Theorem esumdivc

Description: An extended sum divided by a constant. (Contributed by Thierry Arnoux, 6-Jul-2017)

	Ref	Expression
Hypotheses	esumdivc.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	esumdivc.b	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ( 0 [,] +∞ ) )
	esumdivc.c	⊢ ( 𝜑 → 𝐶 ∈ ℝ+ )
Assertion	esumdivc	⊢ ( 𝜑 → ( Σ* 𝑘 ∈ 𝐴 𝐵 /𝑒 𝐶 ) = Σ* 𝑘 ∈ 𝐴 ( 𝐵 /𝑒 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		esumdivc.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
2		esumdivc.b	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ( 0 [,] +∞ ) )
3		esumdivc.c	⊢ ( 𝜑 → 𝐶 ∈ ℝ+ )
4		1red	⊢ ( 𝜑 → 1 ∈ ℝ )
5	3	rpred	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
6	3	rpne0d	⊢ ( 𝜑 → 𝐶 ≠ 0 )
7		rexdiv	⊢ ( ( 1 ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ 𝐶 ≠ 0 ) → ( 1 /𝑒 𝐶 ) = ( 1 / 𝐶 ) )
8	4 5 6 7	syl3anc	⊢ ( 𝜑 → ( 1 /𝑒 𝐶 ) = ( 1 / 𝐶 ) )
9		ioorp	⊢ ( 0 (,) +∞ ) = ℝ+
10		ioossico	⊢ ( 0 (,) +∞ ) ⊆ ( 0 [,) +∞ )
11	9 10	eqsstrri	⊢ ℝ+ ⊆ ( 0 [,) +∞ )
12	3	rpreccld	⊢ ( 𝜑 → ( 1 / 𝐶 ) ∈ ℝ+ )
13	11 12	sseldi	⊢ ( 𝜑 → ( 1 / 𝐶 ) ∈ ( 0 [,) +∞ ) )
14	8 13	eqeltrd	⊢ ( 𝜑 → ( 1 /𝑒 𝐶 ) ∈ ( 0 [,) +∞ ) )
15	1 2 14	esummulc1	⊢ ( 𝜑 → ( Σ* 𝑘 ∈ 𝐴 𝐵 ·e ( 1 /𝑒 𝐶 ) ) = Σ* 𝑘 ∈ 𝐴 ( 𝐵 ·e ( 1 /𝑒 𝐶 ) ) )
16		iccssxr	⊢ ( 0 [,] +∞ ) ⊆ ℝ*
17	2	ralrimiva	⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ( 0 [,] +∞ ) )
18		nfcv	⊢ Ⅎ 𝑘 𝐴
19	18	esumcl	⊢ ( ( 𝐴 ∈ 𝑉 ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ( 0 [,] +∞ ) ) → Σ* 𝑘 ∈ 𝐴 𝐵 ∈ ( 0 [,] +∞ ) )
20	1 17 19	syl2anc	⊢ ( 𝜑 → Σ* 𝑘 ∈ 𝐴 𝐵 ∈ ( 0 [,] +∞ ) )
21	16 20	sseldi	⊢ ( 𝜑 → Σ* 𝑘 ∈ 𝐴 𝐵 ∈ ℝ* )
22		xdivrec	⊢ ( ( Σ* 𝑘 ∈ 𝐴 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ ∧ 𝐶 ≠ 0 ) → ( Σ* 𝑘 ∈ 𝐴 𝐵 /𝑒 𝐶 ) = ( Σ* 𝑘 ∈ 𝐴 𝐵 ·e ( 1 /𝑒 𝐶 ) ) )
23	21 5 6 22	syl3anc	⊢ ( 𝜑 → ( Σ* 𝑘 ∈ 𝐴 𝐵 /𝑒 𝐶 ) = ( Σ* 𝑘 ∈ 𝐴 𝐵 ·e ( 1 /𝑒 𝐶 ) ) )
24	16 2	sseldi	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℝ* )
25	5	adantr	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐶 ∈ ℝ )
26	6	adantr	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐶 ≠ 0 )
27		xdivrec	⊢ ( ( 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ ∧ 𝐶 ≠ 0 ) → ( 𝐵 /𝑒 𝐶 ) = ( 𝐵 ·e ( 1 /𝑒 𝐶 ) ) )
28	24 25 26 27	syl3anc	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( 𝐵 /𝑒 𝐶 ) = ( 𝐵 ·e ( 1 /𝑒 𝐶 ) ) )
29	28	esumeq2dv	⊢ ( 𝜑 → Σ* 𝑘 ∈ 𝐴 ( 𝐵 /𝑒 𝐶 ) = Σ* 𝑘 ∈ 𝐴 ( 𝐵 ·e ( 1 /𝑒 𝐶 ) ) )
30	15 23 29	3eqtr4d	⊢ ( 𝜑 → ( Σ* 𝑘 ∈ 𝐴 𝐵 /𝑒 𝐶 ) = Σ* 𝑘 ∈ 𝐴 ( 𝐵 /𝑒 𝐶 ) )