Theorem repswlen

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

	Ref	Expression
Assertion	repswlen	⊢ ( ( 𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ) → ( ♯ ‘ ( 𝑆 repeatS 𝑁 ) ) = 𝑁 )

Proof

Step	Hyp	Ref	Expression
1		repsf	⊢ ( ( 𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ) → ( 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 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ) → ( ♯ ‘ ( 𝑆 repeatS 𝑁 ) ) = ( ♯ ‘ ( 0 ..^ 𝑁 ) ) )
5		hashfzo0	⊢ ( 𝑁 ∈ ℕ0 → ( ♯ ‘ ( 0 ..^ 𝑁 ) ) = 𝑁 )
6	5	adantl	⊢ ( ( 𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ) → ( ♯ ‘ ( 0 ..^ 𝑁 ) ) = 𝑁 )
7	4 6	eqtrd	⊢ ( ( 𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ) → ( ♯ ‘ ( 𝑆 repeatS 𝑁 ) ) = 𝑁 )