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Metamath Proof Explorer

Theorem onesuc

Description: Exponentiation with a successor exponent. Definition 8.30 of TakeutiZaring p. 67. (Contributed by Mario Carneiro, 14-Nov-2014)

Ref Expression
Assertion onesuc 
|- ( ( A e. On /\ B e. _om ) -> ( A ^o suc B ) = ( ( A ^o B ) .o A ) )

Proof

Step Hyp Ref Expression
1 limom 
 |-  Lim _om
2 frsuc 
 |-  ( B e. _om -> ( ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) |` _om ) ` suc B ) = ( ( x e. _V |-> ( x .o A ) ) ` ( ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) |` _om ) ` B ) ) )
3 peano2 
 |-  ( B e. _om -> suc B e. _om )
4 3 fvresd 
 |-  ( B e. _om -> ( ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) |` _om ) ` suc B ) = ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` suc B ) )
5 fvres 
 |-  ( B e. _om -> ( ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) |` _om ) ` B ) = ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) )
6 5 fveq2d 
 |-  ( B e. _om -> ( ( x e. _V |-> ( x .o A ) ) ` ( ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) |` _om ) ` B ) ) = ( ( x e. _V |-> ( x .o A ) ) ` ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) ) )
7 2 4 6 3eqtr3d 
 |-  ( B e. _om -> ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` suc B ) = ( ( x e. _V |-> ( x .o A ) ) ` ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) ) )
8 1 7 oesuclem 
 |-  ( ( A e. On /\ B e. _om ) -> ( A ^o suc B ) = ( ( A ^o B ) .o A ) )