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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
StepHypRef Expression
1 coei 6430 . 2  classo
2 vx . . 3  setvar  x
3 vy . . 3  setvar  y
4 con0 4375 . . 3  class  On
53cv 1362 . . . 4  class  y
6 vz . . . . . 6  setvar  z
7 cvv 2749 . . . . . 6  class  _V
86cv 1362 . . . . . . 7  class  z
92cv 1362 . . . . . . 7  class  x
10 comu 6429 . . . . . . 7  class  .o
118, 9, 10co 5888 . . . . . 6  class  ( z  .o  x )
126, 7, 11cmpt 4076 . . . . 5  class  ( z  e.  _V  |->  ( z  .o  x ) )
13 c1o 6424 . . . . 5  class  1o
1412, 13crdg 6384 . . . 4  class  rec (
( z  e.  _V  |->  ( z  .o  x
) ) ,  1o )
155, 14cfv 5228 . . 3  class  ( rec ( ( z  e. 
_V  |->  ( z  .o  x ) ) ,  1o ) `  y
)
162, 3, 4, 4, 15cmpo 5890 . 2  class  ( x  e.  On ,  y  e.  On  |->  ( rec ( ( z  e. 
_V  |->  ( z  .o  x ) ) ,  1o ) `  y
) )
171, 16wceq 1363 1  wffo  =  ( x  e.  On ,  y  e.  On  |->  ( rec (
( z  e.  _V  |->  ( z  .o  x
) ) ,  1o ) `  y )
)
Colors of variables: wff set class
This definition is referenced by:  fnoei  6467  oeiexg  6468  oeiv  6471
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