Metamath Proof Explorer


Theorem oe1

Description: Ordinal exponentiation with an exponent of 1. (Contributed by NM, 2-Jan-2005)

Ref Expression
Assertion oe1
|- ( A e. On -> ( A ^o 1o ) = A )

Proof

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 )