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Definition df-omul 6311
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 6304 . 2 class ·o
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 con0 4280 . . 3 class On
53cv 1330 . . . 4 class 𝑦
6 vz . . . . . 6 setvar 𝑧
7 cvv 2681 . . . . . 6 class V
86cv 1330 . . . . . . 7 class 𝑧
92cv 1330 . . . . . . 7 class 𝑥
10 coa 6303 . . . . . . 7 class +o
118, 9, 10co 5767 . . . . . 6 class (𝑧 +o 𝑥)
126, 7, 11cmpt 3984 . . . . 5 class (𝑧 ∈ V ↦ (𝑧 +o 𝑥))
13 c0 3358 . . . . 5 class
1412, 13crdg 6259 . . . 4 class rec((𝑧 ∈ V ↦ (𝑧 +o 𝑥)), ∅)
155, 14cfv 5118 . . 3 class (rec((𝑧 ∈ V ↦ (𝑧 +o 𝑥)), ∅)‘𝑦)
162, 3, 4, 4, 15cmpo 5769 . 2 class (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 +o 𝑥)), ∅)‘𝑦))
171, 16wceq 1331 1 wff ·o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 +o 𝑥)), ∅)‘𝑦))
Colors of variables: wff set class
This definition is referenced by:  fnom  6339  omexg  6340  omv  6344
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