Definition df-omul 6573

Description: Define the ordinal multiplication operation. (Contributed by NM, 26-Aug-1995.)

Assertion

Ref	Expression
df-omul	|- .o = ( x e. On , y e. On |-> ( rec ( ( z e. _V |-> ( z +o x ) ) , (/) ) ` y ) )

Distinct variable group:   x, y, z

Detailed syntax breakdown of Definition df-omul

Step	Hyp	Ref	Expression
1		comu 6566	. 2 class .o
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		con0 4454	. . 3 class On
5	3	cv 1394	. . . 4 class y
6		vz	. . . . . 6 setvar z
7		cvv 2799	. . . . . 6 class _V
8	6	cv 1394	. . . . . . 7 class z
9	2	cv 1394	. . . . . . 7 class x
10		coa 6565	. . . . . . 7 class +o
11	8, 9, 10	co 6007	. . . . . 6 class ( z +o x )
12	6, 7, 11	cmpt 4145	. . . . 5 class ( z e. _V |-> ( z +o x ) )
13		c0 3491	. . . . 5 class (/)
14	12, 13	crdg 6521	. . . 4 class rec ( ( z e. _V |-> ( z +o x ) ) , (/) )
15	5, 14	cfv 5318	. . 3 class ( rec ( ( z e. _V |-> ( z +o x ) ) , (/) ) ` y )
16	2, 3, 4, 4, 15	cmpo 6009	. 2 class ( x e. On , y e. On |-> ( rec ( ( z e. _V |-> ( z +o x ) ) , (/) ) ` y ) )
17	1, 16	wceq 1395	1 wff .o = ( x e. On , y e. On |-> ( rec ( ( z e. _V |-> ( z +o x ) ) , (/) ) ` y ) )

This definition is referenced by:  fnom  6604  omexg  6605  omv  6609