Theorem repswlen

Description: The length of a "repeated symbol word". (Contributed by AV, 4-Nov-2018)

	Ref	Expression
Assertion	repswlen	|- ( ( S e. V /\ N e. NN0 ) -> ( # ` ( S repeatS N ) ) = N )

Proof

Step	Hyp	Ref	Expression
1		repsf	|- ( ( S e. V /\ N e. NN0 ) -> ( S repeatS N ) : ( 0 ..^ N ) --> V )
2		ffn	|- ( ( S repeatS N ) : ( 0 ..^ N ) --> V -> ( S repeatS N ) Fn ( 0 ..^ N ) )
3		hashfn	|- ( ( S repeatS N ) Fn ( 0 ..^ N ) -> ( # ` ( S repeatS N ) ) = ( # ` ( 0 ..^ N ) ) )
4	1 2 3	3syl	|- ( ( S e. V /\ N e. NN0 ) -> ( # ` ( S repeatS N ) ) = ( # ` ( 0 ..^ N ) ) )
5		hashfzo0	|- ( N e. NN0 -> ( # ` ( 0 ..^ N ) ) = N )
6	5	adantl	|- ( ( S e. V /\ N e. NN0 ) -> ( # ` ( 0 ..^ N ) ) = N )
7	4 6	eqtrd	|- ( ( S e. V /\ N e. NN0 ) -> ( # ` ( S repeatS N ) ) = N )