Metamath Proof Explorer

Theorem oa0

Description: Addition with zero. Proposition 8.3 of TakeutiZaring p. 57. (Contributed by NM, 3-May-1995) (Revised by Mario Carneiro, 8-Sep-2013)

Ref Expression
Assertion oa0 
|- ( A e. On -> ( A +o (/) ) = A )

Proof

Step Hyp Ref Expression
1 0elon 
 |-  (/) e. On
2 oav 
 |-  ( ( A e. On /\ (/) e. On ) -> ( A +o (/) ) = ( rec ( ( x e. _V |-> suc x ) , A ) ` (/) ) )
3 1 2 mpan2 
 |-  ( A e. On -> ( A +o (/) ) = ( rec ( ( x e. _V |-> suc x ) , A ) ` (/) ) )
4 rdg0g 
 |-  ( A e. On -> ( rec ( ( x e. _V |-> suc x ) , A ) ` (/) ) = A )
5 3 4 eqtrd 
 |-  ( A e. On -> ( A +o (/) ) = A )