MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-omul Structured version   Visualization version   GIF version

Definition df-omul 8098
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 8091 . 2 class ·o
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 con0 6185 . . 3 class On
53cv 1527 . . . 4 class 𝑦
6 vz . . . . . 6 setvar 𝑧
7 cvv 3495 . . . . . 6 class V
86cv 1527 . . . . . . 7 class 𝑧
92cv 1527 . . . . . . 7 class 𝑥
10 coa 8090 . . . . . . 7 class +o
118, 9, 10co 7145 . . . . . 6 class (𝑧 +o 𝑥)
126, 7, 11cmpt 5138 . . . . 5 class (𝑧 ∈ V ↦ (𝑧 +o 𝑥))
13 c0 4290 . . . . 5 class
1412, 13crdg 8036 . . . 4 class rec((𝑧 ∈ V ↦ (𝑧 +o 𝑥)), ∅)
155, 14cfv 6349 . . 3 class (rec((𝑧 ∈ V ↦ (𝑧 +o 𝑥)), ∅)‘𝑦)
162, 3, 4, 4, 15cmpo 7147 . 2 class (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 +o 𝑥)), ∅)‘𝑦))
171, 16wceq 1528 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  8125  omv  8128
  Copyright terms: Public domain W3C validator