Definition df-omul 7565

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 7558	. 2 class ·𝑜
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		con0 5723	. . 3 class On
5	3	cv 1482	. . . 4 class 𝑦
6		vz	. . . . . 6 setvar 𝑧
7		cvv 3200	. . . . . 6 class V
8	6	cv 1482	. . . . . . 7 class 𝑧
9	2	cv 1482	. . . . . . 7 class 𝑥
10		coa 7557	. . . . . . 7 class +𝑜
11	8, 9, 10	co 6650	. . . . . 6 class (𝑧 +𝑜 𝑥)
12	6, 7, 11	cmpt 4729	. . . . 5 class (𝑧 ∈ V ↦ (𝑧 +𝑜 𝑥))
13		c0 3915	. . . . 5 class ∅
14	12, 13	crdg 7505	. . . 4 class rec((𝑧 ∈ V ↦ (𝑧 +𝑜 𝑥)), ∅)
15	5, 14	cfv 5888	. . 3 class (rec((𝑧 ∈ V ↦ (𝑧 +𝑜 𝑥)), ∅)‘𝑦)
16	2, 3, 4, 4, 15	cmpt2 6652	. 2 class (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 +𝑜 𝑥)), ∅)‘𝑦))
17	1, 16	wceq 1483	1 wff ·𝑜 = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 +𝑜 𝑥)), ∅)‘𝑦))

This definition is referenced by:  fnom  7589  omv  7592