Theorem esummulc2

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

	Ref	Expression
Hypotheses	esummulc2.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	esummulc2.b	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ( 0 [,] +∞ ) )
	esummulc2.c	⊢ ( 𝜑 → 𝐶 ∈ ( 0 [,) +∞ ) )
Assertion	esummulc2	⊢ ( 𝜑 → ( 𝐶 ·e Σ* 𝑘 ∈ 𝐴 𝐵 ) = Σ* 𝑘 ∈ 𝐴 ( 𝐶 ·e 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		esummulc2.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
2		esummulc2.b	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ( 0 [,] +∞ ) )
3		esummulc2.c	⊢ ( 𝜑 → 𝐶 ∈ ( 0 [,) +∞ ) )
4		icossxr	⊢ ( 0 [,) +∞ ) ⊆ ℝ*
5	4 3	sselid	⊢ ( 𝜑 → 𝐶 ∈ ℝ* )
6		iccssxr	⊢ ( 0 [,] +∞ ) ⊆ ℝ*
7	2	ralrimiva	⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ( 0 [,] +∞ ) )
8		nfcv	⊢ Ⅎ 𝑘 𝐴
9	8	esumcl	⊢ ( ( 𝐴 ∈ 𝑉 ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ( 0 [,] +∞ ) ) → Σ* 𝑘 ∈ 𝐴 𝐵 ∈ ( 0 [,] +∞ ) )
10	1 7 9	syl2anc	⊢ ( 𝜑 → Σ* 𝑘 ∈ 𝐴 𝐵 ∈ ( 0 [,] +∞ ) )
11	6 10	sselid	⊢ ( 𝜑 → Σ* 𝑘 ∈ 𝐴 𝐵 ∈ ℝ* )
12		xmulcom	⊢ ( ( 𝐶 ∈ ℝ* ∧ Σ* 𝑘 ∈ 𝐴 𝐵 ∈ ℝ* ) → ( 𝐶 ·e Σ* 𝑘 ∈ 𝐴 𝐵 ) = ( Σ* 𝑘 ∈ 𝐴 𝐵 ·e 𝐶 ) )
13	5 11 12	syl2anc	⊢ ( 𝜑 → ( 𝐶 ·e Σ* 𝑘 ∈ 𝐴 𝐵 ) = ( Σ* 𝑘 ∈ 𝐴 𝐵 ·e 𝐶 ) )
14	1 2 3	esummulc1	⊢ ( 𝜑 → ( Σ* 𝑘 ∈ 𝐴 𝐵 ·e 𝐶 ) = Σ* 𝑘 ∈ 𝐴 ( 𝐵 ·e 𝐶 ) )
15	6 2	sselid	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℝ* )
16	5	adantr	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐶 ∈ ℝ* )
17		xmulcom	⊢ ( ( 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( 𝐵 ·e 𝐶 ) = ( 𝐶 ·e 𝐵 ) )
18	15 16 17	syl2anc	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( 𝐵 ·e 𝐶 ) = ( 𝐶 ·e 𝐵 ) )
19	18	esumeq2dv	⊢ ( 𝜑 → Σ* 𝑘 ∈ 𝐴 ( 𝐵 ·e 𝐶 ) = Σ* 𝑘 ∈ 𝐴 ( 𝐶 ·e 𝐵 ) )
20	13 14 19	3eqtrd	⊢ ( 𝜑 → ( 𝐶 ·e Σ* 𝑘 ∈ 𝐴 𝐵 ) = Σ* 𝑘 ∈ 𝐴 ( 𝐶 ·e 𝐵 ) )