Definition df-oexpi 6437

Description: Define the ordinal exponentiation operation.

This definition is similar to a conventional definition of exponentiation except that it defines (/) ↑_o A to be 1o for all A e. On, in order to avoid having different cases for whether the base is (/) or not.

We do not yet have an extensive development of ordinal exponentiation. For background on ordinal exponentiation without excluded middle, see Tom de Jong, Nicolai Kraus, Fredrik Nordvall Forsberg, and Chuangjie Xu (2025), "Ordinal Exponentiation in Homotopy Type Theory", arXiv:2501.14542 , https://arxiv.org/abs/2501.14542 which is formalized in the TypeTopology proof library at https://ordinal-exponentiation-hott.github.io/.

(Contributed by Mario Carneiro, 4-Jul-2019.)

Assertion

Ref	Expression
df-oexpi	|- ↑o = ( x e. On , y e. On |-> ( rec ( ( z e. _V |-> ( z .o x ) ) , 1o ) ` y ) )

Distinct variable group:   x, y, z

Detailed syntax breakdown of Definition df-oexpi

Step	Hyp	Ref	Expression
1		coei 6430	. 2 class ↑o
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		con0 4375	. . 3 class On
5	3	cv 1362	. . . 4 class y
6		vz	. . . . . 6 setvar z
7		cvv 2749	. . . . . 6 class _V
8	6	cv 1362	. . . . . . 7 class z
9	2	cv 1362	. . . . . . 7 class x
10		comu 6429	. . . . . . 7 class .o
11	8, 9, 10	co 5888	. . . . . 6 class ( z .o x )
12	6, 7, 11	cmpt 4076	. . . . 5 class ( z e. _V |-> ( z .o x ) )
13		c1o 6424	. . . . 5 class 1o
14	12, 13	crdg 6384	. . . 4 class rec ( ( z e. _V |-> ( z .o x ) ) , 1o )
15	5, 14	cfv 5228	. . 3 class ( rec ( ( z e. _V |-> ( z .o x ) ) , 1o ) ` y )
16	2, 3, 4, 4, 15	cmpo 5890	. 2 class ( x e. On , y e. On |-> ( rec ( ( z e. _V |-> ( z .o x ) ) , 1o ) ` y ) )
17	1, 16	wceq 1363	1 wff ↑o = ( x e. On , y e. On |-> ( rec ( ( z e. _V |-> ( z .o x ) ) , 1o ) ` y ) )

This definition is referenced by:  fnoei  6467  oeiexg  6468  oeiv  6471