Step |
Hyp |
Ref |
Expression |
1 |
|
df-1o |
|- 1o = suc (/) |
2 |
1
|
oveq2i |
|- ( A ^o 1o ) = ( A ^o suc (/) ) |
3 |
|
peano1 |
|- (/) e. _om |
4 |
|
onesuc |
|- ( ( A e. On /\ (/) e. _om ) -> ( A ^o suc (/) ) = ( ( A ^o (/) ) .o A ) ) |
5 |
3 4
|
mpan2 |
|- ( A e. On -> ( A ^o suc (/) ) = ( ( A ^o (/) ) .o A ) ) |
6 |
2 5
|
eqtrid |
|- ( A e. On -> ( A ^o 1o ) = ( ( A ^o (/) ) .o A ) ) |
7 |
|
oe0 |
|- ( A e. On -> ( A ^o (/) ) = 1o ) |
8 |
7
|
oveq1d |
|- ( A e. On -> ( ( A ^o (/) ) .o A ) = ( 1o .o A ) ) |
9 |
|
om1r |
|- ( A e. On -> ( 1o .o A ) = A ) |
10 |
6 8 9
|
3eqtrd |
|- ( A e. On -> ( A ^o 1o ) = A ) |