![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > df-oexpi | GIF version |
Description: Define the ordinal
exponentiation operation.
This definition is similar to a conventional definition of exponentiation except that it defines โ โo ๐ด to be 1o for all ๐ด โ On, in order to avoid having different cases for whether the base is โ 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.) |
Ref | Expression |
---|---|
df-oexpi | โข โo = (๐ฅ โ On, ๐ฆ โ On โฆ (rec((๐ง โ V โฆ (๐ง ยทo ๐ฅ)), 1o)โ๐ฆ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coei 6416 | . 2 class โo | |
2 | vx | . . 3 setvar ๐ฅ | |
3 | vy | . . 3 setvar ๐ฆ | |
4 | con0 4364 | . . 3 class On | |
5 | 3 | cv 1352 | . . . 4 class ๐ฆ |
6 | vz | . . . . . 6 setvar ๐ง | |
7 | cvv 2738 | . . . . . 6 class V | |
8 | 6 | cv 1352 | . . . . . . 7 class ๐ง |
9 | 2 | cv 1352 | . . . . . . 7 class ๐ฅ |
10 | comu 6415 | . . . . . . 7 class ยทo | |
11 | 8, 9, 10 | co 5875 | . . . . . 6 class (๐ง ยทo ๐ฅ) |
12 | 6, 7, 11 | cmpt 4065 | . . . . 5 class (๐ง โ V โฆ (๐ง ยทo ๐ฅ)) |
13 | c1o 6410 | . . . . 5 class 1o | |
14 | 12, 13 | crdg 6370 | . . . 4 class rec((๐ง โ V โฆ (๐ง ยทo ๐ฅ)), 1o) |
15 | 5, 14 | cfv 5217 | . . 3 class (rec((๐ง โ V โฆ (๐ง ยทo ๐ฅ)), 1o)โ๐ฆ) |
16 | 2, 3, 4, 4, 15 | cmpo 5877 | . 2 class (๐ฅ โ On, ๐ฆ โ On โฆ (rec((๐ง โ V โฆ (๐ง ยทo ๐ฅ)), 1o)โ๐ฆ)) |
17 | 1, 16 | wceq 1353 | 1 wff โo = (๐ฅ โ On, ๐ฆ โ On โฆ (rec((๐ง โ V โฆ (๐ง ยทo ๐ฅ)), 1o)โ๐ฆ)) |
Colors of variables: wff set class |
This definition is referenced by: fnoei 6453 oeiexg 6454 oeiv 6457 |
Copyright terms: Public domain | W3C validator |