Theorem fsummulc1f

Description: Closure of a finite sum of complex numbers A ( k ) . A version of fsummulc1 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020)

	Ref	Expression
Hypotheses	fsummulc1f.ph	⊢ Ⅎ 𝑘 𝜑
	fsummulclf.a	⊢ ( 𝜑 → 𝐴 ∈ Fin )
	fsummulclf.c	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
	fsummulclf.b	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ )
Assertion	fsummulc1f	⊢ ( 𝜑 → ( Σ 𝑘 ∈ 𝐴 𝐵 · 𝐶 ) = Σ 𝑘 ∈ 𝐴 ( 𝐵 · 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		fsummulc1f.ph	⊢ Ⅎ 𝑘 𝜑
2		fsummulclf.a	⊢ ( 𝜑 → 𝐴 ∈ Fin )
3		fsummulclf.c	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4		fsummulclf.b	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ )
5		csbeq1a	⊢ ( 𝑘 = 𝑗 → 𝐵 = ⦋ 𝑗 / 𝑘 ⦌ 𝐵 )
6		nfcv	⊢ Ⅎ 𝑗 𝐴
7		nfcv	⊢ Ⅎ 𝑘 𝐴
8		nfcv	⊢ Ⅎ 𝑗 𝐵
9		nfcsb1v	⊢ Ⅎ 𝑘 ⦋ 𝑗 / 𝑘 ⦌ 𝐵
10	5 6 7 8 9	cbvsum	⊢ Σ 𝑘 ∈ 𝐴 𝐵 = Σ 𝑗 ∈ 𝐴 ⦋ 𝑗 / 𝑘 ⦌ 𝐵
11	10	oveq1i	⊢ ( Σ 𝑘 ∈ 𝐴 𝐵 · 𝐶 ) = ( Σ 𝑗 ∈ 𝐴 ⦋ 𝑗 / 𝑘 ⦌ 𝐵 · 𝐶 )
12	11	a1i	⊢ ( 𝜑 → ( Σ 𝑘 ∈ 𝐴 𝐵 · 𝐶 ) = ( Σ 𝑗 ∈ 𝐴 ⦋ 𝑗 / 𝑘 ⦌ 𝐵 · 𝐶 ) )
13		nfv	⊢ Ⅎ 𝑘 𝑗 ∈ 𝐴
14	1 13	nfan	⊢ Ⅎ 𝑘 ( 𝜑 ∧ 𝑗 ∈ 𝐴 )
15	9	nfel1	⊢ Ⅎ 𝑘 ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ∈ ℂ
16	14 15	nfim	⊢ Ⅎ 𝑘 ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ∈ ℂ )
17		eleq1w	⊢ ( 𝑘 = 𝑗 → ( 𝑘 ∈ 𝐴 ↔ 𝑗 ∈ 𝐴 ) )
18	17	anbi2d	⊢ ( 𝑘 = 𝑗 → ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ↔ ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) ) )
19	5	eleq1d	⊢ ( 𝑘 = 𝑗 → ( 𝐵 ∈ ℂ ↔ ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ∈ ℂ ) )
20	18 19	imbi12d	⊢ ( 𝑘 = 𝑗 → ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ ) ↔ ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ∈ ℂ ) ) )
21	16 20 4	chvarfv	⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ∈ ℂ )
22	2 3 21	fsummulc1	⊢ ( 𝜑 → ( Σ 𝑗 ∈ 𝐴 ⦋ 𝑗 / 𝑘 ⦌ 𝐵 · 𝐶 ) = Σ 𝑗 ∈ 𝐴 ( ⦋ 𝑗 / 𝑘 ⦌ 𝐵 · 𝐶 ) )
23		eqcom	⊢ ( 𝑘 = 𝑗 ↔ 𝑗 = 𝑘 )
24	23	imbi1i	⊢ ( ( 𝑘 = 𝑗 → 𝐵 = ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ) ↔ ( 𝑗 = 𝑘 → 𝐵 = ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ) )
25		eqcom	⊢ ( 𝐵 = ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ↔ ⦋ 𝑗 / 𝑘 ⦌ 𝐵 = 𝐵 )
26	25	imbi2i	⊢ ( ( 𝑗 = 𝑘 → 𝐵 = ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ) ↔ ( 𝑗 = 𝑘 → ⦋ 𝑗 / 𝑘 ⦌ 𝐵 = 𝐵 ) )
27	24 26	bitri	⊢ ( ( 𝑘 = 𝑗 → 𝐵 = ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ) ↔ ( 𝑗 = 𝑘 → ⦋ 𝑗 / 𝑘 ⦌ 𝐵 = 𝐵 ) )
28	5 27	mpbi	⊢ ( 𝑗 = 𝑘 → ⦋ 𝑗 / 𝑘 ⦌ 𝐵 = 𝐵 )
29	28	oveq1d	⊢ ( 𝑗 = 𝑘 → ( ⦋ 𝑗 / 𝑘 ⦌ 𝐵 · 𝐶 ) = ( 𝐵 · 𝐶 ) )
30		nfcv	⊢ Ⅎ 𝑘 ·
31		nfcv	⊢ Ⅎ 𝑘 𝐶
32	9 30 31	nfov	⊢ Ⅎ 𝑘 ( ⦋ 𝑗 / 𝑘 ⦌ 𝐵 · 𝐶 )
33		nfcv	⊢ Ⅎ 𝑗 ( 𝐵 · 𝐶 )
34	29 7 6 32 33	cbvsum	⊢ Σ 𝑗 ∈ 𝐴 ( ⦋ 𝑗 / 𝑘 ⦌ 𝐵 · 𝐶 ) = Σ 𝑘 ∈ 𝐴 ( 𝐵 · 𝐶 )
35	34	a1i	⊢ ( 𝜑 → Σ 𝑗 ∈ 𝐴 ( ⦋ 𝑗 / 𝑘 ⦌ 𝐵 · 𝐶 ) = Σ 𝑘 ∈ 𝐴 ( 𝐵 · 𝐶 ) )
36	12 22 35	3eqtrd	⊢ ( 𝜑 → ( Σ 𝑘 ∈ 𝐴 𝐵 · 𝐶 ) = Σ 𝑘 ∈ 𝐴 ( 𝐵 · 𝐶 ) )