Definition df-omul 6692

Description: Define the ordinal multiplication operation. (Contributed by NM, 26-Aug-1995.)

Assertion

Ref	Expression
df-omul	⊢ ·o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 +o 𝑥)), ∅)‘𝑦))

Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-omul

Step	Hyp	Ref	Expression
1		comu 6685	. 2 class ·o
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		con0 4508	. . 3 class On
5	3	cv 1401	. . . 4 class 𝑦
6		vz	. . . . . 6 setvar 𝑧
7		cvv 2821	. . . . . 6 class V
8	6	cv 1401	. . . . . . 7 class 𝑧
9	2	cv 1401	. . . . . . 7 class 𝑥
10		coa 6684	. . . . . . 7 class +o
11	8, 9, 10	co 6085	. . . . . 6 class (𝑧 +o 𝑥)
12	6, 7, 11	cmpt 4192	. . . . 5 class (𝑧 ∈ V ↦ (𝑧 +o 𝑥))
13		c0 3520	. . . . 5 class ∅
14	12, 13	crdg 6640	. . . 4 class rec((𝑧 ∈ V ↦ (𝑧 +o 𝑥)), ∅)
15	5, 14	cfv 5377	. . 3 class (rec((𝑧 ∈ V ↦ (𝑧 +o 𝑥)), ∅)‘𝑦)
16	2, 3, 4, 4, 15	cmpo 6087	. 2 class (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 +o 𝑥)), ∅)‘𝑦))
17	1, 16	wceq 1402	1 wff ·o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 +o 𝑥)), ∅)‘𝑦))

This definition is used by:  fnom  6723  omexg  6724  omv  6728