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Metamath Proof Explorer

Theorem oe0

Description: Ordinal exponentiation with zero exponent. Definition 8.30 of TakeutiZaring p. 67. Definition 2.6 of Schloeder p. 4. (Contributed by NM, 31-Dec-2004) (Revised by Mario Carneiro, 8-Sep-2013)

Ref Expression
Assertion oe0 
|- ( A e. On -> ( A ^o (/) ) = 1o )

Proof

Step Hyp Ref Expression
1 oveq1 
 |-  ( A = (/) -> ( A ^o (/) ) = ( (/) ^o (/) ) )
2 oe0m0 
 |-  ( (/) ^o (/) ) = 1o
3 1 2 eqtrdi 
 |-  ( A = (/) -> ( A ^o (/) ) = 1o )
4 3 adantl 
 |-  ( ( A e. On /\ A = (/) ) -> ( A ^o (/) ) = 1o )
5 0elon 
 |-  (/) e. On
6 oevn0 
 |-  ( ( ( A e. On /\ (/) e. On ) /\ (/) e. A ) -> ( A ^o (/) ) = ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` (/) ) )
7 5 6 mpanl2 
 |-  ( ( A e. On /\ (/) e. A ) -> ( A ^o (/) ) = ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` (/) ) )
8 1oex 
 |-  1o e. _V
9 8 rdg0 
 |-  ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` (/) ) = 1o
10 7 9 eqtrdi 
 |-  ( ( A e. On /\ (/) e. A ) -> ( A ^o (/) ) = 1o )
11 10 adantll 
 |-  ( ( ( A e. On /\ A e. On ) /\ (/) e. A ) -> ( A ^o (/) ) = 1o )
12 4 11 oe0lem 
 |-  ( ( A e. On /\ A e. On ) -> ( A ^o (/) ) = 1o )
13 12 anidms 
 |-  ( A e. On -> ( A ^o (/) ) = 1o )