Theorem fsumdivc

Description: A finite sum divided by a constant. (Contributed by NM, 2-Jan-2006) (Revised by Mario Carneiro, 24-Apr-2014)

	Ref	Expression
Hypotheses	fsummulc2.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
	fsummulc2.2	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
	fsummulc2.3	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ )
	fsumdivc.4	⊢ ( 𝜑 → 𝐶 ≠ 0 )
Assertion	fsumdivc	⊢ ( 𝜑 → ( Σ 𝑘 ∈ 𝐴 𝐵 / 𝐶 ) = Σ 𝑘 ∈ 𝐴 ( 𝐵 / 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		fsummulc2.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
2		fsummulc2.2	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
3		fsummulc2.3	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ )
4		fsumdivc.4	⊢ ( 𝜑 → 𝐶 ≠ 0 )
5	2 4	reccld	⊢ ( 𝜑 → ( 1 / 𝐶 ) ∈ ℂ )
6	1 5 3	fsummulc1	⊢ ( 𝜑 → ( Σ 𝑘 ∈ 𝐴 𝐵 · ( 1 / 𝐶 ) ) = Σ 𝑘 ∈ 𝐴 ( 𝐵 · ( 1 / 𝐶 ) ) )
7	1 3	fsumcl	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐵 ∈ ℂ )
8	7 2 4	divrecd	⊢ ( 𝜑 → ( Σ 𝑘 ∈ 𝐴 𝐵 / 𝐶 ) = ( Σ 𝑘 ∈ 𝐴 𝐵 · ( 1 / 𝐶 ) ) )
9	2	adantr	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐶 ∈ ℂ )
10	4	adantr	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐶 ≠ 0 )
11	3 9 10	divrecd	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( 𝐵 / 𝐶 ) = ( 𝐵 · ( 1 / 𝐶 ) ) )
12	11	sumeq2dv	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 ( 𝐵 / 𝐶 ) = Σ 𝑘 ∈ 𝐴 ( 𝐵 · ( 1 / 𝐶 ) ) )
13	6 8 12	3eqtr4d	⊢ ( 𝜑 → ( Σ 𝑘 ∈ 𝐴 𝐵 / 𝐶 ) = Σ 𝑘 ∈ 𝐴 ( 𝐵 / 𝐶 ) )