Theorem fz1sumconst

Description: The sum of N constant terms ( k is not free in C ). (Contributed by SN, 21-Mar-2025)

	Ref	Expression
Hypotheses	fz1sumconst.n	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
	fz1sumconst.c	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
Assertion	fz1sumconst	⊢ ( 𝜑 → Σ 𝑘 ∈ ( 1 ... 𝑁 ) 𝐶 = ( 𝑁 · 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		fz1sumconst.n	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
2		fz1sumconst.c	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
3		fzfi	⊢ ( 1 ... 𝑁 ) ∈ Fin
4		fsumconst	⊢ ( ( ( 1 ... 𝑁 ) ∈ Fin ∧ 𝐶 ∈ ℂ ) → Σ 𝑘 ∈ ( 1 ... 𝑁 ) 𝐶 = ( ( ♯ ‘ ( 1 ... 𝑁 ) ) · 𝐶 ) )
5	3 2 4	sylancr	⊢ ( 𝜑 → Σ 𝑘 ∈ ( 1 ... 𝑁 ) 𝐶 = ( ( ♯ ‘ ( 1 ... 𝑁 ) ) · 𝐶 ) )
6		hashfz1	⊢ ( 𝑁 ∈ ℕ0 → ( ♯ ‘ ( 1 ... 𝑁 ) ) = 𝑁 )
7	1 6	syl	⊢ ( 𝜑 → ( ♯ ‘ ( 1 ... 𝑁 ) ) = 𝑁 )
8	7	oveq1d	⊢ ( 𝜑 → ( ( ♯ ‘ ( 1 ... 𝑁 ) ) · 𝐶 ) = ( 𝑁 · 𝐶 ) )
9	5 8	eqtrd	⊢ ( 𝜑 → Σ 𝑘 ∈ ( 1 ... 𝑁 ) 𝐶 = ( 𝑁 · 𝐶 ) )