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Metamath Proof Explorer

Theorem omv

Description: Value of ordinal multiplication. (Contributed by NM, 17-Sep-1995) (Revised by Mario Carneiro, 23-Aug-2014)

Ref Expression
Assertion omv 
|- ( ( A e. On /\ B e. On ) -> ( A .o B ) = ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) )

Proof

Step Hyp Ref Expression
1 oveq2 
 |-  ( y = A -> ( x +o y ) = ( x +o A ) )
2 1 mpteq2dv 
 |-  ( y = A -> ( x e. _V |-> ( x +o y ) ) = ( x e. _V |-> ( x +o A ) ) )
3 rdgeq1 
 |-  ( ( x e. _V |-> ( x +o y ) ) = ( x e. _V |-> ( x +o A ) ) -> rec ( ( x e. _V |-> ( x +o y ) ) , (/) ) = rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) )
4 2 3 syl 
 |-  ( y = A -> rec ( ( x e. _V |-> ( x +o y ) ) , (/) ) = rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) )
5 4 fveq1d 
 |-  ( y = A -> ( rec ( ( x e. _V |-> ( x +o y ) ) , (/) ) ` z ) = ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` z ) )
6 fveq2 
 |-  ( z = B -> ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` z ) = ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) )
7 df-omul 
 |-  .o = ( y e. On , z e. On |-> ( rec ( ( x e. _V |-> ( x +o y ) ) , (/) ) ` z ) )
8 fvex 
 |-  ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) e. _V
9 5 6 7 8 ovmpo 
 |-  ( ( A e. On /\ B e. On ) -> ( A .o B ) = ( rec ( ( x e. _V |-> ( x +o A ) ) , (/) ) ` B ) )