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Definition df-omul 6037
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 6030 . 2 class ·𝑜
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 con0 4128 . . 3 class On
53cv 1258 . . . 4 class 𝑦
6 vz . . . . . 6 setvar 𝑧
7 cvv 2574 . . . . . 6 class V
86cv 1258 . . . . . . 7 class 𝑧
92cv 1258 . . . . . . 7 class 𝑥
10 coa 6029 . . . . . . 7 class +𝑜
118, 9, 10co 5540 . . . . . 6 class (𝑧 +𝑜 𝑥)
126, 7, 11cmpt 3846 . . . . 5 class (𝑧 ∈ V ↦ (𝑧 +𝑜 𝑥))
13 c0 3252 . . . . 5 class
1412, 13crdg 5987 . . . 4 class rec((𝑧 ∈ V ↦ (𝑧 +𝑜 𝑥)), ∅)
155, 14cfv 4930 . . 3 class (rec((𝑧 ∈ V ↦ (𝑧 +𝑜 𝑥)), ∅)‘𝑦)
162, 3, 4, 4, 15cmpt2 5542 . 2 class (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 +𝑜 𝑥)), ∅)‘𝑦))
171, 16wceq 1259 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  6061  omexg  6062  omv  6066
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