, 𝑔 ∈ (ℂ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔‘𝑥))))
17	1, 16	wceq 1540	1 wff ·fn = (𝑓 ∈ ℂ, 𝑔 ∈ (ℂ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔‘𝑥))))

This definition is referenced by:  hfmmval  31720