Definition df-mulv 45406

Description: Define the operation of scalar multiplication. (Contributed by Andrew Salmon, 27-Jan-2012.)

Assertion

Ref	Expression
df-mulv	⊢ .𝑣 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ (𝑥 · (𝑦‘𝑣))))

Distinct variable group:   𝑥,𝑣,𝑦

Detailed syntax breakdown of Definition df-mulv

Step	Hyp	Ref	Expression
1		ctimesr 45400	. 2 class .𝑣
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cvv 3451	. . 3 class V
5		vv	. . . 4 setvar 𝑣
6		cr 11180	. . . 4 class ℝ
7	2	cv 1569	. . . . 5 class 𝑥
8	5	cv 1569	. . . . . 6 class 𝑣
9	3	cv 1569	. . . . . 6 class 𝑦
10	8, 9	cfv 6531	. . . . 5 class (𝑦‘𝑣)
11		cmul 11186	. . . . 5 class ·
12	7, 10, 11	co 7412	. . . 4 class (𝑥 · (𝑦‘𝑣))
13	5, 6, 12	cmpt 5186	. . 3 class (𝑣 ∈ ℝ ↦ (𝑥 · (𝑦‘𝑣)))
14	2, 3, 4, 4, 13	cmpo 7414	. 2 class (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ (𝑥 · (𝑦‘𝑣))))
15	1, 14	wceq 1570	1 wff .𝑣 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ (𝑥 · (𝑦‘𝑣))))

This definition is used by:  mulvval  45409