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Definition df-omul 7610
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
StepHypRef Expression
1 comu 7603 . 2 class ·𝑜
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 con0 5761 . . 3 class On
53cv 1522 . . . 4 class 𝑦
6 vz . . . . . 6 setvar 𝑧
7 cvv 3231 . . . . . 6 class V
86cv 1522 . . . . . . 7 class 𝑧
92cv 1522 . . . . . . 7 class 𝑥
10 coa 7602 . . . . . . 7 class +𝑜
118, 9, 10co 6690 . . . . . 6 class (𝑧 +𝑜 𝑥)
126, 7, 11cmpt 4762 . . . . 5 class (𝑧 ∈ V ↦ (𝑧 +𝑜 𝑥))
13 c0 3948 . . . . 5 class
1412, 13crdg 7550 . . . 4 class rec((𝑧 ∈ V ↦ (𝑧 +𝑜 𝑥)), ∅)
155, 14cfv 5926 . . 3 class (rec((𝑧 ∈ V ↦ (𝑧 +𝑜 𝑥)), ∅)‘𝑦)
162, 3, 4, 4, 15cmpt2 6692 . 2 class (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 +𝑜 𝑥)), ∅)‘𝑦))
171, 16wceq 1523 1 wff ·𝑜 = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 +𝑜 𝑥)), ∅)‘𝑦))
Colors of variables: wff setvar class
This definition is referenced by:  fnom  7634  omv  7637
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